Mathematica 10 Integration Test Results Integrands involving log functions IntegrationTest@"2 Exponential Functions\\u Ha+b logHc Hd He+f xL^pL^qLL^n"D; Testing Mathematica on 251 integration problems... Problem ð14: Mathematica is able to integrate expression!!!: 9Hg + h xLm Ha + b Log@c Hd He + f xLp Lq DL2 , x, 0, 0= IntAHg + h xLm Ha + b Log@c Hd He + f xLp Lq DL2 , xE Hg + h xLm a2 Hg + h xL h H1 + mL2 a2 m Hg + h xL h H1 + mL2 + + b2 p2 q2 Hg + h xL Log@e + f xD2 - h+hm 2 a b p q Hg + h xL Jf Hg + h xL Hypergeometric2F1B1, 2 + m, 3 + m, f Hg+h xL f g-e h h H- f g + e hL H1 + mL2 H2 + mL 2 a b m p q Hg + h xL Jf Hg + h xL Hypergeometric2F1B1, 2 + m, 3 + m, F + Hf g - e hL H2 + mL Log@e + f xDN f Hg+h xL f g-e h h H- f g + e hL H1 + mL2 H2 + mL 2 b2 p2 q2 Hg + h xL Jf Hg + h xL Hypergeometric2F1B1, 2 + m, 3 + m, 2 b2 g p2 q2 J 2 b2 e p2 q2 J f Hg+h xL h He+f xL f Hg+h xL h He+f xL N -m N -m f Hg+h xL f g-e h h H- f g + e hL H1 + mL2 H2 + mL F + Hf g - e hL H2 + mL Log@e + f xDN F + Hf g - e hL H2 + mL Log@e + f xDN JHypergeometricPFQB8- m, - m, - m<, 81 - m, 1 - m<, -f g+e h h He+f xL h m2 H1 + mL J- HypergeometricPFQB8- m, - m, - m<, 81 - m, 1 - m<, 2 b2 p q2 Hg + h xL Jf Hg + h xL Hypergeometric2F1B1, 2 + m, 3 + m, f g-e h + + F - m Hypergeometric2F1B- m, - m, 1 - m, -f g+e h h He+f xL f m2 H1 + mL f Hg+h xL - F + m Hypergeometric2F1B- m, - m, 1 - m, + + F Log@e + f xDN -f g+e h h He+f xL f Hg+h xL f g-e h + F + Hf g - e hL H2 + mL Log@e + f xDN Hp Log@e + f xD - Log@d He + f xLp DL h H- f g + e hL H1 + mL2 H2 + mL + - + F Log@e + f xDN F + Hf g - e hL H2 + mL Log@e + f xDN Hp Log@e + f xD - Log@d He + f xLp DL h H- f g + e hL H1 + mL2 H2 + mL 2 b2 m p q2 Hg + h xL Jf Hg + h xL Hypergeometric2F1B1, 2 + m, 3 + m, -f g+e h h He+f xL + + 2 2.2 Logarithm Functions.nb 2 a b q Hg + h xL H- p Log@e + f xD + Log@d He + f xLp DL h H1 + mL2 b2 q2 Hg + h xL H- p Log@e + f xD + Log@d He + f xLp DL2 h H1 + mL2 + + 2 a b m q Hg + h xL H- p Log@e + f xD + Log@d He + f xLp DL b2 m q2 Hg + h xL H- p Log@e + f xD + Log@d He + f xLp DL2 2 a b Hg + h xL Hq Log@d He + f xLp D - Log@c Hd He + f xLp Lq DL 1 h H1 + mL2 h H- f g + e hL H1 + mL H2 + mL 2 h H1 + mL2 - h H1 + mL2 1 h H- f g + e hL H1 + mL2 H2 + mL f Hg + h xL Hypergeometric2F1B1, 2 + m, 3 + m, f Hg + h xL fg-eh h H1 + mL2 f Hg + h xL fg-eh 2 b2 m p q Hg + h xL b2 Hg + h xL H- q Log@d He + f xLp D + Log@c Hd He + f xLp Lq DL2 h H1 + mL2 F + Hf g - e hL H2 + mL Log@e + f xD + 2 b2 m q Hg + h xL Hp Log@e + f xD - Log@d He + f xLp DL Hq Log@d He + f xLp D - Log@c Hd He + f xLp Lq DL h H1 + mL2 + F + Hf g - e hL H2 + mL Log@e + f xD Hq Log@d He + f xLp D - Log@c Hd He + f xLp Lq DL + 2 b2 q Hg + h xL Hp Log@e + f xD - Log@d He + f xLp DL Hq Log@d He + f xLp D - Log@c Hd He + f xLp Lq DL h H1 + mL2 - 2 a b m Hg + h xL Hq Log@d He + f xLp D - Log@c Hd He + f xLp Lq DL 2 b2 p q Hg + h xL f Hg + h xL Hypergeometric2F1B1, 2 + m, 3 + m, Hq Log@d He + f xLp D - Log@c Hd He + f xLp Lq DL + + + + b2 m Hg + h xL H- q Log@d He + f xLp D + Log@c Hd He + f xLp Lq DL2 h H1 + mL2 Problem ð19: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL2 , x, 3, 0> g+hx Ha + b Log@c Hd He + f xLp Lq DL2 LogB h 1 h f Hg+h xL f g-e h F + 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, h h He+f xL F f g-e h 2 b2 p2 q2 PolyLogB3, h h He+f xL F f g-e h a2 Log@g + h xD - 2 a b p q Log@e + f xD Log@g + h xD + b2 p2 q2 Log@e + f xD2 Log@g + h xD + 2 a b Log@c Hd He + f xLp Lq D Log@g + h xD - 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + b2 Log@c Hd He + f xLp Lq D2 Log@g + h xD + f Hg + h xL f Hg + h xL f Hg + h xL 2 a b p q Log@e + f xD LogB F - b2 p2 q2 Log@e + f xD2 LogB F + 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB F+ fg-eh fg-eh fg-eh 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, h He + f xL -f g + e h F - 2 b2 p2 q2 PolyLogB3, Problem ð23: Mathematica is able to integrate expression!!!: h He + f xL -f g + e h F 2.2 Logarithm Functions.nb 9Hg + h xLm Ha + b Log@c Hd He + f xLp Lq DL3 , x, 0, 0= IntAHg + h xLm Ha + b Log@c Hd He + f xLp Lq DL3 , xE a3 Hg + h xL1+m h H1 + mL2 3 a2 b p q - + a3 m Hg + h xL1+m h H1 + mL2 + Hg + h xLm b3 g p3 q3 h H1 + mL f Hg+h xL2+m Hypergeometric2F1 B1,2+m,3+m,H-f g+e hL H2+mL f Hg+h xL 3 bmpq - 1 f h H1 + mL f Hg+h xL2+m Hypergeometric2F1 B1,2+m,3+m,H-f g+e hL H2+mL h H1 + mL2 f g - e h + h He + f xL 3 a b2 p2 q2 2 h h h H1 + mL2 a2 F fg e- F fg h h m 1+ f f g - e h + h He + f xL h He + f xL +h e-e fg-eh 1 f h H1 + mL 3 a b2 m p2 q2 2 f g - e h + h He + f xL 1+ h He + f xL -m fg-eh 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, - m<, 82, 2<, f g - e h + h He + f xL m +h e-e fg-eh 1 f h H1 + mL 3 b3 p3 q3 2 f g - e h + h He + f xL m 1+ h He + f xL -m fg-eh m +h e-e fg-eh 3 b3 e p3 q3 1 + f Ig - eh M f h He + f xL -m eh g- + f -f g + e h h He + f xL f m f g - e h + h He + f xL -f g + e h Log@e + f xD2 + fg-eh m F Log@e + f xD + + He + f xL f g - e h + h He + f xL h He + f xL -f g + e h m + He + f xL f g - e h + h He + f xL m Log@e + f xD2 + 2 HypergeometricPFQB8- m, - m, - m, - m<, 81 - m, 1 - m, 1 - m<, - + -f g + e h Log@e + f xD2 - fg-eh m3 h He + f xL m fg-eh F Log@e + f xD + F- m 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, 1, - m<, 82, 2, 2<, f g - e h + h He + f xL fg-eh + He + f xL h He + f xL fg-eh f f g - e h + h He + f xL m F Log@e + f xD + h He + f xL 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, 1, - m<, 82, 2, 2<, f g - e h + h He + f xL 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, - m<, 82, 2<, f g -1 + -f g + e h f g - e h + h He + f xL f f g -1 + h He + f xL fg-eh m + 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, 1, - m<, 82, 2, 2<, -m fg-eh m + + Hg + h xL1+m Log@e + f xD 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, - m<, 82, 2<, f g -1 + Log@e + f xD3 + h+hm + Hg + h xL1+m Log@e + f xD f Hg+h xL e- b3 h p3 q3 x + f Jg- eh f N h He+f xL F h He + f xL -f g + e h F- 1 f H1 + mL - - F- 3 4 2.2 Logarithm Functions.nb 2 HypergeometricPFQB8- m, - m, - m<, 81 - m, 1 - m<, m2 1 3 h H1 + mL 3 3 3b gp q 1+ f Ig - eh M f -m eh h He + f xL g- + f 3 a2 b q Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL 2 h H1 + mL2 2 6ab pq 1 h H1 + mL2 1 - + F Log@e + f xD f Jg- eh f N h He+f xL 1+ f f g - e h + h He + f xL f Hg+h xL Je- h He + f xL -m fg-eh m +h e-e fg-eh H- p Log@e + f xD + Log@d He + f xLp DL + fg h Nh F f h H1 + mL2 f Jg- eh f N h He+f xL F Log@e + f xD2 + m 3 b3 m p2 q3 f Jg- eh f N h He+f xL f Jg- f Hg+h xL Je- fg h Nh F N F - F Log@e + f xD2 + + + Hg + h xL1+m Log@e + f xD H- p Log@e + f xD + Log@d He + f xLp DL + -f g + e h m F Log@e + f xD + + He + f xL f g - e h + h He + f xL 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, - m<, 82, 2<, f - + Hg + h xL1+m Log@e + f xD H- p Log@e + f xD + Log@d He + f xLp DL + h He + f xL f g - e h + h He + f xL fg-eh m 1+ f 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, 1, - m<, 82, 2, 2<, eh h He+f xL 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, 1, - m<, 82, 2, 2<, f g - e h + h He + f xL fg-eh 1 Hypergeometric2F1B- m, - m, 1 - m, - h H1 + mL2 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, - m<, 82, 2<, f g -1 + m 3 a2 b m q Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL H- f g + e hL H2 + mL m + F Log@e + f xD f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, - f g - e h + h He + f xL Hypergeometric2F1B- m, - m, 1 - m, - m3 H- f g + e hL H2 + mL 3 b3 p2 q3 N 2 HypergeometricPFQB8- m, - m, - m, - m<, 81 - m, 1 - m, 1 - m<, - m f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, - 6 a b2 m p q2 - f h H1 + mL2 f f m2 h H1 + mL2 eh h He+f xL h He + f xL 2 HypergeometricPFQB8- m, - m, - m<, 81 - m, 1 - m<, - 1 f Jg- h He + f xL -f g + e h h He + f xL -f g + e h F- F Log@e + f xD + h He + f xL fg-eh -m m Log@e + f xD2 h He + f xL -f g + e h F- 2.2 Logarithm Functions.nb f g -1 + f g - e h + h He + f xL f g - e h + h He + f xL m +h e-e fg-eh fg-eh H- p Log@e + f xD + Log@d He + f xLp DL + h H1 + mL2 H- f g + e hL H2 + mL h H1 + mL2 3 b3 m p q3 - h H1 + mL2 h H1 + mL2 f Hg+h xL Je- f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, H- f g + e hL H2 + mL b3 q3 Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL3 h H1 + mL2 h H1 + mL 2 fg h Nh F f Hg+h xL Je- fg h Nh F + Hg + h xL1+m Log@e + f xD H- p Log@e + f xD + Log@d He + f xLp DL2 + h H1 + mL2 q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D F + 1 q H- p Log@e + f xD + Log@d He + f xLp DL h H1 + mL2 3 a2 b m Hg + h xL1+m - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL 2 6ab pq - p DL q- Log@d He+f xLp D f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, - Log@d He + f xLp D q - H- f g + e hL H2 + mL f Hg+h xL Je- q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xL D p + 1 + Hd He + f xLp L q- Log@e + f xD2 fg-eh + Hg + h xL1+m Log@e + f xD H- p Log@e + f xD + Log@d He + f xLp DL2 + 3 a2 b Hg + h xL1+m - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q p DL m + b3 m q3 Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL3 LogBc ãq H-p Log@e+f xD+Log@d He+f xL f g - e h + h He + f xL 1 + f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, - 1 + He + f xL 3 a b2 q2 Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL2 3 a b2 m q2 Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL2 3 b3 p q3 - m fg h Nh F + F 1 Log@d He + f xLp D + q H- p Log@e + f xD + Log@d He + f xLp DL h H1 + mL2 Log@d He + f xLp D + Hg + h xL1+m Log@e + f xD + LogBc ãq H-p Log@e+f xD+Log@d He+f xL + - q H- p Log@e + f xD + Log@d He + f xLp DL p DL Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D F + 1 h H1 + mL2 5 6 2.2 Logarithm Functions.nb 2 6ab mpq - f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, - Log@d He + f xLp D q 1 f h H1 + mL2 3 b3 p2 q2 H- f g + e hL H2 + mL f Hg+h xL Je- q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D f g - e h + h He + f xL f m 1+ h He + f xL f g - e h + h He + f xL +h e-e fg-eh 1 f h H1 + mL 3 b3 m p2 q2 2 Hd He + f xLp L q- f g - e h + h He + f xL m 1+ f f g - e h + h He + f xL -m fg-eh +h e-e fg-eh fg-eh - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL 1 h H1 + mL2 h H1 + mL2 Hd He + f xLp L q- q- + He + f xL F + h He + f xL -f g + e h f g - e h + h He + f xL Log@d He + f xLp D m + He + f xL h He + f xL -f g + e h F + F- + f g - e h + h He + f xL h He + f xL -f g + e h m Log@e + f xD2 fg-eh Log@d He + f xLp D + 6 a b2 m q Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL - q- + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D F + Log@e + f xD2 q- Log@d He + f xLp D q - Log@d He+f xLp D m F Log@e + f xD + q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q H-p Log@e+f xD+Log@d He+f xLp DL fg-eh q H- p Log@e + f xD + Log@d He + f xLp DL q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D m F Log@e + f xD + 6 a b2 q Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q 1 p DL Hd He + f xLp L 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, 1, - m<, 82, 2, 2<, f g - e h + h He + f xL m p DL - q H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He+f xLp D 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, - m<, 82, 2<, f g -1 + + Hg + h xL1+m Log@e + f xD -f g + e h q H-p Log@e+f xD+Log@d He+f xLp DL h He + f xL F h He + f xL f g - e h + h He + f xL m - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q p DL Nh 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, 1, - m<, 82, 2, 2<, -m fg-eh fg-eh LogBc ãq H-p Log@e+f xD+Log@d He+f xL h + LogBc ãq H-p Log@e+f xD+Log@d He+f xL 2 h H1 + mL He + f xL HypergeometricPFQB81, 1, - m<, 82, 2<, f g -1 + fg p DL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + Log@d He+f xLp D Log@d He+f xLp D F- 2.2 Logarithm Functions.nb 1 h H1 + mL2 3 2 6b pq - f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, H- f g + e hL H2 + mL f Hg+h xL Je- fg h Nh F + Hg + h xL1+m Log@e + f xD H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q 1 h H1 + mL2 3 2 6b mpq - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, H- f g + e hL H2 + mL f Hg+h xL Je- fg h Nh F 1 h H1 + mL2 h H1 + mL2 h H1 + mL2 + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + q- 3 b3 m q2 Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL2 - q H- p Log@e + f xD + Log@d He + f xLp DL - q- Log@d He + f xLp D Log@d He + f xLp D 3 a b2 Hg + h xL1+m - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL q- Log@d He+f xLp D F 2 LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL q- Log@d He+f xLp D F 2 h H1 + mL2 f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, H- f g + e hL H2 + mL f Hg+h xL Je- fg h Nh F 1 + Log@d He+f xLp D p DL p DL Log@d He+f xLp D Log@d He+f xLp D Log@d He+f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D 1 + 3 a b2 m Hg + h xL1+m - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - 3 b3 p q - p DL q- Log@d He + f xLp D q 1 F + q- 3 b3 q2 Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL2 - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q 1 q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L + Hg + h xL1+m Log@e + f xD H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - p DL q H- p Log@e + f xD + Log@d He + f xLp DL h H1 + mL2 Log@d He + f xLp D + Hg + h xL1+m Log@e + f xD + + - q H- p Log@e + f xD + Log@d He + f xLp DL 2 + + 7 8 2.2 Logarithm Functions.nb Log@d He + f xLp D q 1 h H1 + mL2 3 3b mpq - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D f Hg + h xL2+m Hypergeometric2F1B1, 2 + m, 3 + m, H- f g + e hL H2 + mL - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL 1 h H1 + mL2 h H1 + mL2 h H1 + mL2 Hd He + f xLp L q- Je- fg h Nh F q H-p Log@e+f xD+Log@d He+f xLp DL F 2 Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F 2 Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F 2 Hd He + f xLp L q- q H- p Log@e + f xD + Log@d He + f xLp DL q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D F 2 Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D b3 Hg + h xL1+m - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL q- Log@d He+f xLp D F 3 F 3 1 + h H1 + mL2 b3 m Hg + h xL1+m - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL q- Log@d He+f xLp D + p DL 6 a b2 p2 q2 x - 6 b3 p3 q3 x + 6 b3 p2 q2 He + f xL Log@c Hd He + f xLp Lq D f 3 b p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 f + - He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 f p DL q- q- Log@d He+f xLp D Log@d He+f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL Problem ð27: Valid but suboptimal antiderivative: 9Ha + b Log@c Hd He + f xLp Lq DL3 , x, 4, 0= + + q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D Log@d He+f xLp D + Hg + h xL1+m Log@e + f xD 3 b3 m q Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - 1 p DL f Hg+h xL p DL 3 b3 q Hg + h xL1+m H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - 1 + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D + + + + 2.2 Logarithm Functions.nb 1 f Ib3 e p3 q3 Log@e + f xD3 - 3 b2 e p2 q2 Log@e + f xD2 Ha - b p q + b Log@c Hd He + f xLp Lq DL + 3 b e p q Log@e + f xD Ia2 - 2 a b p q + 2 b2 p2 q2 + 2 b Ha - b p qL Log@c Hd He + f xLp Lq D + b2 Log@c Hd He + f xLp Lq D2 M + f x Ia3 - 3 a2 b p q + 6 a b2 p2 q2 6 b3 p3 q3 + 3 b Ia2 - 2 a b p q + 2 b2 p2 q2 M Log@c Hd He + f xLp Lq D + 3 b2 Ha - b p qL Log@c Hd He + f xLp Lq D2 + b3 Log@c Hd He + f xLp Lq D3 MM Problem ð28: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL3 , x, 4, 0> g+hx Ha + b Log@c Hd He + f xLp Lq DL3 LogB f Hg+h xL f g-e h h F + 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - 6 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, h 1 h h He+f xL F f g-e h h 6 b3 p3 q3 PolyLogB4, + h h He+f xL F f g-e h h He+f xL F f g-e h a3 Log@g + h xD - 3 a2 b p q Log@e + f xD Log@g + h xD + 3 a b2 p2 q2 Log@e + f xD2 Log@g + h xD b3 p3 q3 Log@e + f xD3 Log@g + h xD + 3 a2 b Log@c Hd He + f xLp Lq D Log@g + h xD - 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 3 a b2 Log@c Hd He + f xLp Lq D2 Log@g + h xD 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + b3 Log@c Hd He + f xLp Lq D3 Log@g + h xD + f Hg + h xL f Hg + h xL f Hg + h xL 3 a2 b p q Log@e + f xD LogB F - 3 a b2 p2 q2 Log@e + f xD2 LogB F + b3 p3 q3 Log@e + f xD3 LogB F+ fg-eh fg-eh fg-eh 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB f Hg + h xL fg-eh f Hg + h xL fg-eh 6 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, F - 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB F + 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, h He + f xL -f g + e h F + 6 b3 p3 q3 PolyLogB4, h He + f xL -f g + e h Problem ð29: Valid but suboptimal antiderivative: : - Ha + b Log@c Hd He + f xLp Lq DL3 Hg + h xL2 , x, 4, 0> He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 Hf g - e hL Hg + h xL - 3 b f p q Ha + b Log@c Hd He + f xLp Lq DL2 LogB 6 b2 f p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, h Hf g - e hL h Hf g - e hL h He+f xL F f g-e h f Hg+h xL f g-e h 6 b3 f p3 q3 PolyLogB3, - + h Hf g - e hL F - h He+f xL F f g-e h F f Hg + h xL fg-eh h He + f xL -f g + e h F+ F- 9 10 2.2 Logarithm Functions.nb 1 h 3 b f p q Log@e + f xD Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL2 fg-eh Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL3 - g+hx 1 Hf g - e hL Hg + h xL Hf g - e hL Hg + h xL 3 b p q Log@e + f xD Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL2 g+hx 3 b f p q Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL2 Log@g + h xD 3 b2 p2 q2 H- a + b p q Log@e + f xD - b Log@c Hd He + f xLp Lq DL f Hg + h xL fg-eh F + 2 f Hg + h xL PolyLogB2, b3 p3 q3 Log@e + f xD2 h He + f xL Log@e + f xD - 3 f Hg + h xL LogB 6 f Hg + h xL Log@e + f xD PolyLogB2, h He + f xL -f g + e h + fg-eh - Log@e + f xD h He + f xL Log@e + f xD - 2 f Hg + h xL LogB 1 - F + 6 f Hg + h xL PolyLogB3, f Hg + h xL fg-eh h He + f xL -f g + e h F h He + f xL F - -f g + e h F + Problem ð33: Valid but suboptimal antiderivative: 9Ha + b Log@c Hd He + f xLp Lq DL4 , x, 5, 0= - 24 a b3 p3 q3 x + 24 b4 p4 q4 x - 24 b4 p3 q3 He + f xL Log@c Hd He + f xLp Lq D 4 b p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 f 1 f f + + 12 b2 p2 q2 He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 He + f xL Ha + b Log@c Hd He + f xLp Lq DL4 - f f I- b4 e p4 q4 Log@e + f xD4 + 4 b3 e p3 q3 Log@e + f xD3 Ha - b p q + b Log@c Hd He + f xLp Lq DL - 6 b2 e p2 q2 Log@e + f xD2 Ia2 - 2 a b p q + 2 b2 p2 q2 + 2 b Ha - b p qL Log@c Hd He + f xLp Lq D + b2 Log@c Hd He + f xLp Lq D2 M + 4 b e p q Log@e + f xD Ia3 - 3 a2 b p q + 6 a b2 p2 q2 - 6 b3 p3 q3 + 3 b Ia2 - 2 a b p q + 2 b2 p2 q2 M Log@c Hd He + f xLp Lq D + 3 b2 Ha - b p qL Log@c Hd He + f xLp Lq D2 + b3 Log@c Hd He + f xLp Lq D3 M + f x Ia4 - 4 a3 b p q + 12 a2 b2 p2 q2 - 24 a b3 p3 q3 + 24 b4 p4 q4 + 4 b Ia3 - 3 a2 b p q + 6 a b2 p2 q2 - 6 b3 p3 q3 M Log@c Hd He + f xLp Lq D + 6 b2 Ia2 - 2 a b p q + 2 b2 p2 q2 M Log@c Hd He + f xLp Lq D2 + 4 b3 Ha - b p qL Log@c Hd He + f xLp Lq D3 + b4 Log@c Hd He + f xLp Lq D4 MM Problem ð34: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL4 g+hx , x, 5, 0> - 2.2 Logarithm Functions.nb Ha + b Log@c Hd He + f xLp Lq DL4 LogB f Hg+h xL f g-e h h F + 4 b p q Ha + b Log@c Hd He + f xLp Lq DL3 PolyLogB2, - 12 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB3, h 24 b3 p3 q3 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB4, h 1 h h He+f xL F f g-e h h He+f xL F f g-e h - + 24 b4 p4 q4 PolyLogB5, - h h He+f xL F f g-e h a4 Log@g + h xD - 4 a3 b p q Log@e + f xD Log@g + h xD + 6 a2 b2 p2 q2 Log@e + f xD2 Log@g + h xD - 4 a b3 p3 q3 Log@e + f xD3 Log@g + h xD + b4 p4 q4 Log@e + f xD4 Log@g + h xD + 4 a3 b Log@c Hd He + f xLp Lq D Log@g + h xD - 12 a2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 12 a b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD - 4 b4 p3 q3 Log@e + f xD3 Log@c Hd He + f xLp Lq D Log@g + h xD + 6 a2 b2 Log@c Hd He + f xLp Lq D2 Log@g + h xD - 12 a b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + 6 b4 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D2 Log@g + h xD + 4 a b3 Log@c Hd He + f xLp Lq D3 Log@g + h xD f Hg + h xL 4 b4 p q Log@e + f xD Log@c Hd He + f xLp Lq D3 Log@g + h xD + b4 Log@c Hd He + f xLp Lq D4 Log@g + h xD + 4 a3 b p q Log@e + f xD LogB Ffg-eh 6 a2 b2 p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh F + 4 a b3 p3 q3 Log@e + f xD3 LogB 12 a2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB 4 b4 p3 q3 Log@e + f xD3 Log@c Hd He + f xLp Lq D LogB 4 b p q Ha + b Log@c Hd He + f xLp Lq DL3 PolyLogB2, 24 a b3 p3 q3 PolyLogB4, h He + f xL -f g + e h f Hg + h xL fg-eh f Hg + h xL fg-eh 6 b4 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D2 LogB fg-eh h He + f xL -f g + e h Ha + b Log@c Hd He + f xLp Lq DL4 Hg + h xL2 f Hg + h xL fg-eh f Hg + h xL fg-eh F - 12 a b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB F + 4 b4 p q Log@e + f xD Log@c Hd He + f xLp Lq D3 LogB h He + f xL -f g + e h F+ f Hg + h xL fg-eh f Hg + h xL fg-eh f Hg + h xL fg-eh F - 12 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB3, F + 24 b4 p3 q3 Log@c Hd He + f xLp Lq D PolyLogB4, , x, 5, 0> F - b4 p4 q4 Log@e + f xD4 LogB F + 12 a b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB f Hg + h xL Problem ð35: Valid but suboptimal antiderivative: : h h He+f xL F f g-e h F- F+ h He + f xL -f g + e h F - 24 b4 p4 q4 PolyLogB5, F+ F+ h He + f xL -f g + e h F 11 12 2.2 Logarithm Functions.nb He + f xL Ha + b Log@c Hd He + f xLp Lq DL4 Hf g - e hL Hg + h xL - 4 b f p q Ha + b Log@c Hd He + f xLp Lq DL3 LogB 12 b2 f p2 q2 Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, h Hf g - e hL 24 b3 f p3 q3 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, 1 h H- f g + e hL Hg + h xL h Hf g - e hL h Hf g - e hL h He+f xL F f g-e h h He+f xL F f g-e h f Hg+h xL f g-e h F - + 24 b4 f p4 q4 PolyLogB4, h Hf g - e hL - h He+f xL F f g-e h a4 f g - a4 e h - 4 a3 b f g p q Log@e + f xD - 4 a3 b f h p q x Log@e + f xD + 6 a2 b2 f g p2 q2 Log@e + f xD2 + 6 a2 b2 f h p2 q2 x Log@e + f xD2 4 a b3 f g p3 q3 Log@e + f xD3 - 4 a b3 f h p3 q3 x Log@e + f xD3 + b4 f g p4 q4 Log@e + f xD4 + b4 f h p4 q4 x Log@e + f xD4 + 4 a3 b f g Log@c Hd He + f xLp Lq D 4 a3 b e h Log@c Hd He + f xLp Lq D - 12 a2 b2 f g p q Log@e + f xD Log@c Hd He + f xLp Lq D - 12 a2 b2 f h p q x Log@e + f xD Log@c Hd He + f xLp Lq D + 12 a b3 f g p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D + 12 a b3 f h p2 q2 x Log@e + f xD2 Log@c Hd He + f xLp Lq D 4 b4 f g p3 q3 Log@e + f xD3 Log@c Hd He + f xLp Lq D - 4 b4 f h p3 q3 x Log@e + f xD3 Log@c Hd He + f xLp Lq D + 6 a2 b2 f g Log@c Hd He + f xLp Lq D2 6 a2 b2 e h Log@c Hd He + f xLp Lq D2 - 12 a b3 f g p q Log@e + f xD Log@c Hd He + f xLp Lq D2 - 12 a b3 f h p q x Log@e + f xD Log@c Hd He + f xLp Lq D2 + 6 b4 f g p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D2 + 6 b4 f h p2 q2 x Log@e + f xD2 Log@c Hd He + f xLp Lq D2 + 4 a b3 f g Log@c Hd He + f xLp Lq D3 - 4 a b3 e h Log@c Hd He + f xLp Lq D3 - 4 b4 f g p q Log@e + f xD Log@c Hd He + f xLp Lq D3 4 b4 f h p q x Log@e + f xD Log@c Hd He + f xLp Lq D3 + b4 f g Log@c Hd He + f xLp Lq D4 - b4 e h Log@c Hd He + f xLp Lq D4 + f Hg + h xL f Hg + h xL f Hg + h xL 4 a3 b f g p q LogB F + 4 a3 b f h p q x LogB F + 12 a2 b2 f g p q Log@c Hd He + f xLp Lq D LogB F+ fg-eh fg-eh fg-eh 12 a2 b2 f h p q x Log@c Hd He + f xLp Lq D LogB 12 a b3 f h p q x Log@c Hd He + f xLp Lq D2 LogB 4 b4 f h p q x Log@c Hd He + f xLp Lq D3 LogB f Hg + h xL fg-eh f Hg + h xL fg-eh f Hg + h xL fg-eh F + 12 a b3 f g p q Log@c Hd He + f xLp Lq D2 LogB F + 4 b4 f g p q Log@c Hd He + f xLp Lq D3 LogB h He + f xL -f g + e h : g Hd+e xL e f-d g f+gx F , x, 1, 0> h He + f xL -f g + e h F + 24 b4 f h p4 q4 x PolyLogB4, Problem ð42: Valid but suboptimal antiderivative: LogB- fg-eh f Hg + h xL fg-eh F+ F+ F + 12 b2 f p2 q2 Hg + h xL Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, 24 b3 f p3 q3 Hg + h xL Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, 24 b4 f g p4 q4 PolyLogB4, f Hg + h xL F+ h He + f xL -f g + e h F h He + f xL -f g + e h F- 2.2 Logarithm Functions.nb PolyLogB2, - e Hf+g xL e f-d g g g Hd+e xL LogB -e f+d g F LogB F e Hf+g xL e f-d g g g Hd+e xL F + PolyLogB2, -e f+d g F Problem ð55: Valid but suboptimal antiderivative: : Hg + h xL3 Ha + b Log@c Hd He + f xLp Lq DL2 Hf g - e hL3 He + f xL Hc Hd He + f xLp Lq L a - ã , x, 12, 0> bpq - 1 pq ExpIntegralEiB a+b Log@c Hd He+f xLp Lq D bpq b2 f4 p2 q2 - 6ã 2a bpq h Hf g - e hL2 He + f xL2 Hc Hd He + f xLp Lq L - 2 pq ExpIntegralEiB F + 2 Ha+b Log@c Hd He+f xLp Lq DL F 3 Ha+b Log@c Hd He+f xLp Lq DL F bpq b2 f4 p2 q2 - 9ã 3a bpq h2 Hf g - e hL He + f xL3 Hc Hd He + f xLp Lq L - 3 pq ExpIntegralEiB bpq b2 f4 p2 q2 - 4ã 4a bpq h3 He + f xL4 Hc Hd He + f xLp Lq L - 4 pq ExpIntegralEiB 4 Ha+b Log@c Hd He+f xLp Lq DL bpq b2 f4 p2 q2 F - 1 b2 f4 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL - ã 4a bpq Hc Hd He + f xLp Lq L - 4 pq + + He + f xL Hg + h xL3 b f p q Ha + b Log@c Hd He + f xLp Lq DL - b e ã b p q f3 g3 p q Hc Hd He + f xLp Lq L p q - b ã b p q f4 g3 p q x Hc Hd He + f xLp Lq L p q - 3 b e ã b p q f3 g2 h p q x Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a 4 3 b ã b p q f4 g2 h p q x2 Hc Hd He + f xLp Lq L p q - 3 b e ã b p q f3 g h2 p q x2 Hc Hd He + f xLp Lq L p q - 3 b ã b p q f4 g h2 p q x3 Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a 4 b e ã b p q f3 h3 p q x3 Hc Hd He + f xLp Lq L p q - b ã b p q f4 h3 p q x4 Hc Hd He + f xLp Lq L p q + a ã b p q f3 g3 He + f xL Hc Hd He + f xLp Lq L p q 4a 4 ExpIntegralEiB 4a a + b Log@c Hd He + f xLp Lq D bpq 4 3 a + b Log@c Hd He + f xL L D p q ExpIntegralEiB bpq 3a 3 a + b Log@c Hd He + f xLp Lq D bpq 2 a + b Log@c Hd He + f xLp Lq D bpq F - a e3 ã b p q h3 He + f xL Hc Hd He + f xLp Lq L p q 3a F + 6 a ã b p q f2 g2 h He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 12 a e ã b p q f g h2 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 3 F - 3 a e ã b p q f2 g2 h He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3 a e2 ã b p q f g h2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3a 2 2 Ha + b Log@c Hd He + f xLp Lq DL bpq + F+ 3 F+ 2 Ha + b Log@c Hd He + f xLp Lq DL bpq F- 13 14 2.2 Logarithm Functions.nb 6 a e2 ã b p q h3 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 2 Ha + b Log@c Hd He + f xLp Lq DL bpq 9 a ã b p q f g h2 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 1 3 Ha + b Log@c Hd He + f xL L DL p q ExpIntegralEiB bpq 3a 3 3 Ha + b Log@c Hd He + f xLp Lq DL bpq F + 4 a h3 He + f xL4 ExpIntegralEiB b ã b p q f3 g3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB a + b Log@c Hd He + f xLp Lq D bpq 3 b e ã b p q f2 g2 h He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 3 b e2 ã b p q f g h2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 b e3 ã b p q h3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 2 2a bpq 2a 2 a 1 9 b e ã b p q h3 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 1 4 b h3 He + f xL4 ExpIntegralEiB bpq Problem ð56: Valid but suboptimal antiderivative: : Hg + h xL2 Ha + b Log@c Hd He + f xLp Lq DL2 , x, 10, 0> F Log@c Hd He + f xLp Lq D + bpq F Log@c Hd He + f xLp Lq D - 2 Ha + b Log@c Hd He + f xLp Lq DL bpq bpq bpq 3 Ha + b Log@c Hd He + f xLp Lq DL bpq F Log@c Hd He + f xLp Lq D F Log@c Hd He + f xLp Lq D + F Log@c Hd He + f xLp Lq D + 3 Ha + b Log@c Hd He + f xLp Lq DL 4 Ha + b Log@c Hd He + f xLp Lq DL F+ F Log@c Hd He + f xLp Lq D - 2 Ha + b Log@c Hd He + f xLp Lq DL 9 b ã b p q f g h2 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 1 F Log@c Hd He + f xLp Lq D + 2 Ha + b Log@c Hd He + f xLp Lq DL 2 6 b e2 ã b p q h3 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB F Log@c Hd He + f xLp Lq D - a + b Log@c Hd He + f xLp Lq D 12 b e ã b p q f g h2 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a bpq bpq bpq F - 9 a e ã b p q h3 He + f xL3 Hc Hd He + f xLp Lq L p q 4 Ha + b Log@c Hd He + f xLp Lq DL a + b Log@c Hd He + f xLp Lq D a + b Log@c Hd He + f xLp Lq D 6 b ã b p q f2 g2 h He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a F+ F Log@c Hd He + f xLp Lq D - F Log@c Hd He + f xLp Lq D + 2.2 Logarithm Functions.nb Hf g - e hL2 He + f xL Hc Hd He + f xLp Lq L a - ã - bpq 1 pq ExpIntegralEiB a+b Log@c Hd He+f xLp Lq D bpq b2 f3 p2 q2 - 4ã 2a bpq h Hf g - e hL He + f xL2 Hc Hd He + f xLp Lq L - 2 pq ExpIntegralEiB F + 2 Ha+b Log@c Hd He+f xLp Lq DL bpq b2 f3 p2 q2 - 3ã 3a bpq h2 He + f xL3 Hc Hd He + f xLp Lq L - 3 pq ExpIntegralEiB 3 Ha+b Log@c Hd He+f xLp Lq DL bpq b2 f3 p2 q2 F - 1 b2 f3 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL - ã 3a bpq Hc Hd He + f xLp Lq L - 3 pq F + He + f xL Hg + h xL2 b f p q Ha + b Log@c Hd He + f xLp Lq DL - b e ã b p q f2 g2 p q Hc Hd He + f xLp Lq L p q - b ã b p q f3 g2 p q x Hc Hd He + f xLp Lq L p q - 2 b e ã b p q f2 g h p q x Hc Hd He + f xLp Lq L p q 3a 3 3a 3 3a 3 2 b ã b p q f3 g h p q x2 Hc Hd He + f xLp Lq L p q - b e ã b p q f2 h2 p q x2 Hc Hd He + f xLp Lq L p q - b ã b p q f3 h2 p q x3 Hc Hd He + f xLp Lq L p q + 3a 3 3a 3 a ã b p q f2 g2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 a + b Log@c Hd He + f xL L D p q ExpIntegralEiB bpq a + b Log@c Hd He + f xLp Lq D bpq 1 2 Ha + b Log@c Hd He + f xL L DL p q ExpIntegralEiB bpq 2 bpq a + b Log@c Hd He + f xLp Lq D bpq 2 b e2 ã b p q h2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 bpq 1 4 b e ã b p q h2 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 3 b h2 He + f xL3 ExpIntegralEiB 1 bpq F Log@c Hd He + f xLp Lq D + 2 Ha + b Log@c Hd He + f xLp Lq DL bpq F Log@c Hd He + f xLp Lq D F+ 1 F+ F Log@c Hd He + f xLp Lq D + 2 Ha + b Log@c Hd He + f xLp Lq DL 3 Ha + b Log@c Hd He + f xLp Lq DL a F Log@c Hd He + f xLp Lq D - bpq bpq F - 4 a e ã b p q h2 He + f xL2 Hc Hd He + f xLp Lq L p q bpq bpq 2 a + b Log@c Hd He + f xLp Lq D 3 Ha + b Log@c Hd He + f xLp Lq DL a + b Log@c Hd He + f xLp Lq D a + b Log@c Hd He + f xLp Lq D 4 b ã b p q f g h He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 2a 2 Ha + b Log@c Hd He + f xLp Lq DL 2 b e ã b p q f g h He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a F - 2 a e ã b p q f g h He + f xL Hc Hd He + f xLp Lq L p q 2 F + 3 a h2 He + f xL3 ExpIntegralEiB b ã b p q f2 g2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 3 F + a e2 ã b p q h2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 4 a ã b p q f g h He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 3a F Log@c Hd He + f xLp Lq D - F Log@c Hd He + f xLp Lq D + 15 16 2.2 Logarithm Functions.nb Problem ð62: Valid but suboptimal antiderivative: : Hg + h xL3 Ha + b Log@c Hd He + f xLp Lq DL3 - ã , x, 23, 0> Hf g - e hL3 He + f xL Hc Hd He + f xLp Lq L a bpq - 1 ExpIntegralEiB pq a+b Log@c Hd He+f xLp Lq D bpq 2 b3 f4 p3 q3 - 6ã 2a bpq h Hf g - e hL2 He + f xL2 Hc Hd He + f xLp Lq L - 2 ExpIntegralEiB pq F + 2 Ha+b Log@c Hd He+f xLp Lq DL bpq b3 f4 p3 q3 - 27 ã 3a bpq h2 Hf g - e hL He + f xL3 Hc Hd He + f xLp Lq L - 3 pq ExpIntegralEiB 3 Ha+b Log@c Hd He+f xLp Lq DL bpq 2 b3 f4 p3 q3 - 8ã 4a bpq h3 He + f xL4 Hc Hd He + f xLp Lq L - 4 ExpIntegralEiB pq 4 Ha+b Log@c Hd He+f xLp Lq DL bpq b3 f4 p3 q3 3 Hf g - e hL He + f xL Hg + h xL2 2 b2 f2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL - 1 2 b3 f4 p3 q3 Ha + b Log@c Hd He + f xLp Lq DL2 - ã 2 He + f xL Hg + h xL3 F - b2 f p2 q2 Ha + b Log@c Hd He + f xLp Lq DL 4a bpq Hc Hd He + f xLp Lq L - F + F + He + f xL Hg + h xL3 2 b f p q Ha + b Log@c Hd He + f xLp Lq DL2 + 4 pq - a b e ã b p q f3 g3 p q Hc Hd He + f xLp Lq L p q - 3 a b e2 ã b p q f2 g2 h p q Hc Hd He + f xLp Lq L p q - b2 e ã b p q f3 g3 p2 q2 Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a 4 a b ã b p q f4 g3 p q x Hc Hd He + f xLp Lq L p q - 9 a b e ã b p q f3 g2 h p q x Hc Hd He + f xLp Lq L p q - 6 a b e2 ã b p q f2 g h2 p q x Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a 4 b2 ã b p q f4 g3 p2 q2 x Hc Hd He + f xLp Lq L p q - 3 b2 e ã b p q f3 g2 h p2 q2 x Hc Hd He + f xLp Lq L p q - 6 a b ã b p q f4 g2 h p q x2 Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a 4 15 a b e ã b p q f3 g h2 p q x2 Hc Hd He + f xLp Lq L p q - 3 a b e2 ã b p q f2 h3 p q x2 Hc Hd He + f xLp Lq L p q - 3 b2 ã b p q f4 g2 h p2 q2 x2 Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a 4 3 b2 e ã b p q f3 g h2 p2 q2 x2 Hc Hd He + f xLp Lq L p q - 9 a b ã b p q f4 g h2 p q x3 Hc Hd He + f xLp Lq L p q - 7 a b e ã b p q f3 h3 p q x3 Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a 3 b2 ã b p q f4 g h2 p2 q2 x3 Hc Hd He + f xLp Lq L p q - b2 e ã b p q f3 h3 p2 q2 x3 Hc Hd He + f xLp Lq L p q - 4 a b ã b p q f4 h3 p q x4 Hc Hd He + f xLp Lq L p q 4a 4 4a 4 4a b2 ã b p q f4 h3 p2 q2 x4 Hc Hd He + f xLp Lq L p q + a2 ã b p q f3 g3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 4a 4 3a 3 3 a2 e ã b p q f2 g2 h He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 3 a2 e2 ã b p q f g h2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 a2 e3 ã b p q h3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 a + b Log@c Hd He + f xLp Lq D bpq a + b Log@c Hd He + f xLp Lq D bpq a + b Log@c Hd He + f xLp Lq D bpq F+ F+ F- - 4 a + b Log@c Hd He + f xLp Lq D bpq F- 4 2.2 Logarithm Functions.nb 12 a2 ã b p q f2 g2 h He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 2 Ha + b Log@c Hd He + f xLp Lq DL bpq 24 a2 e ã b p q f g h2 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 12 a2 e2 ã b p q h3 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 a 1 27 a2 e ã b p q h3 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 1 16 a2 h3 He + f xL4 ExpIntegralEiB 2 Ha + b Log@c Hd He + f xLp Lq DL bpq 2 Ha + b Log@c Hd He + f xLp Lq DL 27 a2 ã b p q f g h2 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB F- bpq F+ 3 Ha + b Log@c Hd He + f xLp Lq DL bpq 3 Ha + b Log@c Hd He + f xLp Lq DL 4 Ha + b Log@c Hd He + f xLp Lq DL bpq bpq F+ F- F+ F - b2 e ã b p q f3 g3 p q Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D 4a 4 3 b2 e2 ã b p q f2 g2 h p q Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D - b2 ã b p q f4 g3 p q x Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D 4a 4 4a 4 9 b2 e ã b p q f3 g2 h p q x Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D - 6 b2 e2 ã b p q f2 g h2 p q x Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D 4a 4 4a 4 6 b2 ã b p q f4 g2 h p q x2 Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D - 15 b2 e ã b p q f3 g h2 p q x2 Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D 4a 4 4a 4 3 b2 e2 ã b p q f2 h3 p q x2 Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D - 9 b2 ã b p q f4 g h2 p q x3 Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D 4a 4 4a 4 7 b2 e ã b p q f3 h3 p q x3 Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D - 4 b2 ã b p q f4 h3 p q x4 Hc Hd He + f xLp Lq L p q Log@c Hd He + f xLp Lq D + 4a 4 4a 2 a b ã b p q f3 g3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 a + b Log@c Hd He + f xLp Lq D bpq 6 a b e ã b p q f2 g2 h He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 6 a b e2 ã b p q f g h2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 2 a b e3 ã b p q h3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 bpq a + b Log@c Hd He + f xLp Lq D bpq a + b Log@c Hd He + f xLp Lq D bpq 2 2 24 a b e2 ã b p q h3 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 1 F Log@c Hd He + f xLp Lq D - F Log@c Hd He + f xLp Lq D + bpq F Log@c Hd He + f xLp Lq D - 2 Ha + b Log@c Hd He + f xLp Lq DL bpq 2 Ha + b Log@c Hd He + f xLp Lq DL 54 a b ã b p q f g h2 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a F Log@c Hd He + f xLp Lq D + 2 Ha + b Log@c Hd He + f xLp Lq DL 48 a b e ã b p q f g h2 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a F Log@c Hd He + f xLp Lq D - a + b Log@c Hd He + f xLp Lq D 24 a b ã b p q f2 g2 h He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 4 bpq F Log@c Hd He + f xLp Lq D + 3 Ha + b Log@c Hd He + f xLp Lq DL bpq F Log@c Hd He + f xLp Lq D + F Log@c Hd He + f xLp Lq D + 17 18 2.2 Logarithm Functions.nb 54 a b e ã b p q h3 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 1 32 a b h3 He + f xL4 ExpIntegralEiB 4 Ha + b Log@c Hd He + f xLp Lq DL bpq b2 ã b p q f3 g3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 Ha + b Log@c Hd He + f xLp Lq DL 3 bpq 3 3 b2 e2 ã b p q f g h2 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 b2 e3 ã b p q h3 He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a 3 bpq a + b Log@c Hd He + f xLp Lq D bpq a + b Log@c Hd He + f xLp Lq D bpq 2 2 12 b2 e2 ã b p q h3 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a 2 1 27 b2 e ã b p q h3 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a 16 b2 h3 He + f xL4 ExpIntegralEiB 1 Problem ð69: Unable to integrate: :Hg + h xL4 a + b Log@c Hd He + f xLp Lq D , x, 12, 0> F Log@c Hd He + f xLp Lq D2 + bpq F Log@c Hd He + f xLp Lq D2 - 2 Ha + b Log@c Hd He + f xLp Lq DL bpq bpq bpq 3 Ha + b Log@c Hd He + f xLp Lq DL bpq F Log@c Hd He + f xLp Lq D2 F Log@c Hd He + f xLp Lq D2 + F Log@c Hd He + f xLp Lq D2 + 3 Ha + b Log@c Hd He + f xLp Lq DL 4 Ha + b Log@c Hd He + f xLp Lq DL bpq F Log@c Hd He + f xLp Lq D2 - 2 Ha + b Log@c Hd He + f xLp Lq DL 27 b2 ã b p q f g h2 He + f xL3 Hc Hd He + f xLp Lq L p q ExpIntegralEiB a F Log@c Hd He + f xLp Lq D2 + 2 Ha + b Log@c Hd He + f xLp Lq DL 24 b2 e ã b p q f g h2 He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a F Log@c Hd He + f xLp Lq D + F Log@c Hd He + f xLp Lq D2 - a + b Log@c Hd He + f xLp Lq D 12 b2 ã b p q f2 g2 h He + f xL2 Hc Hd He + f xLp Lq L p q ExpIntegralEiB 2a F Log@c Hd He + f xLp Lq D + a + b Log@c Hd He + f xLp Lq D 3 b2 e ã b p q f2 g2 h He + f xL Hc Hd He + f xLp Lq L p q ExpIntegralEiB 3a bpq F Log@c Hd He + f xLp Lq D2 - F Log@c Hd He + f xLp Lq D2 + 2.2 Logarithm Functions.nb - b ã a bpq - - b ã - b ã - b ã - b ã 4a bpq 2a bpq 3a bpq Hf g - e hL4 p Π 5 1 ErfiB pq a+b Log@c Hd He+f xLp Lq D b q He + f xL4 Hc Hd He + f xLp Lq L 4 - Π ErfiB pq p q 2 - pq p 4 p Π 3 b q He + f xL3 Hc Hd He + f xLp Lq L - F p - q 3 pq a+b Log@c Hd He+f xLp Lq D 3 ErfiB b p q f5 q He + f xL5 Hc Hd He + f xLp Lq L - a + b Log@c Hd He + f xLp Lq D f5 a + b Log@c Hd He + f xLp Lq D f5 a + b Log@c Hd He + f xLp Lq D â x Problem ð70: Unable to integrate: :Hg + h xL3 q f5 2 h Hf g - e hL3 He + f xL2 à Hg + h xL - a+b Log@c Hd He+f xLp Lq D 2 ErfiB 5 pq ErfiB 5 a+b Log@c Hd He+f xLp Lq D b p q 10 f5 h3 Hf g - e hL He + f xL4 F a+b Log@c Hd He+f xLp Lq D 2 b q He + f xL2 Hc Hd He + f xLp Lq L Π 2 p h2 Hf g - e hL2 p - 4 f5 h Hf g - e hL3 h4 q He + f xL Hc Hd He + f xLp Lq L Π 2 f5 h3 Hf g - e hL 5a bpq p a + b Log@c Hd He + f xLp Lq D , x, 10, 0> + + 2 h2 Hf g - e hL2 He + f xL3 h4 He + f xL5 F + - F - Hf g - e hL4 He + f xL a + b Log@c Hd He + f xLp Lq D f5 a + b Log@c Hd He + f xLp Lq D 5 f5 F a + b Log@c Hd He + f xLp Lq D f5 + + 19 20 2.2 Logarithm Functions.nb - b ã a bpq - - b ã Hf g - e hL3 4a bpq h3 p Π p q He + f xL Hc Hd He + f xLp Lq L - Π 1 ErfiB pq a+b Log@c Hd He+f xLp Lq D b p q 2 f4 q He + f xL4 Hc Hd He + f xLp Lq L 4 - pq ErfiB a+b Log@c Hd He+f xLp Lq D 2 b p q 16 f4 3 - b ã - b ã 2a bpq 3a bpq h Hf g - e hL2 h2 Hf g - e hL He + f xL3 à Hg + h xL q He + f xL2 Hc Hd He + f xLp Lq L - Π 3 p q He + f xL3 Hc Hd He + f xLp Lq L - a+b Log@c Hd He+f xLp Lq D 2 ErfiB - b p F q 3 pq a+b Log@c Hd He+f xLp Lq D 3 ErfiB b p q 2 f4 a + b Log@c Hd He + f xLp Lq D + f4 a + b Log@c Hd He + f xLp Lq D 3 h Hf g - e hL2 He + f xL2 + f4 a + b Log@c Hd He + f xLp Lq D â x 3 2 pq - 4 f4 h2 Hf g - e hL Hf g - e hL3 He + f xL Π 2 p F F h3 He + f xL4 F - + a + b Log@c Hd He + f xLp Lq D + 2 f4 a + b Log@c Hd He + f xLp Lq D 4 f4 Problem ð71: Unable to integrate: :Hg + h xL2 - b ã a bpq - - b ã - b ã 2a bpq a + b Log@c Hd He + f xLp Lq D , x, 8, 0> Hf g - e hL2 h Hf g - e hL 3a bpq h2 p Π 3 p q He + f xL Hc Hd He + f xLp Lq L Π - 1 pq ErfiB a+b Log@c Hd He+f xLp Lq D b q 2 f3 p Π 2 q He + f xL2 Hc Hd He + f xLp Lq L - 2 pq ErfiB 2 F - a+b Log@c Hd He+f xLp Lq D b p q 2 f3 q He + f xL3 Hc Hd He + f xLp Lq L - 3 pq ErfiB 3 a+b Log@c Hd He+f xLp Lq D b p 6 f3 h Hf g - e hL He + f xL2 p a + b Log@c Hd He + f xLp Lq D f3 + h2 He + f xL3 q F + - Hf g - e hL2 He + f xL a + b Log@c Hd He + f xLp Lq D 3 f3 F a + b Log@c Hd He + f xLp Lq D f3 + 2.2 Logarithm Functions.nb à Hg + h xL a + b Log@c Hd He + f xLp Lq D â x 2 Problem ð72: Unable to integrate: :Hg + h xL - b ã a bpq - - b ã a + b Log@c Hd He + f xLp Lq D , x, 6, 0> Hf g - e hL 2a bpq h Π 2 p p Π q He + f xL Hc Hd He + f xLp Lq L - 1 pq ErfiB a+b Log@c Hd He+f xLp Lq D b p q 2 f2 q He + f xL2 Hc Hd He + f xLp Lq L - 2 pq ErfiB a+b Log@c Hd He+f xLp Lq D 2 b p q 4 f2 Hf g - e hL He + f xL à Hg + h xL a + b Log@c Hd He + f xLp Lq D + f2 a + b Log@c Hd He + f xLp Lq D â x h He + f xL2 F F - + a + b Log@c Hd He + f xLp Lq D 2 f2 Problem ð73: Unable to integrate: : - a + b Log@c Hd He + f xLp Lq D , x, 2, 0> - b ã a bpq p Π q He + f xL Hc Hd He + f xLp Lq L - 1 pq 2f à a + b Log@c Hd He + f xLp Lq D â x Problem ð80: Unable to integrate: 9Hg + h xL3 Ha + b Log@c Hd He + f xLp Lq DL32 , x, 14, 0= ErfiB a+b Log@c Hd He+f xLp Lq D b p q F + He + f xL a + b Log@c Hd He + f xLp Lq D f 21 22 2.2 Logarithm Functions.nb - 3 b32 ã a bpq - 3 b32 ã Hf g - e hL3 p32 4a bpq h3 p32 Π q32 He + f xL Hc Hd He + f xLp Lq L - a+b Log@c Hd He+f xLp Lq D 1 ErfiB pq b p q 4 f4 Π q32 He + f xL4 Hc Hd He + f xLp Lq L - 4 pq ErfiB a+b Log@c Hd He+f xLp Lq D 2 b p q 128 f4 - 9 b32 ã - b32 ã 2a bpq 3a bpq h Hf g - e hL2 p32 h2 Hf g - e hL p32 Π 2 q32 He + f xL2 Hc Hd He + f xLp Lq L - 2 pq 2 ErfiB F + a+b Log@c Hd He+f xLp Lq D b p q q32 He + f xL3 Hc Hd He + f xLp Lq L - 3 pq ErfiB a+b Log@c Hd He+f xLp Lq D 3 b p q 4 f4 a + b Log@c Hd He + f xLp Lq D 2 f4 b h2 Hf g - e hL p q He + f xL3 + 16 f4 Π 3 3 b Hf g - e hL3 p q He + f xL F a + b Log@c Hd He + f xLp Lq D 2 f4 Hf g - e hL3 He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 f4 + h2 Hf g - e hL He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL32 f4 32 3 p q âx à Hg + h xL Ha + b Log@c Hd He + f xL L DL Problem ð81: Unable to integrate: 9Hg + h xL2 Ha + b Log@c Hd He + f xLp Lq DL32 , x, 11, 0= - 9 b h Hf g - e hL2 p q He + f xL2 8 f4 - 3 b h3 p q He + f xL4 F F + - a + b Log@c Hd He + f xLp Lq D a + b Log@c Hd He + f xLp Lq D + 32 f4 3 h Hf g - e hL2 He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL32 2 f4 + h3 He + f xL4 Ha + b Log@c Hd He + f xLp Lq DL32 4 f4 + - 2.2 Logarithm Functions.nb - 3 b32 ã a bpq - 3 b32 ã - b32 ã Hf g - e hL2 p32 2a bpq - h2 p32 Π 2 ErfiB b q32 He + f xL2 Hc Hd He + f xLp Lq L - p 2 pq - 3 b Hf g - e hL2 p q He + f xL + a+b Log@c Hd He+f xLp Lq D 2 ErfiB b p 3 pq ErfiB 3 a+b Log@c Hd He+f xLp Lq D b p q F q 12 f3 a + b Log@c Hd He + f xLp Lq D - 2 f3 b h2 p q He + f xL3 F q 8 f3 q32 He + f xL3 Hc Hd He + f xLp Lq L Π 3 a+b Log@c Hd He+f xLp Lq D 1 pq 4 f3 h Hf g - e hL p32 3a bpq Π q32 He + f xL Hc Hd He + f xLp Lq L a + b Log@c Hd He + f xLp Lq D + 6 f3 F + - 3 b h Hf g - e hL p q He + f xL2 a + b Log@c Hd He + f xLp Lq D - 4 f3 Hf g - e hL2 He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 h Hf g - e hL He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL32 23 + f3 + f3 h2 He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL32 3 f3 32 2 p q âx à Hg + h xL Ha + b Log@c Hd He + f xL L DL Problem ð82: Unable to integrate: 9Hg + h xL Ha + b Log@c Hd He + f xLp Lq DL32 , x, 8, 0= - 3 b32 ã a bpq - 3 b32 ã Hf g - e hL p32 2a bpq h p32 Π 2 Π q32 He + f xL Hc Hd He + f xLp Lq L - 1 pq ErfiB a+b Log@c Hd He+f xLp Lq D b q32 He + f xL2 Hc Hd He + f xLp Lq L - a + b Log@c Hd He + f xLp Lq D 8 f2 32 p q âx à Hg + h xL Ha + b Log@c Hd He + f xL L DL Problem ð83: Unable to integrate: q 4 f2 2 pq ErfiB 2 a+b Log@c Hd He+f xLp Lq D b p 16 f2 3 b h p q He + f xL2 p + q F F - + 3 b Hf g - e hL p q He + f xL Hf g - e hL He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 f2 + a + b Log@c Hd He + f xLp Lq D 2 f2 h He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL32 2 f2 - 24 2.2 Logarithm Functions.nb 9Ha + b Log@c Hd He + f xLp Lq DL32 , x, 3, 0= - 3 b32 ã a bpq p32 Π q32 He + f xL Hc Hd He + f xLp Lq L - 1 pq ErfiB a+b Log@c Hd He+f xLp Lq D b p 4f 3 b p q He + f xL a + b Log@c Hd He + f xLp Lq D 2f 32 p q âx à Ha + b Log@c Hd He + f xL L DL + q F - He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 Problem ð89: Unable to integrate: 9Hg + h xL3 Ha + b Log@c Hd He + f xLp Lq DL52 , x, 18, 0= f 2.2 Logarithm Functions.nb - 15 b52 ã a bpq - - 15 b52 ã Hf g - e hL3 p52 4a bpq h3 p52 Π q52 He + f xL Hc Hd He + f xLp Lq L - 1 ErfiB pq a+b Log@c Hd He+f xLp Lq D b p q 8 f4 Π q52 He + f xL4 Hc Hd He + f xLp Lq L - 4 pq ErfiB a+b Log@c Hd He+f xLp Lq D 2 b p q 1024 f4 - 45 b52 ã - 5 b52 ã 2a bpq 3a bpq h Hf g - e hL2 p52 h2 Hf g - e hL p52 Π 2 q52 He + f xL2 Hc Hd He + f xLp Lq L - 2 pq - - a+b Log@c Hd He+f xLp Lq D 2 ErfiB F F b p q 64 f4 q52 He + f xL3 Hc Hd He + f xLp Lq L - Π 3 3 pq ErfiB 3 a+b Log@c Hd He+f xLp Lq D b p q 24 f4 15 b2 Hf g - e hL3 p2 q2 He + f xL 4 f4 5 b2 h2 Hf g - e hL p2 q2 He + f xL3 a + b Log@c Hd He + f xLp Lq D 12 f4 a + b Log@c Hd He + f xLp Lq D 5 b Hf g - e hL3 p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 2 f4 - 5 b h2 Hf g - e hL p q He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL32 6 f4 Hf g - e hL3 He + f xL Ha + b Log@c Hd He + f xLp Lq DL52 f4 + h2 Hf g - e hL He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL52 f4 52 3 p q âx à Hg + h xL Ha + b Log@c Hd He + f xL L DL Problem ð90: Unable to integrate: 9Hg + h xL2 Ha + b Log@c Hd He + f xLp Lq DL52 , x, 14, 0= 45 b2 h Hf g - e hL2 p2 q2 He + f xL2 + 32 f4 + 15 b2 h3 p2 q2 He + f xL4 F F - + a + b Log@c Hd He + f xLp Lq D a + b Log@c Hd He + f xLp Lq D 15 b h Hf g - e hL2 p q He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL32 8 f4 - 5 b h3 p q He + f xL4 Ha + b Log@c Hd He + f xLp Lq DL32 32 f4 3 h Hf g - e hL2 He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL52 2 f4 + - 256 f4 h3 He + f xL4 Ha + b Log@c Hd He + f xLp Lq DL52 4 f4 + + - + 25 26 2.2 Logarithm Functions.nb - 15 b52 ã a bpq - - 15 b52 ã - 5 b52 ã 2a bpq Hf g - e hL2 p52 - h2 p52 Π 3 a+b Log@c Hd He+f xLp Lq D 1 pq ErfiB b p q 8 f3 h Hf g - e hL p52 3a bpq Π q52 He + f xL Hc Hd He + f xLp Lq L q52 He + f xL2 Hc Hd He + f xLp Lq L - Π 2 2 pq ErfiB - 3 pq ErfiB b 3 p a+b Log@c Hd He+f xLp Lq D b p q 72 f3 15 b2 Hf g - e hL2 p2 q2 He + f xL 4 f3 15 b2 h Hf g - e hL p2 q2 He + f xL2 a + b Log@c Hd He + f xLp Lq D 16 f3 q 2 f3 5 b h2 p q He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL32 18 f3 + 52 2 p q âx à Hg + h xL Ha + b Log@c Hd He + f xL L DL Problem ð91: Unable to integrate: 9Hg + h xL Ha + b Log@c Hd He + f xLp Lq DL52 , x, 10, 0= - + + 5 b2 h2 p2 q2 He + f xL3 a + b Log@c Hd He + f xLp Lq D 5 b h Hf g - e hL p q He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL32 4 f3 f3 + - 36 f3 Hf g - e hL2 He + f xL Ha + b Log@c Hd He + f xLp Lq DL52 h Hf g - e hL He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL52 f3 - F F + a + b Log@c Hd He + f xLp Lq D 5 b Hf g - e hL2 p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 - a+b Log@c Hd He+f xLp Lq D 2 32 f3 q52 He + f xL3 Hc Hd He + f xLp Lq L F h2 He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL52 3 f3 + - 2.2 Logarithm Functions.nb - 15 b52 ã a bpq - - 15 b52 ã Hf g - e hL p52 2a bpq Π 2 h p52 Π q52 He + f xL Hc Hd He + f xLp Lq L - 1 pq ErfiB a+b Log@c Hd He+f xLp Lq D b p q 8 f2 q52 He + f xL2 Hc Hd He + f xLp Lq L - 2 ErfiB pq a+b Log@c Hd He+f xLp Lq D 2 b p q 64 f2 15 b2 Hf g - e hL p2 q2 He + f xL 4 f2 a + b Log@c Hd He + f xLp Lq D 5 b Hf g - e hL p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 2 f2 Hf g - e hL He + f xL Ha + b Log@c Hd He + f xLp Lq DL52 + f2 - + 15 b2 h p2 q2 He + f xL2 F F - + a + b Log@c Hd He + f xLp Lq D - 32 f2 5 b h p q He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL32 + 8 f2 h He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL52 2 f2 52 p q âx à Hg + h xL Ha + b Log@c Hd He + f xL L DL Problem ð92: Unable to integrate: 9Ha + b Log@c Hd He + f xLp Lq DL52 , x, 4, 0= - 15 b52 ã - a bpq p52 Π q52 He + f xL Hc Hd He + f xLp Lq L - 1 pq ErfiB a+b Log@c Hd He+f xLp Lq D b p q 8f 15 b2 p2 q2 He + f xL a + b Log@c Hd He + f xLp Lq D 4f 52 p q âx à Ha + b Log@c Hd He + f xL L DL - Hg + h xL3 Ha + b Log@c Hd He + f xLp Lq DL32 , x, 12, 0> + 5 b p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL32 Problem ð105: Valid but suboptimal antiderivative: : F 2f + He + f xL Ha + b Log@c Hd He + f xLp Lq DL52 f 27 28 2.2 Logarithm Functions.nb - 2ã a bpq - 4ã Hf g - e hL3 4a bpq h3 Π He + f xL Hc Hd He + f xLp Lq L - a+b Log@c Hd He+f xLp Lq D 1 ErfiB pq b p q b32 f4 p32 q32 Π He + f xL4 Hc Hd He + f xLp Lq L - 4 pq ErfiB 2 a+b Log@c Hd He+f xLp Lq D b p q b32 f4 p32 q32 - 6ã - 6ã 2a bpq 3a bpq h Hf g - e hL2 2 Π He + f xL2 Hc Hd He + f xLp Lq L h2 Hf g - e hL 3 Π He + f xL3 Hc Hd He + f xLp Lq L - 2 pq ErfiB 2 F F + + a+b Log@c Hd He+f xLp Lq D F a+b Log@c Hd He+f xLp Lq D F b p q b32 f4 p32 q32 - b32 f4 p32 q32 3 pq ErfiB 3 b p q + bfpq 2 He + f xL Hg + h xL3 a + b Log@c Hd He + f xLp Lq D 2.2 Logarithm Functions.nb 1 a + b Log@c Hd He + f xLp Lq D b32 f4 p32 q32 4a b e ã b p q f3 g3 - - 4a b ã b p q f4 g2 h 3 b ã b p q f4 g h2 p 4a 3a ã 3 3 f g bpq - 4 p Π He + f xL Hc Hd He + f xL L L 4a b e ã b p q f3 g h2 3 pq ErfiB 3a b 2 p b F q 2a 2a 3 e2 ã b p q h3 a 2 Π He + f xL2 Hc Hd He + f xLp Lq L p q ErfiB 2 a 3 e ã b p q h3 1 p F a + b Log@c Hd He + f xLp Lq D - a + b Log@c Hd He + f xLp Lq D + 3 q p p q a + b Log@c Hd He + f xLp Lq D b p p F q a + b Log@c Hd He + f xLp Lq D b F q a + b Log@c Hd He + f xLp Lq D b 3 F a + b Log@c Hd He + f xLp Lq D + a + b Log@c Hd He + f xLp Lq D b 2 3 Π He + f xL3 Hc Hd He + f xLp Lq L p q ErfiB 3 Π He + f xL3 Hc Hd He + f xLp Lq L p q ErfiB q F a + b Log@c Hd He + f xLp Lq D 2 2 2 Π He + f xL2 Hc Hd He + f xLp Lq L p q ErfiB p b 1 3 ã b p q f g h2 q q x4 Hc Hd He + f xLp Lq L p q + 4 p a + b Log@c Hd He + f xLp Lq D + 2 2 6 e ã b p q f g h2 p 4a b ã b p q f4 h3 a + b Log@c Hd He + f xLp Lq D - q a + b Log@c Hd He + f xLp Lq D 2 Π He + f xL2 Hc Hd He + f xLp Lq L p q ErfiB 2a 3 ã b p q f2 g2 h p b a + b Log@c Hd He + f xLp Lq D b F a + b Log@c Hd He + f xLp Lq D 3 3 q b Π He + f xL Hc Hd He + f xLp Lq L p q ErfiB Π He + f xL Hc Hd He + f xLp Lq L p q ErfiB p a + b Log@c Hd He + f xLp Lq D Π He + f xL Hc Hd He + f xLp Lq L p q ErfiB Π He + f xL4 ErfiB 4 a + b Log@c Hd He + f xLp Lq D 4 p 4 q x3 Hc Hd He + f xLp Lq L p q - p q x Hc Hd He + f xLp Lq L p q - 4a b e ã b p q f3 g2 h q x2 Hc Hd He + f xLp Lq L p q - p 4a b e ã b p q f3 h3 3 3 e2 ã b p q f g h2 e3 ã b p q h3 4 p 4 q x3 Hc Hd He + f xLp Lq L p q - 4 pq q x Hc Hd He + f xLp Lq L p q - 3 4a b ã b p q f4 g3 q x2 Hc Hd He + f xLp Lq L p q - 3 p q 3a 2 h3 Hc Hd He + f xLp Lq L 4 3 e ã b p q f2 g2 h 3a 4a bpq q Hc Hd He + f xLp Lq L p q - p 3 2ã 29 q F F F a + b Log@c Hd He + f xLp Lq D a + b Log@c Hd He + f xLp Lq D + a + b Log@c Hd He + f xLp Lq D + a + b Log@c Hd He + f xLp Lq D - a + b Log@c Hd He + f xLp Lq D 30 2.2 Logarithm Functions.nb Problem ð111: Valid but suboptimal antiderivative: : Hg + h xL3 Ha + b Log@c Hd He + f xLp Lq DL52 - 4ã a bpq - 32 ã Hf g - e hL3 4a bpq h3 , x, 23, 0> Π He + f xL Hc Hd He + f xLp Lq L - 1 pq ErfiB a+b Log@c Hd He+f xLp Lq D b p q 3 b52 f4 p52 q52 Π He + f xL4 Hc Hd He + f xLp Lq L - 4 pq ErfiB a+b Log@c Hd He+f xLp Lq D 2 b p q 3 b52 f4 p52 q52 - 8ã h Hf g - e hL2 2a bpq - 12 ã 3a bpq h2 Hf g - e hL 2 Π He + f xL2 Hc Hd He + f xLp Lq L - + a+b Log@c Hd He+f xLp Lq D 2 b 3 Π He + f xL3 Hc Hd He + f xLp Lq L - p q 3 pq ErfiB 3 + b p q 4 Hf g - e hL He + f xL Hg + h xL2 b2 f2 p2 q2 a + b Log@c Hd He + f xLp Lq D Result of integration not displayed since its leaf count is 10139 Problem ð112: Valid but suboptimal antiderivative: Hg + h xL2 Ha + b Log@c Hd He + f xLp Lq DL52 , x, 17, 0> F a+b Log@c Hd He+f xLp Lq D b52 f4 p52 q52 3 b f p q Ha + b Log@c Hd He + f xLp Lq DL32 : ErfiB + b52 f4 p52 q52 2 He + f xL Hg + h xL3 * * * 2 pq F F + F - 16 He + f xL Hg + h xL3 3 b2 f p2 q2 * * * a + b Log@c Hd He + f xLp Lq D 2.2 Logarithm Functions.nb - 4ã a bpq - 16 ã - 4ã Hf g - e hL2 2a bpq - h2 a+b Log@c Hd He+f xLp Lq D 1 pq ErfiB b p 2 Π He + f xL2 Hc Hd He + f xLp Lq L - 2 pq b p q 3 b52 f3 p52 q52 3 Π He + f xL3 Hc Hd He + f xLp Lq L - 8 Hf g - e hL He + f xL Hg + h xL * * * + a+b Log@c Hd He+f xLp Lq D 2 ErfiB 3 pq a+b Log@c Hd He+f xLp Lq D 3 ErfiB b p q b52 f3 p52 q52 3 b2 f2 p2 q2 F q 3 b52 f3 p52 q52 h Hf g - e hL 3a bpq Π He + f xL Hc Hd He + f xLp Lq L - 4 He + f xL Hg + h xL2 - a + b Log@c Hd He + f xLp Lq D F + 2 He + f xL Hg + h xL2 3 b f p q Ha + b Log@c Hd He + f xLp Lq DL32 a + b Log@c Hd He + f xLp Lq D b2 f p2 q2 F Result of integration not displayed since its leaf count is 6106 + * * * Problem ð123: Valid but suboptimal antiderivative: 9Hg + h xL32 Ha + b Log@c Hd He + f xLp Lq DL2 , x, 13, 0= 368 b2 Hf g - e hL 2 p2 q2 g+hx 128 b2 + 75 f2 h Hf g - e hL p2 q2 225 f h 8 b2 Hf g - e hL52 p2 q2 ArcTanhB f g+h x f g-e h 5 f52 h F 2 - Hg + h xL 32 8 b Hf g - e hL2 p q 8 b Hf g - e hL p q Hg + h xL32 Ha + b Log@c Hd He + f xLp Lq DL - 15 f h 8 b Hf g - e hL52 p q ArcTanhB f g+h x f g-e h 5 f52 h 16 b2 Hf g - e hL52 p2 q2 ArcTanhB 5 f52 h f 16 f g-e h F LogB 2 1- f g+h x f g-e h p2 + q2 Hg + h xL 52 125 h - 368 b2 Hf g - e hL52 p2 q2 ArcTanhB 75 f52 h g + h x Ha + b Log@c Hd He + f xLp Lq DL - 5 f2 h 8 b p q Hg + h xL52 Ha + b Log@c Hd He + f xLp Lq DL + 25 h F Ha + b Log@c Hd He + f xLp Lq DL g+h x b2 F + + 2 Hg + h xL52 Ha + b Log@c Hd He + f xLp Lq DL2 + 5h 8 b2 Hf g - e hL52 p2 q2 PolyLogB2, 5 f52 h f g-e h+ f f g-e h g+h x f g-e h- f f g-e h g+h x F f g+h x f g-e h F - 31 32 2.2 Logarithm Functions.nb 1 3fh 1+ f g - e h + h He + f xL 2 b2 g p2 q2 h He+f xL f g-e h f 3 h He + f xL HypergeometricPFQB:fg 1+ h He + f xL 1 2 , 1, 1>, 82, 2<, Log@e + f xD2 - e h 1+ fg-eh 1 15 f2 h 1+ h He + f xL fg-eh 2 b2 p2 q2 h He+f xL f g-e h f g - e h + h He + f xL f 10 e h2 He + f xL HypergeometricPFQB:15 e h2 He + f xL HypergeometricPFQB:4 f2 g2 3 h He + f xL HypergeometricPFQB:- f g - e h + h He + f xL 3 2 1 2 4 h2 He + f xL2 -f g + e h 2 , 1, 1, 1>, 82, 2, 2<, Log@e + f xD2 + h He + f xL , 1, 1, 1>, 82, 2, 2<, , 1, 1, 1>, 82, 2, 2<, Log@e + f xD - 8 e f g h 1+ h He + f xL fg-eh f g - e h + h He + f xL fg-eh -f g + e h F- h He + f xL -f g + e h h He + f xL -f g + e h Log@e + f xD2 - fg-eh 3 2 , 1, 1, 1>, 82, 2, 2<, -f g + e h F- F - 4 f2 g2 Log@e + f xD + 8 e f g h Log@e + f xD - 4 e2 h2 Log@e + f xD + f g - e h + h He + f xL Log@e + f xD - 8 e h2 He + f xL h He + f xL F+ f g - e h + h He + f xL Log@e + f xD + 4 e2 h2 fg-eh f g - e h + h He + f xL h He + f xL F Log@e + f xD - f g Log@e + f xD2 + e h Log@e + f xD2 + 10 f g h He + f xL HypergeometricPFQB:- fg-eh 8 f g h He + f xL h He + f xL 1 Log@e + f xD + fg-eh f g - e h + h He + f xL Log@e + f xD + fg-eh Log@e + f xD - 15 e h2 He + f xL HypergeometricPFQB:- 2 f2 g2 Log@e + f xD2 - e f g h Log@e + f xD2 + 3 e2 h2 Log@e + f xD2 + 2 f2 g2 1 2 , 1, 1>, 82, 2<, f g - e h + h He + f xL h He + f xL -f g + e h F Log@e + f xD - Log@e + f xD2 + fg-eh efgh f g - e h + h He + f xL Log@e + f xD2 - 3 e2 h2 fg-eh 6 e h2 He + f xL f g - e h + h He + f xL fg-eh f g - e h + h He + f xL fg-eh Log@e + f xD2 - 3 h2 He + f xL2 Log@e + f xD2 - f g h He + f xL f g - e h + h He + f xL Log@e + f xD2 + fg-eh + f g - e h + h He + f xL fg-eh Log@e + f xD2 + 2.2 Logarithm Functions.nb 10 h H- f g + e hL He + f xL HypergeometricPFQB:6 Hf g - e hL32 ArcTanhB 1 4bgpq 9f f f g-e h+h He+f xL f f g-e h f h 3 2 F , 1, 1>, 82, 2<, h He + f xL -f g + e h f g - e h + h He + f xL + f F H1 + Log@e + f xDL + Hf g - e hL H- 8 + 3 Log@e + f xDL h a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - 1 4bpq 225 f2 h q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D 30 Hf g - e hL32 H2 f g + 3 e hL ArcTanhB f + LogBc ãq H-p Log@e+f xD+Log@d He+f xL f g-e h+h He+f xL f f g-e h f F + p DL + He + f xL H- 2 + 3 Log@e + f xDL Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL F q H-p Log@e+f xD+Log@d He+f xLp DL F Log@d He+f xLp D + f g - e h + h He + f xL f I2 f2 g2 H31 - 15 Log@e + f xDL + f g h He H76 - 15 Log@e + f xDL + He + f xL H- 16 + 15 Log@e + f xDLL + 3 h2 I3 He + f xL2 H- 2 + 5 Log@e + f xDL + e2 H- 46 + 15 Log@e + f xDL - 2 e He + f xL H- 11 + 15 Log@e + f xDLMM a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q 1 g+hx 5h 4 5 q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- 2 g2 a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L g x a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D + Log@d He+f xLp D q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D 2 + + F 2 + 33 34 2.2 Logarithm Functions.nb q2 5 q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL q- Log@d He+f xLp D h x2 a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q- Log@d He + f xLp D p DL q- Log@d He+f xLp D F 2 F 2 + Problem ð124: Valid but suboptimal antiderivative: : g + h x Ha + b Log@c Hd He + f xLp Lq DL2 , x, 14, 0> 64 b2 Hf g - e hL p2 q2 g+hx 16 b2 p2 + 9fh 8 b Hf g - e hL p q q2 Hg + h xL 64 b2 Hf g - e hL32 p2 q2 ArcTanhB 32 - 27 h 9 g + h x Ha + b Log@c Hd He + f xLp Lq DL - 3fh 8 b Hf g - e hL32 p q ArcTanhB f g+h x f g-e h 3 f32 h 16 b2 Hf g - e hL32 p2 q2 ArcTanhB 3 f32 h f g+h x F LogB 2 1- f g+h x f g-e h g+h x f g-e h h F - 8 b2 Hf g - e hL32 p2 q2 ArcTanhB 8 b p q Hg + h xL32 Ha + b Log@c Hd He + f xLp Lq DL 3 f32 + 9h F Ha + b Log@c Hd He + f xLp Lq DL f g-e h f32 f F + + 2 Hg + h xL32 Ha + b Log@c Hd He + f xLp Lq DL2 + 3h 8 b2 Hf g - e hL32 p2 q2 PolyLogB2, 3 f32 h f g-e h+ f f g-e h g+h x f g-e h- f f g-e h g+h x F h f g+h x f g-e h F 2 - 2.2 Logarithm Functions.nb 1 1 3 b2 p2 q2 2 3 h He + f xL HypergeometricPFQB:- g+hx f Hg+h xL 9h f f g-e h - 3 h He + f xL HypergeometricPFQB:1 f32 2 b p q 6 Hf g - e hL32 ArcTanhB f 1 2 , 1, 1>, 82, 2<, g+hx fg-eh F+ 1 2 , 1, 1, 1>, 82, 2, 2<, h He + f xL -f g + e h F+ eh+f hx h He + f xL -f g + e h f Hg + h xL F + Log@e + f xD + g -1 + fg-eh f Hg + h xL fg-eh g + h x H6 e h - 2 f H4 g + h xL + 3 f Hg + h xL Log@e + f xDL f H- a + b p q Log@e + f xD - b Log@c Hd He + f xLp Lq DL + 3 Hg + h xL32 Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL2 Problem ð125: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL2 , x, 12, 0> g+hx 16 b2 p2 q2 16 b2 g+hx f g - e h p2 q2 ArcTanhB f g+h x f g-e h h 8 b2 f h f f g - e h p2 q2 ArcTanhB g+h x f g-e h f h 8b f g - e h p q ArcTanhB f g+h x f g-e h f h 16 b2 f g - e h p2 q2 ArcTanhB f 2 - g + h x Ha + b Log@c Hd He + f xLp Lq DL 8bpq - g+h x F LogB + h F Ha + b Log@c Hd He + f xLp Lq DL f g-e h f h F F 2 1- f g+h x f g-e h F 8 b2 2 + g + h x Ha + b Log@c Hd He + f xLp Lq DL2 + h f g - e h p2 q2 PolyLogB2, - + f h 35 f g-e h+ f f g-e h g+h x f g-e h- f f g-e h g+h x F Log@e + f xD - 36 2.2 Logarithm Functions.nb f 1 fh 2 a2 f g - 4 a b f g p q + a2 f h x - 4 a b f h p q x + 4 a b f fg-eh pq g+hx g + h x ArcTanhB g+hx fg-eh f Hg + h xL b2 h p2 q2 He + f xL fg-eh 1 h He + f xL HypergeometricPFQB: , 1, 1, 1>, 82, 2, 2<, F + 4 b2 f g p2 q2 Log@e + f xD + 2 -f g + e h f 4 b2 f h p2 q2 x Log@e + f xD - 4 b2 f g - e h p2 q2 f g+hx g + h x ArcTanhB fg-eh f Hg + h xL b2 h p2 q2 He + f xL b2 f g p2 q2 fg-eh f Hg + h xL F+ F Log@e + f xD - 1 h He + f xL HypergeometricPFQB: , 1, 1>, 82, 2<, F Log@e + f xD 2 -f g + e h Log@e + f xD2 + b2 e h p2 q2 fg-eh f Hg + h xL fg-eh Log@e + f xD2 + 2 a b f g Log@c Hd He + f xLp Lq D - 4 b2 f g p q Log@c Hd He + f xLp Lq D + 2 a b f h x Log@c Hd He + f xLp Lq D - 4 b2 f h p q x Log@c Hd He + f xLp Lq D + f 4 b2 f fg-eh pq g+hx g + h x ArcTanhB fg-eh F Log@c Hd He + f xLp Lq D + b2 f g Log@c Hd He + f xLp Lq D2 + b2 f h x Log@c Hd He + f xLp Lq D2 Problem ð126: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL2 8 b2 Hg + h xL32 f p2 q2 ArcTanhB f , x, 9, 0> g+h x f g-e h h 16 b2 fg-eh f p2 q2 ArcTanhB f F g+h x f g-e h h 2 fg-eh 8b f p q ArcTanhB f g+h x f g-e h h F LogB 2 1- f g+h x f g-e h F 8 b2 F Ha + b Log@c Hd He + f xLp Lq DL fg-eh f p2 q2 PolyLogB2, - 2 Ha + b Log@c Hd He + f xLp Lq DL2 h f g-e h+ f f g-e h g+h x f g-e h- f f g-e h g+h x h - fg-eh F g+hx - 2.2 Logarithm Functions.nb 1 h 2 1 K f g - e h Hg + h xLO2 b p q 2 f Hg + h xL ArcTanhB H- a + b p q Log@e + f xD - b Log@c Hd He + f xLp Lq DL - 1 KHf g - e hL g + h x Ob2 p2 q2 h He + f xL Hf g - e hL Log@e + f xD g+hx fg-eh F+ fg-eh g + h x Log@e + f xD Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL2 f Hg + h xL + g+hx 3 HypergeometricPFQB:1, 1, 1, fg-eh f Hg + h xL -1 + f 2 Log@e + f xD - 4 fg-eh f Hg + h xL >, 82, 2, 2<, 1 LogB fg-eh 1+ 2 f Hg + h xL fg-eh h He + f xL -f g + e h F Problem ð127: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL2 Hg + h xL52 f 16 b2 f32 p2 q2 ArcTanhB , x, 11, 0> g+h x f g-e h 3 h Hf g - e hL32 8 b f32 p q ArcTanhB f g+h x f g-e h F 8 b2 f32 p2 q2 ArcTanhB f 3 h Hf g - e hL32 F Ha + b Log@c Hd He + f xLp Lq DL g+h x f g-e h 3 h Hf g - e hL32 g+h x f g-e h + 3 h Hf g - e hL32 16 b2 f32 p2 q2 ArcTanhB f F LogB 2 1- f g+h x f g-e h F - F 2 + 8 b f p q Ha + b Log@c Hd He + f xLp Lq DL 2 Ha + b Log@c Hd He + f xLp Lq DL2 3 h Hg + h xL32 8 b2 f32 p2 q2 PolyLogB2, - 3 h Hf g - e hL g+hx - f g-e h+ f f g-e h g+h x f g-e h- f f g-e h g+h x 3 h Hf g - e hL32 F - F+ 37 38 2.2 Logarithm Functions.nb f 2 ArcTanhB 1 f g-e h+h He+f xL F f f g-e h 4 b f32 p q - Hf g - e hL32 3h f + f g-e h+h He+f xL f Hh H2 He + f xL + e H- 2 + Log@e + f xDLL - f g H- 2 + Log@e + f xDLL Hf g - e hL Hf g + f h xL2 a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL 1 Log@d He + f xLp D q - 3 h Hg + h xL32 q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F q- 2 a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - Log@d He + f xLp D 1 f g-e h+h He+f xL 3 h Hf g - e hL2 Hf g + f h xL f 5 HypergeometricPFQB:1, 1, 1, 2 4 f g - 4 e h + 4 h He + f xL - 4 f g 2 b2 f p2 q2 3 h He + f xL Hf g + f h xL >, 82, 2, 2<, h He + f xL -f g + e h f g - e h + h He + f xL +4eh f g - e h + h He + f xL fg-eh Log@e + f xD - e h f g - e h + h He + f xL fg-eh Log@e + f xD - 4 Hf g - e hL Problem ð128: Valid but suboptimal antiderivative: Ha + b Log@c Hd He + f xLp Lq DL2 Hg + h xL72 , x, 12, 0> q- Log@d He+f xLp D Log@d He+f xLp D fg-eh f g - e h + h He + f xL fg-eh h He + f xL p DL f g - e h + h He + f xL F + Hf g - e hL Log@e + f xD fg-eh f g Log@e + f xD + e h Log@e + f xD + f g : p DL - 4 h He + f xL f g - e h + h He + f xL fg-eh f g - e h + h He + f xL Log@e + f xD + fg-eh f g - e h + h He + f xL fg-eh 32 1 LogB 1+ 2 1+ h He + f xL fg-eh F - - 2 + 2.2 Logarithm Functions.nb 16 - b2 f2 p2 15 h Hf g - e hL2 64 b2 f52 p2 q2 ArcTanhB q2 F g+h x f g-e h 15 h Hf g - e hL52 + g+hx 8 b f p q Ha + b Log@c Hd He + f xL L DL p q 15 h Hf g - e hL Hg + h xL32 2 Ha + b Log@c Hd He + f xL L DL p q 5 h Hg + h xL52 f 8b + f2 8 b2 f52 p2 q2 ArcTanhB f g+h x 2 5 h Hf g - e hL52 F LogB 1- 2 + f g+h x f g-e h g+h x f g-e h F F Ha + b Log@c Hd He + f xLp Lq DL 5 h Hf g - e hL52 - 2 f F 8 b f52 p q ArcTanhB g+hx f g-e h - 5 h Hf g - e hL52 p q Ha + b Log@c Hd He + f xL L DL 16 b2 f52 p2 q2 ArcTanhB g+h x f g-e h + p q 5 h Hf g - e hL2 f 8 b2 f52 p2 q2 PolyLogB2, - f g-e h+ f f g-e h g+h x f g-e h- f f g-e h g+h x 5 h Hf g - e hL52 F - 39 40 2.2 Logarithm Functions.nb 1 f g-e h+h He+f xL 5 h Hf g - e hL3 Hf g + f h xL2 f f g - e h + h He + f xL 2 b2 f2 p2 q2 5 h He + f xL Hf g + f h xL2 5 h He + f xL Hf g + f h xL2 Hf g - e hL f2 g2 - 1 + h2 - 2 e He + f xL f g - e h + h He + f xL f g - e h + h He + f xL f g - e h + h He + f xL fg-eh f g-e h+h He+f xL f f g-e h 52 4bf pq - Hf g - e hL 52 15 h 2 7 HypergeometricPFQB:1, 1, fg-eh f 1 fg-eh fg-eh 6 ArcTanhB 7 HypergeometricPFQB:1, 1, 1, F 2 f g - e h + h He + f xL - 2 f g h - He + f xL + He + f xL2 f g - e h + h He + f xL + e2 - 1 + fg-eh Hf g - e hL Hf g + f h xL 2 h He + f xL -f g + e h + e -1 + fg-eh 1 + >, 82, 2<, >, 82, 2, 2<, 3 f h He + f xL -f g + e h F- F Log@e + f xD + f g - e h + h He + f xL + fg-eh f g - e h + h He + f xL Log@e + f xD2 + fg-eh f g - e h + h He + f xL f Ih2 I- 14 e He + f xL + 6 He + f xL2 + e2 H8 - 3 Log@e + f xDLM + f2 g2 H8 - 3 Log@e + f xDL + 2 f g h H7 He + f xL + e H- 8 + 3 Log@e + f xDLLM a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL 1 Log@d He + f xLp D q - 5 h Hg + h xL52 q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F q- 2 a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - Log@d He + f xLp D p DL q- Log@d He+f xLp D Log@d He+f xLp D - 2 2.2 Logarithm Functions.nb 41 Problem ð129: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL2 Hg + h xL92 16 - b2 f2 p2 , x, 13, 0> q2 105 h Hf g - e hL2 Hg + h xL32 f 8 b2 f72 p2 q2 ArcTanhB 128 - F 2 g+h x + p q Ha + b Log@c Hd He + f xL L DL f g+h x f g-e h 1 7 h Hf g - e hL72 7 h Hf g - e hL4 Hg + h xL72 g+hx 8 b f72 p q ArcTanhB f g+h x f g-e h 2 1- f g+h x f g-e h f Hg + h xL F 1 105 h g+h x f g-e h 4 b f72 p q - Hf g - e hL72 F f Hg + h xL 2 f Hg + h xL 6 f Hg+h xL f g-e h Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL - 2 Ha - b p q Log@e + f xD + b Log@c Hd He + f xLp Lq DL2 7 h Hg + h xL72 + - f g-e h+ f f g-e h g+h x f g-e h- f f g-e h g+h x 7 h Hg + h xL72 9 + 3 g h2 x2 2 h He + f xL -f g + e h f Hg + h xL Hf g-e hL2 + 30 f3 Hg+h xL3 Hf g-e hL3 f72 Hg + h xL72 >, 82, 2, 2<, h He + f xL -f g + e h F- F Log@e + f xD + Hf g - e hL + h3 x3 fg-eh 10 f2 Hg+h xL2 - F HypergeometricPFQB:1, 1, 1, >, 82, 2<, + 2 Ha + b Log@c Hd He + f xLp Lq DL2 fg-eh 9 + 21 h Hf g - e hL2 Hg + h xL32 7 h Hf g - e hL72 - fg-eh f 8 b f2 p q Ha + b Log@c Hd He + f xLp Lq DL + 8 b2 f72 p2 q2 PolyLogB2, - fg-eh 30 ArcTanhB + F Ha + b Log@c Hd He + f xLp Lq DL HypergeometricPFQB:1, 1, 3 e f2 g2 h - 3 e2 f g h2 + e3 h3 + f3 3 g2 h x F 7 h Hf g - e hL72 - 2 b2 p2 q2 7 f3 h He + f xL Hg + h xL3 7 f3 h He + f xL Hg + h xL3 105 h Hf g - e hL72 + 35 h Hf g - e hL Hg + h xL52 F LogB g+h x f g-e h g+hx 16 b2 f72 p2 q2 ArcTanhB f 368 b2 f72 p2 q2 ArcTanhB q2 8 b f p q Ha + b Log@c Hd He + f xLp Lq DL p q 7 h Hf g - e hL3 p2 105 h Hf g - e hL 7 h Hf g - e hL72 8b f3 3 f g-e h f3 b2 - 15 Log@e + f xD f Hg + h xL fg-eh + g3 - 1 + f Hg + h xL fg-eh Log@e + f xD2 + 42 2.2 Logarithm Functions.nb Problem ð142: Mathematica is able to integrate expression!!!: : Hi + j xLp Ha + b Log@c He + f xLDL , x, 0, 0> de+dfx IntB Hi + j xLp Ha + b Log@c He + f xLDL , xF de+dfx 1 d Hi + j xL p - a Hi + j xL Hypergeometric2F1B1, 1 + p, 2 + p, Hf i - e jL H1 + pL - HypergeometricPFQB8- p, - p, - p<, 81 - p, 1 - p<, f Hi+j xL f i-e j F -f i + e j j He + f xL + 1 If p2 Mb f Hi + j xL -p j He + f xL F + p Hypergeometric2F1B- p, - p, 1 - p, -f i + e j j He + f xL F Log@c He + f xLD Problem ð151: Mathematica is able to integrate expression!!!: : Hi + j xLp Ha + b Log@c He + f xLDL2 , x, 0, 0> de+dfx IntB Hi + j xLp Ha + b Log@c He + f xLDL2 , xF de+dfx 1 d Hi + j xL p - a2 Hi + j xL Hypergeometric2F1B1, 1 + p, 2 + p, Hf i - e jL H1 + pL - HypergeometricPFQB8- p, - p, - p<, 81 - p, 1 - p<, 1 b2 f p3 f Hi + j xL -p j He + f xL f Hi+j xL f i-e j -f i + e j j He + f xL Problem ð156: Valid but suboptimal antiderivative: Ha + b Log@c He + f xLDL2 de+dfx Ha + b Log@c He + f xLDL3 3bdf 1 2ab + f p2 f Hi + j xL , x, 1, 0> -p j He + f xL F + p Hypergeometric2F1B- p, - p, 1 - p, 2 HypergeometricPFQB8- p, - p, - p, - p<, 81 - p, 1 - p, 1 - p<, - 2 HypergeometricPFQB8- p, - p, - p<, 81 - p, 1 - p<, : F -f i + e j j He + f xL -f i + e j j He + f xL -f i + e j j He + f xL F Log@c He + f xLD + F + p Log@c He + f xLD F + p Hypergeometric2F1B- p, - p, 1 - p, -f i + e j j He + f xL F Log@c He + f xLD 2.2 Logarithm Functions.nb a2 Log@c He + f xLD + df a b Log@c He + f xLD2 + df b2 Log@c He + f xLD3 3df Problem ð169: Valid but suboptimal antiderivative: : Hf + g xL52 Ha + b Log@c Hd + e xLn DL - , x, 12, 0> d+ex 92 b He f - d gL2 n 32 b He f - d gL n Hf + g xL 32 f+gx - 15 e3 e f+g x e f-d g e72 2 Hf + g xL 52 - 45 e2 2 b He f - d gL52 n ArcTanhB Ha + b Log@c Hd + e xL DL F 2 n - 5e 4 b He f - d gL52 n ArcTanhB e f+g x e f-d g + 2 He f - d gL2 2 1- e f+g x e f-d g F e g d+ex - g 2 b f2 n e f-d g e72 e f-d g + + 2 He f - d gL Hf + g xL32 Ha + b Log@c Hd + e xLn DL + 3 e2 F Ha + b Log@c Hd + e xLn DL 2 b He f - d gL52 n PolyLogB2, - F - e f-d g+ e e f-d g f+g x e f-d g- e e f-d g f+g x F e f - d g + g Hd + e xL e e f+e g x g Hd+e xL d + e x HypergeometricPFQB:- 1 ,- 2 ef-dg e f - d g ArcSinhB g 2bn f+g x f+g x 15 e72 e3 e e e72 1 -2 92 b He f - d gL52 n ArcTanhB f + g x Ha + b Log@c Hd + e xLn DL 2 He f - d gL52 ArcTanhB F LogB + 25 e e72 1 4 b n Hf + g xL 52 e f - d g + g Hd + e xL d+ex 2 1 1 -e f + d g >, : , >, F+ 2 2 2 g Hd + e xL F Log@d + e xD - - 2 e2 f2 d+ex e + e f - d g + g Hd + e xL 1 ,- g g Hd + e xL d+ex 1 15 e3 d+ex e f - d g + g Hd + e xL g Hd + e xL e f+e g x g Hd+e xL +4defg 1+ Log@d + e xD - g Hd+e xL e f-d g d+ex e f - d g + g Hd + e xL g Hd + e xL - - + 43 44 2.2 Logarithm Functions.nb 2 d2 g2 2 d2 g2 e f - d g + g Hd + e xL g Hd + e xL d+ex ef+egx d+ex 4 d g2 Hd + e xL32 g Hd + e xL g Hd + e xL ef-dg ef+egx g Hd + e xL 3 HypergeometricPFQB:2 30 d2 g2 1+ d+ex 1+ + 2 e2 f2 1+ g Hd + e xL g Hd + e xL g Hd + e xL + 4 e f g Hd + e xL32 g Hd + e xL -e f + d g 1 HypergeometricPFQB:- 6defg 2 e2 f2 2 ef+egx g Hd + e xL 1 HypergeometricPFQB:2 e f - d g + g Hd + e xL g Hd + e xL d+ex ef+egx g Hd + e xL d+ex 23 d2 g2 ef-dg g Hd + e xL 1+ 1+ g Hd + e xL g Hd + e xL 2bfn e f - d g + g Hd + e xL e 12 d g , 1, 1>, 82, 2<, g Hd + e xL ef-dg - 5 e f g Hd + e xL32 3 2 F - 2 e2 f2 g Hd + e xL g d+ex F Log@d + e xD + e f - d g + g Hd + e xL 3 e2 g Hd + e xL + ef-dg - HypergeometricPFQB:- e f - d g + g Hd + e xL Log@d + e xD - 3 g2 Hd + e xL52 ef-dg d+ex 1+ -e f + d g d+ex 1+ ef-dg g Hd + e xL Log@d + e xD - e f g Hd + e xL32 ArcSinhB ef-dg g Hd + e xL 1 1 -e f + d g >, : , >, F2 2 2 g Hd + e xL d+ex g Hd + e xL d+ex g Hd + e xL g Hd + e xL 1+ g Hd + e xL 1+ Log@d + e xD - g Hd + e xL Log@d + e xD + g Hd + e xL Log@d + e xD + ef-dg 1 e f+e g x 1 + g Hd + e xL g Hd+e xL 1 ,- 2 g Hd + e xL ef-dg ef+egx ef-dg , 1, 1>, 82, 2<, ef-dg ef+egx HypergeometricPFQB:- g Hd + e xL Log@d + e xD + 1+ d+ex ef+egx e f - d g + g Hd + e xL d+ex ef+egx ef-dg ef-dg 1+ g Hd + e xL 1 Log@d + e xD + 6 d e f g g Hd + e xL 1+ ef+egx -4defg ef+egx ef-dg ef+egx g Hd + e xL g Hd + e xL ,2 Log@d + e xD + 8 d2 g2 g Hd + e xL ef+egx d+ex 11 d g2 Hd + e xL32 15 d2 g32 1+ ef+egx ef+egx 1 ,- g Hd + e xL ef-dg F + 5 d g2 Hd + e xL32 ef-dg 15 d g2 Hd + e xL32 1+ g Hd + e xL + 2 g2 Hd + e xL52 ef-dg , 1, 1>, 82, 2<, ef+egx d+ex g Hd+e xL e f-d g 1 1 -e f + d g >, : , >, F2 2 2 g Hd + e xL 1 ,2 1+ g Hd + e xL -e f + d g F+ Log@d + e xD - 2.2 Logarithm Functions.nb 3 g Hd + e xL32 ef+egx 1 g Hd + e xL HypergeometricPFQB:2 e f - d g + g Hd + e xL ef+egx 2 d+ex 3d g g Hd + e xL ef-dg e f -1 + e g Hd + e xL -e f + d g e f - d g + g Hd + e xL f+g x e f-d g F+ e f - d g + g Hd + e xL +g d-4d ef-dg ef-dg ef-dg ArcSinhB ef-dg 2 He f - d gL52 ArcTanhB , 1, 1>, 82, 2<, g d+ex e f - d g + g Hd + e xL + Hd + e xL 45 + ef-dg F Log@d + e xD - F Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL + e72 f+gx 2 I23 e2 f2 - 35 d e f g + 15 d2 g2 M Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL + 15 e3 2 g H11 e f - 5 d gL x Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL + 15 e2 2 g2 x2 Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL 5e Problem ð170: Valid but suboptimal antiderivative: : - Hf + g xL32 Ha + b Log@c Hd + e xLn DL , x, 13, 0> d+ex 16 b He f - d gL n 3 2 He f - d gL 4 b n Hf + g xL 32 f+gx - e2 + 9e 16 b He f - d gL32 n ArcTanhB 3 f + g x Ha + b Log@c Hd + e xLn DL + e2 4 b He f - d gL32 n ArcTanhB e52 e f+g x e f-d g F LogB f+g x e f-d g e52 2 Hf + g xL32 Ha + b Log@c Hd + e xLn DL F - 3e 2 1- e e f+g x e f-d g F - + e f+g x e f-d g e52 2 He f - d gL32 ArcTanhB 2 b He f - d gL32 n PolyLogB2, e52 2 b He f - d gL32 n ArcTanhB e f-d g+ e e f-d g f+g x e f-d g- e e f-d g f+g x F e f+g x e f-d g e52 F 2 + F Ha + b Log@c Hd + e xLn DL - 46 2.2 Logarithm Functions.nb e f - d g + g Hd + e xL 1 2bfn e g e e f+e g x g Hd+e xL d+ex 1 -2 g d + e x HypergeometricPFQB:- 1 ,- 2 ef-dg e f - d g ArcSinhB g d+ex e f - d g + g Hd + e xL bn 12 d g 2 F Log@d + e xD + d+ex 3d g 3 e2 ef+egx 1 g Hd + e xL g Hd + e xL ef-dg e HypergeometricPFQB:2 e f -1 + e f - d g + g Hd + e xL 1+ e f - d g + g Hd + e xL e f-d g g Hd + e xL g Hd + e xL -e f + d g F+ ef-dg g d+ex 1 1 -e f + d g >, : , >, F2 2 2 g Hd + e xL 1 ,2 e f - d g + g Hd + e xL ef-dg ArcSinhB 1 ,- 2 +g d-4d + Hd + e xL F Log@d + e xD - + e52 3 e2 Problem ð171: Valid but suboptimal antiderivative: f + g x Ha + b Log@c Hd + e xLn DL , x, 11, 0> + Log@d + e xD - e f-d g 1 F Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL f+g x e f - d g + g Hd + e xL g Hd+e xL HypergeometricPFQB:- ef-dg 2 H4 e f - 3 d gL Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL d+ex g Hd+e xL , 1, 1>, 82, 2<, f+gx : e f+e g x d+ex ef-dg 2 He f - d gL32 ArcTanhB d+ex ef-dg ef+egx 2 g 1 e f - d g + g Hd + e xL d+ex e 3 g Hd + e xL32 1 1 -e f + d g >, : , >, F+ 2 2 2 g Hd + e xL 1 ,- 2 g x Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL 3e e f - d g + g Hd + e xL ef-dg + 2.2 Logarithm Functions.nb 4bn 4b f+gx - e f - d g n ArcTanhB e f+g x e f-d g + e32 e f + g x Ha + b Log@c Hd + e xLn DL 2 2 F 2b e f - d g n ArcTanhB e f+g x e f-d g e32 e e f - d g ArcTanhB f+g x e f-d g e32 e f-d g F LogB 2 1- e f+g x e f-d g F 2b g e f - d g n PolyLogB2, - 2 - e f-d g+ e e f-d g f+g x e f-d g- e e f-d g f+g x 1 -2 g d + e x HypergeometricPFQB:- e e f+e g x g Hd+e xL e f - d g + g Hd + e xL g Hd + e xL d+ex + - e f - d g + g Hd + e xL 2bn d+ex 2 e32 1 g F F Ha + b Log@c Hd + e xLn DL e32 e f+g x + e 4b e e f - d g n ArcTanhB 1 ,- 2 ef-dg Log@d + e xD - e f - d g ArcSinhB g f + g x Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL 2 e f - d g ArcTanhB d+ex e f+g x e f-d g - 47 F 1 1 -e f + d g >, : , >, F+ 2 2 2 g Hd + e xL 1 ,2 F Log@d + e xD + F Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL e32 e Problem ð173: Valid but suboptimal antiderivative: : a + b Log@c Hd + e xLn D Hd + e xL Hf + g xL32 4b e n ArcTanhB e , x, 10, 0> f+g x e f-d g He f - d gL32 2 e ArcTanhB e f+g x e f-d g F 2b e n ArcTanhB e f+g x e f-d g + He f - d gL32 F Ha + b Log@c Hd + e xL DL He f - d gL32 4b n F 2 + 2 Ha + b Log@c Hd + e xLn DL He f - d gL e n ArcTanhB e - f+gx f+g x e f-d g - F LogB He f - d gL32 2 1- e f+g x e f-d g F 2b - e n PolyLogB2, - e f-d g+ e e f-d g f+g x e f-d g- e e f-d g f+g x He f - d gL32 F 48 2.2 Logarithm Functions.nb 2 9 3 3 3 5 5 -e f + d g - b n 2 e He f - d gL2 Hf + g xL HypergeometricPFQB: , , >, : , >, F + 9 g32 Hd + e xL32 2 2 2 2 2 g Hd + e xL e f - d g Hf + g xL ArcSinhB 9 Ha - b n Log@d + e xD + b Log@c Hd + e xL DL ef-dg g 9 n He f - d gL d+ex F Log@d + e xD e ArcTanhB e f+g x e f-d g g2 He f - d gL2 Hd + e xL2 g H- e f + d gL f+gx e Hf + g xL g Hd + e xL e Hf + g xL g Hd + e xL d+ex +e + F Ha - b n Log@d + e xD + b Log@c Hd + e xLn DL He f - d gL32 - f+gx Problem ð174: Valid but suboptimal antiderivative: : a + b Log@c Hd + e xLn D Hd + e xL Hf + g xL52 , x, 11, 0> 16 b e32 n ArcTanhB 4ben - 3 He f - d gL 2 2 e32 ArcTanhB e f+g x e f-d g 3 He f - d gL52 + f+gx e f+g x e f-d g F Ha + b Log@c Hd + e xLn DL He f - d gL52 F 2 b e32 n ArcTanhB e e f-d g + He f - d gL52 4 b e32 n ArcTanhB e f+g x e f-d g - f+g x F 2 F LogB He f - d gL52 + 2 Ha + b Log@c Hd + e xLn DL 3 He f - d gL Hf + g xL32 2 1- e f+g x e f-d g F + 2 e Ha + b Log@c Hd + e xLn DL He f - d gL 2 2 b e32 n PolyLogB2, - f+gx e f-d g+ e e f-d g f+g x e f-d g- e e f-d g f+g x He f - d gL52 - F 2.2 Logarithm Functions.nb 49 1 eJ e f-d g+g Hd+e xL e N 52 52 ef+egx bn g Hd + e xL g2 - 5 5 5 7 7 -e f + d g HypergeometricPFQB: , , >, : , >, F+1 25 2 2 2 2 2 g Hd + e xL 4 - H- e f + d gL Hd + e xL H4 e f - 4 d g + 3 g Hd + e xLL He f + e g xL2 2 e32 ArcTanhB e f+g x e f-d g f+gx 3 g32 2 - - e f - d g Hd + e xL32 ArcSinhB 2 Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL 3 H- e f + d gL Hf + g xL2 + ef+egx g Hd + e xL e f-d g g d+e x e f+e g x F 32 2 g Hd + e xL - 1 + -e f + d g 2 g Hd + e xL Log@d + e xD - g Hd+e xL F Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL He f - d gL52 3 He f - d gL3 + 2 e Ha + b H- n Log@d + e xD + Log@c Hd + e xLn DLL He f - d gL2 Hf + g xL Problem ð175: Valid but suboptimal antiderivative: : d + e x Log@a + b xD , x, 11, 0> a+bx 4 4 d+ex - b d - a e ArcTanhB b b d-a e + b32 b 2 b d - a e ArcTanhB b d+e x b d-a e b32 - 2 Hd + e xL32 2 - e d+e x F F Log@a + b xD 2 a+bx b32 4 - F 2 2 d + e x Log@a + b xD + F LogB b 2 1- b d+e x b d-a e F 2 b d - a e PolyLogB2, - - b32 1 ,2 1 1 -b d + a e >, : , >, F+ 2 2 2 e Ha + b xL b d - a e ArcSinhB e a+bx F Log@a + b xD b d-a e+ b b d-a e d+e x b d-a e- b b d-a e d+e x b32 1 ,- bd-ae + d+e x b d-a e 2 e Ha + b xL b b d - a e ArcTanhB a + b x HypergeometricPFQB:- b Hd + e xL d+e x b d-a e + 1 e b b d - a e ArcTanhB e32 Ha + b xL32 b Hd + e xL e Ha + b xL 32 F 50 2.2 Logarithm Functions.nb Problem ð181: Valid but suboptimal antiderivative: : a + b Log@c Hd He + f xLp Lq D Hg + h xL Hi + j xL2 b f p q Log@e + f xD - Hf i - e jL Hh i - g jL + a Hh i - g jL Hi + j xL a h Log@g + h xD Hh i - g jL2 - a h Log@i + j xD Hh i - g jL2 - a + b Log@c Hd He + f xLp Lq D Hh i - g jL Hi + j xL Hh i - g jL2 - b e j p q Log@e + f xD Hf i - e jL Hh i - g jL Hi + j xL Hh i - g jL2 + Hf i - e jL Hh i - g jL Hi + j xL b h Log@c Hd He + f xLp Lq D Log@i + j xD Hh i - g jL2 - F Hh i - g jL2 + Hh i - g jL2 , x, 28, 0> f Hi+j xL f i-e j F F + h He+f xL F f g-e h b h p q PolyLogB2, - b f j p q x Log@e + f xD - Hf i - e jL Hh i - g jL Hi + j xL Hf i - e jL Hh i - g jL Hi + j xL Hh i - g jL2 f g-e h b h p q PolyLogB2, - b f j p q x Log@i + j xD + f Hg+h xL b h Log@c Hd He + f xLp Lq D Log@g + h xD b h p q Log@e + f xD LogB Hi + j xL3 Ha + b Log@c Hd He + f xLp Lq DL2 g+hx f i-e j H- h i + g jL Hi + j xL Problem ð183: Valid but suboptimal antiderivative: : f Hi+j xL Hh i - g jL2 b p q Log@e + f xD + b h p q Log@e + f xD Log@g + h xD b f i p q Log@i + j xD + + h Ha + b Log@c Hd He + f xLp Lq DL LogB h Ha + b Log@c Hd He + f xLp Lq DL LogB b f p q Log@i + j xD Hf i - e jL Hh i - g jL , x, 9, 0> + f i-e j F b Log@c Hd He + f xLp Lq D Hh i - g jL Hi + j xL b h p q Log@e + f xD LogB Hh i - g jL2 f Hg+h xL f g-e h F + - b h p q Log@e + f xD Log@i + j xD + Hh i - g jL2 b h p q PolyLogB2, + + Hh i - g jL2 j He+f xL Hh i - g jL2 h He+f xL F -f g+e h - - b h p q PolyLogB2, Hh i - g jL2 j He+f xL -f i+e j F 2.2 Logarithm Functions.nb - 2 a b j Hf i - e jL2 p q x - 3 f2 h 11 b2 j Hf i - e jL2 p2 q2 x 9 f2 h b2 j2 Hh i - g jL p2 q2 x2 + 4 h2 2 a b j Hf i - e jL Hh i - g jL p q x f h2 + b2 e j2 Hh i - g jL p2 q2 x + 2 f h2 2 b2 j Hf i - e jL2 p q He + f xL Log@c Hd He + f xLp Lq D - 3 f3 h 2 b2 j Hh i - g jL2 p q He + f xL Log@c Hd He + f xLp Lq D 3 f3 h + Hi + j xL3 Ha + b Log@c Hd He + f xLp Lq DL2 3h + + 2 b2 p2 q2 Hi + j xL3 27 h - + h3 5 b2 Hf i - e jL3 p2 q2 Log@e + f xD - 9 f3 h b j2 Hh i - g jL p q He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL - - 2 f2 h2 - 2 b p q Hi + j xL3 Ha + b Log@c Hd He + f xLp Lq DL - 9h + f2 h2 + j2 Hh i - g jL He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL2 2 f2 h2 Hh i - g jL3 Ha + b Log@c Hd He + f xLp Lq DL2 LogB 2 b Hh i - g jL3 p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, h4 + 2 b2 j Hh i - g jL2 p2 q2 x j Hf i - e jL Hh i - g jL He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 j Hh i - g jL2 He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 f h3 f h2 + 2 b2 j Hf i - e jL Hh i - g jL p q He + f xL Log@c Hd He + f xLp Lq D b Hf i - e jL p q Hi + j xL2 Ha + b Log@c Hd He + f xLp Lq DL Hf i - e jL3 Ha + b Log@c Hd He + f xLp Lq DL2 + h3 f2 h2 f h3 3fh 2 a b j Hh i - g jL2 p q x 2 b2 j Hf i - e jL Hh i - g jL p2 q2 x 5 b2 Hf i - e jL p2 q2 Hi + j xL2 18 f h - f Hg+h xL f g-e h h4 h He+f xL F f g-e h - F + 2 b2 Hh i - g jL3 p2 q2 PolyLogB3, h4 + h He+f xL F f g-e h 1 108 f3 h4 - 648 a b e f2 h3 i2 j p q + 648 a b e f2 g h2 i j2 p q - 216 a b e f2 g2 h j3 p q + 648 b2 e f2 h3 i2 j p2 q2 - 648 b2 e f2 g h2 i j2 p2 q2 + 216 b2 e f2 g2 h j3 p2 q2 + 324 a2 f3 h3 i2 j x - 324 a2 f3 g h2 i j2 x + 108 a2 f3 g2 h j3 x - 648 a b f3 h3 i2 j p q x + 648 a b f3 g h2 i j2 p q x + 324 a b e f2 h3 i j2 p q x 216 a b f3 g2 h j3 p q x - 108 a b e f2 g h2 j3 p q x - 72 a b e2 f h3 j3 p q x + 648 b2 f3 h3 i2 j p2 q2 x - 648 b2 f3 g h2 i j2 p2 q2 x 486 b2 e f2 h3 i j2 p2 q2 x + 216 b2 f3 g2 h j3 p2 q2 x + 162 b2 e f2 g h2 j3 p2 q2 x + 132 b2 e2 f h3 j3 p2 q2 x + 162 a2 f3 h3 i j2 x2 54 a2 f3 g h2 j3 x2 - 162 a b f3 h3 i j2 p q x2 + 54 a b f3 g h2 j3 p q x2 + 36 a b e f2 h3 j3 p q x2 + 81 b2 f3 h3 i j2 p2 q2 x2 - 27 b2 f3 g h2 j3 p2 q2 x2 30 b2 e f2 h3 j3 p2 q2 x2 + 36 a2 f3 h3 j3 x3 - 24 a b f3 h3 j3 p q x3 + 8 b2 f3 h3 j3 p2 q2 x3 + 648 a b e f2 h3 i2 j p q Log@e + f xD 648 a b e f2 g h2 i j2 p q Log@e + f xD - 324 a b e2 f h3 i j2 p q Log@e + f xD + 216 a b e f2 g2 h j3 p q Log@e + f xD + 108 a b e2 f g h2 j3 p q Log@e + f xD + 72 a b e3 h3 j3 p q Log@e + f xD + 486 b2 e2 f h3 i j2 p2 q2 Log@e + f xD - 162 b2 e2 f g h2 j3 p2 q2 Log@e + f xD 132 b2 e3 h3 j3 p2 q2 Log@e + f xD - 324 b2 e f2 h3 i2 j p2 q2 Log@e + f xD2 + 324 b2 e f2 g h2 i j2 p2 q2 Log@e + f xD2 + 162 b2 e2 f h3 i j2 p2 q2 Log@e + f xD2 - 108 b2 e f2 g2 h j3 p2 q2 Log@e + f xD2 - 54 b2 e2 f g h2 j3 p2 q2 Log@e + f xD2 36 b2 e3 h3 j3 p2 q2 Log@e + f xD2 - 648 b2 e f2 h3 i2 j p q Log@c Hd He + f xLp Lq D + 648 b2 e f2 g h2 i j2 p q Log@c Hd He + f xLp Lq D 216 b2 e f2 g2 h j3 p q Log@c Hd He + f xLp Lq D + 648 a b f3 h3 i2 j x Log@c Hd He + f xLp Lq D - 648 a b f3 g h2 i j2 x Log@c Hd He + f xLp Lq D + 216 a b f3 g2 h j3 x Log@c Hd He + f xLp Lq D - 648 b2 f3 h3 i2 j p q x Log@c Hd He + f xLp Lq D + 648 b2 f3 g h2 i j2 p q x Log@c Hd He + f xLp Lq D + 324 b2 e f2 h3 i j2 p q x Log@c Hd He + f xLp Lq D - 216 b2 f3 g2 h j3 p q x Log@c Hd He + f xLp Lq D - 108 b2 e f2 g h2 j3 p q x Log@c Hd He + f xLp Lq D 72 b2 e2 f h3 j3 p q x Log@c Hd He + f xLp Lq D + 324 a b f3 h3 i j2 x2 Log@c Hd He + f xLp Lq D - 108 a b f3 g h2 j3 x2 Log@c Hd He + f xLp Lq D + + + + - 51 52 2.2 Logarithm Functions.nb 72 b2 e2 f h3 j3 p q x Log@c Hd He + f xLp Lq D + 324 a b f3 h3 i j2 x2 Log@c Hd He + f xLp Lq D - 108 a b f3 g h2 j3 x2 Log@c Hd He + f xLp Lq D 162 b2 f3 h3 i j2 p q x2 Log@c Hd He + f xLp Lq D + 54 b2 f3 g h2 j3 p q x2 Log@c Hd He + f xLp Lq D + 36 b2 e f2 h3 j3 p q x2 Log@c Hd He + f xLp Lq D + 72 a b f3 h3 j3 x3 Log@c Hd He + f xLp Lq D - 24 b2 f3 h3 j3 p q x3 Log@c Hd He + f xLp Lq D + 648 b2 e f2 h3 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D 648 b2 e f2 g h2 i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D - 324 b2 e2 f h3 i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 216 b2 e f2 g2 h j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 108 b2 e2 f g h2 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 72 b2 e3 h3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 324 b2 f3 h3 i2 j x Log@c Hd He + f xLp Lq D2 - 324 b2 f3 g h2 i j2 x Log@c Hd He + f xLp Lq D2 + 108 b2 f3 g2 h j3 x Log@c Hd He + f xLp Lq D2 + 162 b2 f3 h3 i j2 x2 Log@c Hd He + f xLp Lq D2 - 54 b2 f3 g h2 j3 x2 Log@c Hd He + f xLp Lq D2 + 36 b2 f3 h3 j3 x3 Log@c Hd He + f xLp Lq D2 + 108 a2 f3 h3 i3 Log@g + h xD - 324 a2 f3 g h2 i2 j Log@g + h xD + 324 a2 f3 g2 h i j2 Log@g + h xD 108 a2 f3 g3 j3 Log@g + h xD - 216 a b f3 h3 i3 p q Log@e + f xD Log@g + h xD + 648 a b f3 g h2 i2 j p q Log@e + f xD Log@g + h xD 648 a b f3 g2 h i j2 p q Log@e + f xD Log@g + h xD + 216 a b f3 g3 j3 p q Log@e + f xD Log@g + h xD + 108 b2 f3 h3 i3 p2 q2 Log@e + f xD2 Log@g + h xD 324 b2 f3 g h2 i2 j p2 q2 Log@e + f xD2 Log@g + h xD + 324 b2 f3 g2 h i j2 p2 q2 Log@e + f xD2 Log@g + h xD - 108 b2 f3 g3 j3 p2 q2 Log@e + f xD2 Log@g + h xD + 216 a b f3 h3 i3 Log@c Hd He + f xLp Lq D Log@g + h xD - 648 a b f3 g h2 i2 j Log@c Hd He + f xLp Lq D Log@g + h xD + 648 a b f3 g2 h i j2 Log@c Hd He + f xLp Lq D Log@g + h xD - 216 a b f3 g3 j3 Log@c Hd He + f xLp Lq D Log@g + h xD 216 b2 f3 h3 i3 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 648 b2 f3 g h2 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD 648 b2 f3 g2 h i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 216 b2 f3 g3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 108 b2 f3 h3 i3 Log@c Hd He + f xLp Lq D2 Log@g + h xD - 324 b2 f3 g h2 i2 j Log@c Hd He + f xLp Lq D2 Log@g + h xD + 324 b2 f3 g2 h i j2 Log@c Hd He + f xLp Lq D2 Log@g + h xD - 108 b2 f3 g3 j3 Log@c Hd He + f xLp Lq D2 Log@g + h xD + f Hg + h xL f Hg + h xL 216 a b f3 h3 i3 p q Log@e + f xD LogB F - 648 a b f3 g h2 i2 j p q Log@e + f xD LogB F+ fg-eh fg-eh 648 a b f3 g2 h i j2 p q Log@e + f xD LogB f Hg + h xL fg-eh 108 b2 f3 h3 i3 p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh 324 b2 f3 g2 h i j2 p2 q2 Log@e + f xD2 LogB F - 216 a b f3 g3 j3 p q Log@e + f xD LogB f Hg + h xL F + 108 b2 f3 g3 j3 p2 q2 Log@e + f xD2 LogB 216 b2 f3 h3 i3 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh 648 b2 f3 g2 h i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB Problem ð185: Valid but suboptimal antiderivative: g+hx , x, 8, 0> F- f Hg + h xL fg-eh f Hg + h xL fg-eh FF+ F - 648 b2 f3 g h2 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh 216 b f3 Hh i - g jL3 p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, Hi + j xL Ha + b Log@c Hd He + f xLp Lq DL2 fg-eh F + 324 b2 f3 g h2 i2 j p2 q2 Log@e + f xD2 LogB fg-eh : f Hg + h xL F - 216 b2 f3 g3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB h He + f xL -f g + e h F - 216 b2 f3 Hh i - g jL3 p2 q2 PolyLogB3, h He + f xL -f g + e h F f Hg + h xL fg-eh f Hg + h xL fg-eh F+ F+ 2.2 Logarithm Functions.nb 2 b2 j p2 q2 x 2abjpqx - + h h 2 b2 j p q He + f xL Log@c Hd He + f xLp Lq D + fh j He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 Hh i - g jL Ha + b Log@c Hd He + f xLp Lq DL2 LogB + f Hg+h xL f g-e h h2 fh 2 b Hh i - g jL p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, h2 h He+f xL F f g-e h - F + 2 b2 Hh i - g jL p2 q2 PolyLogB3, h2 h He+f xL F f g-e h 1 - 2 a b e h j p q + 2 b2 e h j p2 q2 + a2 f h j x - 2 a b f h j p q x + 2 b2 f h j p2 q2 x + 2 a b e h j p q Log@e + f xD f h2 b2 e h j p2 q2 Log@e + f xD2 - 2 b2 e h j p q Log@c Hd He + f xLp Lq D + 2 a b f h j x Log@c Hd He + f xLp Lq D - 2 b2 f h j p q x Log@c Hd He + f xLp Lq D + 2 b2 e h j p q Log@e + f xD Log@c Hd He + f xLp Lq D + b2 f h j x Log@c Hd He + f xLp Lq D2 + a2 f h i Log@g + h xD - a2 f g j Log@g + h xD 2 a b f h i p q Log@e + f xD Log@g + h xD + 2 a b f g j p q Log@e + f xD Log@g + h xD + b2 f h i p2 q2 Log@e + f xD2 Log@g + h xD b2 f g j p2 q2 Log@e + f xD2 Log@g + h xD + 2 a b f h i Log@c Hd He + f xLp Lq D Log@g + h xD - 2 a b f g j Log@c Hd He + f xLp Lq D Log@g + h xD 2 b2 f h i p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 2 b2 f g j p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + f Hg + h xL b2 f h i Log@c Hd He + f xLp Lq D2 Log@g + h xD - b2 f g j Log@c Hd He + f xLp Lq D2 Log@g + h xD + 2 a b f h i p q Log@e + f xD LogB Ffg-eh 2 a b f g j p q Log@e + f xD LogB f Hg + h xL fg-eh F - b2 f h i p2 q2 Log@e + f xD2 LogB 2 b2 f h i p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh 2 b f Hh i - g jL p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, f Hg + h xL fg-eh F + b2 f g j p2 q2 Log@e + f xD2 LogB F - 2 b2 f g j p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB h He + f xL -f g + e h F + 2 b2 f H- h i + g jL p2 q2 PolyLogB3, f Hg + h xL fg-eh f Hg + h xL fg-eh h He + f xL -f g + e h F F+ F+ Problem ð186: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL2 , x, 3, 0> g+hx Ha + b Log@c Hd He + f xLp Lq DL2 LogB h 1 h f Hg+h xL f g-e h F + 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, h h He+f xL F f g-e h 2 b2 p2 q2 PolyLogB3, h h He+f xL F f g-e h a2 Log@g + h xD - 2 a b p q Log@e + f xD Log@g + h xD + b2 p2 q2 Log@e + f xD2 Log@g + h xD + 2 a b Log@c Hd He + f xLp Lq D Log@g + h xD - 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + b2 Log@c Hd He + f xLp Lq D2 Log@g + h xD + f Hg + h xL f Hg + h xL f Hg + h xL 2 a b p q Log@e + f xD LogB F - b2 p2 q2 Log@e + f xD2 LogB F + 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB F+ fg-eh fg-eh fg-eh 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, h He + f xL -f g + e h F - 2 b2 p2 q2 PolyLogB3, h He + f xL -f g + e h F 53 54 2.2 Logarithm Functions.nb Problem ð187: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL2 Hg + h xL Hi + j xL , x, 8, 0> Ha + b Log@c Hd He + f xLp Lq DL2 LogB f Hg+h xL f g-e h hi-gj F - Ha + b Log@c Hd He + f xLp Lq DL2 LogB f i-e j hi-gj 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, - j He+f xL f i-e j hi-gj 1 f Hi+j xL F 2 b2 p2 q2 PolyLogB3, hi-gj F + 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, - h He+f xL F f g-e h hi-gj 2 b2 p2 q2 PolyLogB3, + j He+f xL f i-e j hi-gj F a2 Log@g + h xD - 2 a b p q Log@e + f xD Log@g + h xD + b2 p2 q2 Log@e + f xD2 Log@g + h xD + hi-gj 2 a b Log@c Hd He + f xLp Lq D Log@g + h xD - 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + f Hg + h xL f Hg + h xL b2 Log@c Hd He + f xLp Lq D2 Log@g + h xD + 2 a b p q Log@e + f xD LogB F - b2 p2 q2 Log@e + f xD2 LogB F+ fg-eh fg-eh 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh F - a2 Log@i + j xD + 2 a b p q Log@e + f xD Log@i + j xD - b2 p2 q2 Log@e + f xD2 Log@i + j xD - 2 a b Log@c Hd He + f xLp Lq D Log@i + j xD + 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@i + j xD f Hi + j xL f Hi + j xL b2 Log@c Hd He + f xLp Lq D2 Log@i + j xD - 2 a b p q Log@e + f xD LogB F + b2 p2 q2 Log@e + f xD2 LogB Ffi-ej fi-ej 2 b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hi + j xL 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, fi-ej j He + f xL -f i + e j Problem ð190: Valid but suboptimal antiderivative: : Hi + j xL3 Ha + b Log@c Hd He + f xLp Lq DL3 g+hx F + 2 b p q Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, , x, 34, 0> F - 2 b2 p2 q2 PolyLogB3, h He + f xL -f g + e h h He + f xL -f g + e h F + 2 b2 p2 q2 PolyLogB3, F- j He + f xL -f i + e j F h He+f xL F f g-e h - 2.2 Logarithm Functions.nb 6 a b2 j Hf i - e jL2 p2 q2 x f2 h + 3 b3 e j2 Hh i - g jL p3 q3 x 4 f h2 2 b3 j3 p3 q3 He + f xL3 27 f3 h + 6 a b2 j Hf i - e jL Hh i - g jL p2 q2 x + f h2 6 b3 j Hf i - e jL Hh i - g jL p3 q3 x - h3 - f h2 6 b3 j Hh i - g jL2 p3 q3 x h3 6 b3 j Hf i - e jL2 p2 q2 He + f xL Log@c Hd He + f xLp Lq D f3 h 6 b3 j Hh i - g jL2 p2 q2 He + f xL Log@c Hd He + f xLp Lq D + f h3 4 f2 h2 3 b j Hf i - e jL2 p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 - f3 h 3 b j Hh i - g jL2 p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 f h3 j Hf i - e jL2 He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 + f3 h j Hh i - g jL2 He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 + j2 Hh i - g jL He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL3 Hh i - g jL3 Ha + b Log@c Hd He + f xLp Lq DL3 LogB h4 f Hg+h xL f g-e h 3 b3 j2 Hf i - e jL p3 q3 x2 6 b3 j Hf i - e jL2 p3 q3 x 3 b3 j2 Hh i - g jL p3 q3 x2 - - f2 h - 8 h2 4fh 6 b3 j Hf i - e jL Hh i - g jL p2 q2 He + f xL Log@c Hd He + f xLp Lq D + f2 h2 + 2 f3 h + 2 b2 j3 p2 q2 He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL - 9 f3 h 3 b j Hf i - e jL Hh i - g jL p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 - 3 b j2 Hf i - e jL p q He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL2 - - 2 f3 h - b j3 p q He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL2 + 3 f3 h j Hf i - e jL Hh i - g jL He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 j2 Hf i - e jL He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL3 + + f3 h + F j3 He + f xL3 Ha + b Log@c Hd He + f xLp Lq DL3 + 3 f3 h + 3 b Hh i - g jL3 p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - 6 b2 Hh i - g jL3 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, h4 2 f2 h - f2 h2 f h3 2 f2 h2 3 b3 e j2 Hf i - e jL p3 q3 x f2 h2 3 b j2 Hh i - g jL p q He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL2 4 f2 h2 + - - 3 b2 j2 Hf i - e jL p2 q2 He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL 3 b2 j2 Hh i - g jL p2 q2 He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL 1 6 a b2 j Hh i - g jL2 p2 q2 x h4 h He+f xL F f g-e h + 6 b3 Hh i - g jL3 p3 q3 PolyLogB4, h4 h He+f xL F f g-e h h He+f xL F f g-e h - - 1944 a2 b e f2 h3 i2 j p q + 1944 a2 b e f2 g h2 i j2 p q - 648 a2 b e f2 g2 h j3 p q + 3888 a b2 e f2 h3 i2 j p2 q2 - 3888 a b2 e f2 g h2 i j2 p2 q2 + 216 f3 h4 1296 a b2 e f2 g2 h j3 p2 q2 - 3888 b3 e f2 h3 i2 j p3 q3 + 3888 b3 e f2 g h2 i j2 p3 q3 - 1296 b3 e f2 g2 h j3 p3 q3 + 648 a3 f3 h3 i2 j x - 648 a3 f3 g h2 i j2 x + 216 a3 f3 g2 h j3 x - 1944 a2 b f3 h3 i2 j p q x + 1944 a2 b f3 g h2 i j2 p q x + 972 a2 b e f2 h3 i j2 p q x - 648 a2 b f3 g2 h j3 p q x - 324 a2 b e f2 g h2 j3 p q x 216 a2 b e2 f h3 j3 p q x + 3888 a b2 f3 h3 i2 j p2 q2 x - 3888 a b2 f3 g h2 i j2 p2 q2 x - 2916 a b2 e f2 h3 i j2 p2 q2 x + 1296 a b2 f3 g2 h j3 p2 q2 x + 972 a b2 e f2 g h2 j3 p2 q2 x + 792 a b2 e2 f h3 j3 p2 q2 x - 3888 b3 f3 h3 i2 j p3 q3 x + 3888 b3 f3 g h2 i j2 p3 q3 x + 3402 b3 e f2 h3 i j2 p3 q3 x 1296 b3 f3 g2 h j3 p3 q3 x - 1134 b3 e f2 g h2 j3 p3 q3 x - 1020 b3 e2 f h3 j3 p3 q3 x + 324 a3 f3 h3 i j2 x2 - 108 a3 f3 g h2 j3 x2 - 486 a2 b f3 h3 i j2 p q x2 + 162 a2 b f3 g h2 j3 p q x2 + 108 a2 b e f2 h3 j3 p q x2 + 486 a b2 f3 h3 i j2 p2 q2 x2 - 162 a b2 f3 g h2 j3 p2 q2 x2 - 180 a b2 e f2 h3 j3 p2 q2 x2 243 b3 f3 h3 i j2 p3 q3 x2 + 81 b3 f3 g h2 j3 p3 q3 x2 + 114 b3 e f2 h3 j3 p3 q3 x2 + 72 a3 f3 h3 j3 x3 - 72 a2 b f3 h3 j3 p q x3 + 48 a b2 f3 h3 j3 p2 q2 x3 16 b3 f3 h3 j3 p3 q3 x3 + 1944 a2 b e f2 h3 i2 j p q Log@e + f xD - 1944 a2 b e f2 g h2 i j2 p q Log@e + f xD - 972 a2 b e2 f h3 i j2 p q Log@e + f xD + 648 a2 b e f2 g2 h j3 p q Log@e + f xD + 324 a2 b e2 f g h2 j3 p q Log@e + f xD + 216 a2 b e3 h3 j3 p q Log@e + f xD + 2916 a b2 e2 f h3 i j2 p2 q2 Log@e + f xD + + + 55 56 2.2 Logarithm Functions.nb 648 a2 b e f2 g2 h j3 p q Log@e + f xD + 324 a2 b e2 f g h2 j3 p q Log@e + f xD + 216 a2 b e3 h3 j3 p q Log@e + f xD + 2916 a b2 e2 f h3 i j2 p2 q2 Log@e + f xD 972 a b2 e2 f g h2 j3 p2 q2 Log@e + f xD - 792 a b2 e3 h3 j3 p2 q2 Log@e + f xD - 3402 b3 e2 f h3 i j2 p3 q3 Log@e + f xD + 1134 b3 e2 f g h2 j3 p3 q3 Log@e + f xD + 1020 b3 e3 h3 j3 p3 q3 Log@e + f xD - 1944 a b2 e f2 h3 i2 j p2 q2 Log@e + f xD2 + 1944 a b2 e f2 g h2 i j2 p2 q2 Log@e + f xD2 + 972 a b2 e2 f h3 i j2 p2 q2 Log@e + f xD2 - 648 a b2 e f2 g2 h j3 p2 q2 Log@e + f xD2 324 a b2 e2 f g h2 j3 p2 q2 Log@e + f xD2 - 216 a b2 e3 h3 j3 p2 q2 Log@e + f xD2 - 1458 b3 e2 f h3 i j2 p3 q3 Log@e + f xD2 + 486 b3 e2 f g h2 j3 p3 q3 Log@e + f xD2 + 396 b3 e3 h3 j3 p3 q3 Log@e + f xD2 + 648 b3 e f2 h3 i2 j p3 q3 Log@e + f xD3 - 648 b3 e f2 g h2 i j2 p3 q3 Log@e + f xD3 324 b3 e2 f h3 i j2 p3 q3 Log@e + f xD3 + 216 b3 e f2 g2 h j3 p3 q3 Log@e + f xD3 + 108 b3 e2 f g h2 j3 p3 q3 Log@e + f xD3 + 72 b3 e3 h3 j3 p3 q3 Log@e + f xD3 3888 a b2 e f2 h3 i2 j p q Log@c Hd He + f xLp Lq D + 3888 a b2 e f2 g h2 i j2 p q Log@c Hd He + f xLp Lq D - 1296 a b2 e f2 g2 h j3 p q Log@c Hd He + f xLp Lq D + 3888 b3 e f2 h3 i2 j p2 q2 Log@c Hd He + f xLp Lq D - 3888 b3 e f2 g h2 i j2 p2 q2 Log@c Hd He + f xLp Lq D + 1296 b3 e f2 g2 h j3 p2 q2 Log@c Hd He + f xLp Lq D + 1944 a2 b f3 h3 i2 j x Log@c Hd He + f xLp Lq D - 1944 a2 b f3 g h2 i j2 x Log@c Hd He + f xLp Lq D + 648 a2 b f3 g2 h j3 x Log@c Hd He + f xLp Lq D 3888 a b2 f3 h3 i2 j p q x Log@c Hd He + f xLp Lq D + 3888 a b2 f3 g h2 i j2 p q x Log@c Hd He + f xLp Lq D + 1944 a b2 e f2 h3 i j2 p q x Log@c Hd He + f xLp Lq D 1296 a b2 f3 g2 h j3 p q x Log@c Hd He + f xLp Lq D - 648 a b2 e f2 g h2 j3 p q x Log@c Hd He + f xLp Lq D - 432 a b2 e2 f h3 j3 p q x Log@c Hd He + f xLp Lq D + 3888 b3 f3 h3 i2 j p2 q2 x Log@c Hd He + f xLp Lq D - 3888 b3 f3 g h2 i j2 p2 q2 x Log@c Hd He + f xLp Lq D - 2916 b3 e f2 h3 i j2 p2 q2 x Log@c Hd He + f xLp Lq D + 1296 b3 f3 g2 h j3 p2 q2 x Log@c Hd He + f xLp Lq D + 972 b3 e f2 g h2 j3 p2 q2 x Log@c Hd He + f xLp Lq D + 792 b3 e2 f h3 j3 p2 q2 x Log@c Hd He + f xLp Lq D + 972 a2 b f3 h3 i j2 x2 Log@c Hd He + f xLp Lq D - 324 a2 b f3 g h2 j3 x2 Log@c Hd He + f xLp Lq D - 972 a b2 f3 h3 i j2 p q x2 Log@c Hd He + f xLp Lq D + 324 a b2 f3 g h2 j3 p q x2 Log@c Hd He + f xLp Lq D + 216 a b2 e f2 h3 j3 p q x2 Log@c Hd He + f xLp Lq D + 486 b3 f3 h3 i j2 p2 q2 x2 Log@c Hd He + f xLp Lq D 162 b3 f3 g h2 j3 p2 q2 x2 Log@c Hd He + f xLp Lq D - 180 b3 e f2 h3 j3 p2 q2 x2 Log@c Hd He + f xLp Lq D + 216 a2 b f3 h3 j3 x3 Log@c Hd He + f xLp Lq D 144 a b2 f3 h3 j3 p q x3 Log@c Hd He + f xLp Lq D + 48 b3 f3 h3 j3 p2 q2 x3 Log@c Hd He + f xLp Lq D + 3888 a b2 e f2 h3 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D 3888 a b2 e f2 g h2 i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D - 1944 a b2 e2 f h3 i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 1296 a b2 e f2 g2 h j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 648 a b2 e2 f g h2 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 432 a b2 e3 h3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 2916 b3 e2 f h3 i j2 p2 q2 Log@e + f xD Log@c Hd He + f xLp Lq D 972 b3 e2 f g h2 j3 p2 q2 Log@e + f xD Log@c Hd He + f xLp Lq D - 792 b3 e3 h3 j3 p2 q2 Log@e + f xD Log@c Hd He + f xLp Lq D 1944 b3 e f2 h3 i2 j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D + 1944 b3 e f2 g h2 i j2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D + 972 b3 e2 f h3 i j2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D - 648 b3 e f2 g2 h j3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D 324 b3 e2 f g h2 j3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D - 216 b3 e3 h3 j3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D 1944 b3 e f2 h3 i2 j p q Log@c Hd He + f xLp Lq D2 + 1944 b3 e f2 g h2 i j2 p q Log@c Hd He + f xLp Lq D2 - 648 b3 e f2 g2 h j3 p q Log@c Hd He + f xLp Lq D2 + 1944 a b2 f3 h3 i2 j x Log@c Hd He + f xLp Lq D2 - 1944 a b2 f3 g h2 i j2 x Log@c Hd He + f xLp Lq D2 + 648 a b2 f3 g2 h j3 x Log@c Hd He + f xLp Lq D2 1944 b3 f3 h3 i2 j p q x Log@c Hd He + f xLp Lq D2 + 1944 b3 f3 g h2 i j2 p q x Log@c Hd He + f xLp Lq D2 + 972 b3 e f2 h3 i j2 p q x Log@c Hd He + f xLp Lq D2 648 b3 f3 g2 h j3 p q x Log@c Hd He + f xLp Lq D2 - 324 b3 e f2 g h2 j3 p q x Log@c Hd He + f xLp Lq D2 - 216 b3 e2 f h3 j3 p q x Log@c Hd He + f xLp Lq D2 + 972 a b2 f3 h3 i j2 x2 Log@c Hd He + f xLp Lq D2 - 324 a b2 f3 g h2 j3 x2 Log@c Hd He + f xLp Lq D2 - 486 b3 f3 h3 i j2 p q x2 Log@c Hd He + f xLp Lq D2 + 162 b3 f3 g h2 j3 p q x2 Log@c Hd He + f xLp Lq D2 + 108 b3 e f2 h3 j3 p q x2 Log@c Hd He + f xLp Lq D2 + 216 a b2 f3 h3 j3 x3 Log@c Hd He + f xLp Lq D2 72 b3 f3 h3 j3 p q x3 Log@c Hd He + f xLp Lq D2 + 1944 b3 e f2 h3 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 1944 b3 e f2 g h2 i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 - 972 b3 e2 f h3 i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 + 648 b3 e f2 g2 h j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 + 324 b3 e2 f g h2 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 + 216 b3 e3 h3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 + 648 b3 f3 h3 i2 j x Log@c Hd He + f xLp Lq D3 - 648 b3 f3 g h2 i j2 x Log@c Hd He + f xLp Lq D3 + 216 b3 f3 g2 h j3 x Log@c Hd He + f xLp Lq D3 + 324 b3 f3 h3 i j2 x2 Log@c Hd He + f xLp Lq D3 - 108 b3 f3 g h2 j3 x2 Log@c Hd He + f xLp Lq D3 + 72 b3 f3 h3 j3 x3 Log@c Hd He + f xLp Lq D3 + 216 a3 f3 h3 i3 Log@g + h xD - 648 a3 f3 g h2 i2 j Log@g + h xD + 648 a3 f3 g2 h i j2 Log@g + h xD 216 a3 f3 g3 j3 Log@g + h xD - 648 a2 b f3 h3 i3 p q Log@e + f xD Log@g + h xD + 1944 a2 b f3 g h2 i2 j p q Log@e + f xD Log@g + h xD 1944 a2 b f3 g2 h i j2 p q Log@e + f xD Log@g + h xD + 648 a2 b f3 g3 j3 p q Log@e + f xD Log@g + h xD + 648 a b2 f3 h3 i3 p2 q2 Log@e + f xD2 Log@g + h xD 1944 a b2 f3 g h2 i2 j p2 q2 Log@e + f xD2 Log@g + h xD + 1944 a b2 f3 g2 h i j2 p2 q2 Log@e + f xD2 Log@g + h xD 648 a b2 f3 g3 j3 p2 q2 Log@e + f xD2 Log@g + h xD - 216 b3 f3 h3 i3 p3 q3 Log@e + f xD3 Log@g + h xD + 648 b3 f3 g h2 i2 j p3 q3 Log@e + f xD3 Log@g + h xD 648 b3 f3 g2 h i j2 p3 q3 Log@e + f xD3 Log@g + h xD + 216 b3 f3 g3 j3 p3 q3 Log@e + f xD3 Log@g + h xD + 648 a2 b f3 h3 i3 Log@c Hd He + f xLp Lq D Log@g + h xD 1944 a2 b f3 g h2 i2 j Log@c Hd He + f xLp Lq D Log@g + h xD + 1944 a2 b f3 g2 h i j2 Log@c Hd He + f xLp Lq D Log@g + h xD 648 a2 b f3 g3 j3 Log@c Hd He + f xLp Lq D Log@g + h xD - 1296 a b2 f3 h3 i3 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 3888 a b2 f3 g h2 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD - 3888 a b2 f3 g2 h i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + + + - 2.2 Logarithm Functions.nb 3888 a b2 f3 g h2 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD - 3888 a b2 f3 g2 h i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 1296 a b2 f3 g3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 648 b3 f3 h3 i3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD 1944 b3 f3 g h2 i2 j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 1944 b3 f3 g2 h i j2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD 648 b3 f3 g3 j3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 648 a b2 f3 h3 i3 Log@c Hd He + f xLp Lq D2 Log@g + h xD 1944 a b2 f3 g h2 i2 j Log@c Hd He + f xLp Lq D2 Log@g + h xD + 1944 a b2 f3 g2 h i j2 Log@c Hd He + f xLp Lq D2 Log@g + h xD 648 a b2 f3 g3 j3 Log@c Hd He + f xLp Lq D2 Log@g + h xD - 648 b3 f3 h3 i3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + 1944 b3 f3 g h2 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD - 1944 b3 f3 g2 h i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + 648 b3 f3 g3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + 216 b3 f3 h3 i3 Log@c Hd He + f xLp Lq D3 Log@g + h xD 648 b3 f3 g h2 i2 j Log@c Hd He + f xLp Lq D3 Log@g + h xD + 648 b3 f3 g2 h i j2 Log@c Hd He + f xLp Lq D3 Log@g + h xD f Hg + h xL 216 b3 f3 g3 j3 Log@c Hd He + f xLp Lq D3 Log@g + h xD + 648 a2 b f3 h3 i3 p q Log@e + f xD LogB Ffg-eh 1944 a2 b f3 g h2 i2 j p q Log@e + f xD LogB f Hg + h xL fg-eh 648 a2 b f3 g3 j3 p q Log@e + f xD LogB f Hg + h xL fg-eh 1944 a b2 f3 g h2 i2 j p2 q2 Log@e + f xD2 LogB F + 1944 a2 b f3 g2 h i j2 p q Log@e + f xD LogB F - 648 a b2 f3 h3 i3 p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh 648 a b2 f3 g3 j3 p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh 648 b3 f3 g h2 i2 j p3 q3 Log@e + f xD3 LogB fg-eh 216 b3 f3 g3 j3 p3 q3 Log@e + f xD3 LogB f Hg + h xL fg-eh fg-eh F- f Hg + h xL fg-eh F- f Hg + h xL fg-eh F- F + 1296 a b2 f3 h3 i3 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB 1296 a b2 f3 g3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh f Hg + h xL fg-eh f Hg + h xL fg-eh 1944 b3 f3 g h2 i2 j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB fg-eh F+ f Hg + h xL F + 648 b3 f3 g2 h i j2 p3 q3 Log@e + f xD3 LogB 3888 a b2 f3 g2 h i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh F - 1944 a b2 f3 g2 h i j2 p2 q2 Log@e + f xD2 LogB 3888 a b2 f3 g h2 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB Log@c Hd He + f xLp Lq D LogB fg-eh f Hg + h xL F + 216 b3 f3 h3 i3 p3 q3 Log@e + f xD3 LogB f Hg + h xL f Hg + h xL F+ fg-eh fg-eh 1944 b3 f3 g2 h i j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB fg-eh F- F - 648 b3 f3 h3 i3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB f Hg + h xL f Hg + h xL f Hg + h xL F- F - 1944 b3 f3 g2 h i j2 p2 q2 Log@e + f xD2 F + 648 b3 f3 g3 j3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB 648 b3 f3 h3 i3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB F+ f Hg + h xL fg-eh f Hg + h xL F+ F - 1944 b3 f3 g h2 i2 j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB f Hg + h xL fg-eh F - 648 b3 f3 g3 j3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB - fg-eh F+ f Hg + h xL fg-eh f Hg + h xL fg-eh F+ F+ 57 58 2.2 Logarithm Functions.nb 648 b f3 Hh i - g jL3 p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, h He + f xL -f g + e h h He + f xL 1296 b2 f3 Hh i - g jL3 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, 3888 b3 f3 g h2 i2 j p3 q3 PolyLogB4, h He + f xL -f g + e h F- -f g + e h F + 1296 b3 f3 h3 i3 p3 q3 PolyLogB4, F + 3888 b3 f3 g2 h i j2 p3 q3 PolyLogB4, h He + f xL -f g + e h h He + f xL -f g + e h F- F - 1296 b3 f3 g3 j3 p3 q3 PolyLogB4, h He + f xL -f g + e h F Problem ð191: Valid but suboptimal antiderivative: : Hi + j xL2 Ha + b Log@c Hd He + f xLp Lq DL3 , x, 20, 0> g+hx 6 a b2 j Hf i - e jL p2 q2 x fh 3 b3 j2 p3 q3 x2 + 8h + 6 a b2 j Hh i - g jL p2 q2 x 3 b3 e j2 p3 q3 x - - 4fh h2 3 2 2 6 b j Hf i - e jL p q He + f xL Log@c Hd He + f xLp Lq D f2 h 3 b2 j2 p2 q2 He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL - 4 f2 h j Hf i - e jL He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 f2 h Hh i - g jL2 Ha + b Log@c Hd He + f xLp Lq DL3 LogB h3 + f Hg+h xL f g-e h - 3 b j2 p q He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL2 4 f2 h j Hh i - g jL He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 F f h2 + + f h2 - f2 h + + j2 He + f xL2 Ha + b Log@c Hd He + f xLp Lq DL3 3 b Hh i - g jL2 p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - 6 b2 Hh i - g jL2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, h3 + 6 b3 j Hh i - g jL p3 q3 x h2 6 b3 j Hh i - g jL p2 q2 He + f xL Log@c Hd He + f xLp Lq D fh - 3 b j Hf i - e jL p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 3 b j Hh i - g jL p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 f h2 6 b3 j Hf i - e jL p3 q3 x h3 h He+f xL F f g-e h + 6 b3 Hh i - g jL2 p3 q3 PolyLogB4, h3 h He+f xL F f g-e h 2 f2 h h He+f xL F f g-e h - 1 - 48 a2 b e f h2 i j p q + 24 a2 b e f g h j2 p q + 96 a b2 e f h2 i j p2 q2 - 48 a b2 e f g h j2 p2 q2 - 96 b3 e f h2 i j p3 q3 + 48 b3 e f g h j2 p3 q3 + 8 f2 h3 16 a3 f2 h2 i j x - 8 a3 f2 g h j2 x - 48 a2 b f2 h2 i j p q x + 24 a2 b f2 g h j2 p q x + 12 a2 b e f h2 j2 p q x + 96 a b2 f2 h2 i j p2 q2 x 48 a b2 f2 g h j2 p2 q2 x - 36 a b2 e f h2 j2 p2 q2 x - 96 b3 f2 h2 i j p3 q3 x + 48 b3 f2 g h j2 p3 q3 x + 42 b3 e f h2 j2 p3 q3 x + 4 a3 f2 h2 j2 x2 6 a2 b f2 h2 j2 p q x2 + 6 a b2 f2 h2 j2 p2 q2 x2 - 3 b3 f2 h2 j2 p3 q3 x2 + 48 a2 b e f h2 i j p q Log@e + f xD - 24 a2 b e f g h j2 p q Log@e + f xD 12 a2 b e2 h2 j2 p q Log@e + f xD + 36 a b2 e2 h2 j2 p2 q2 Log@e + f xD - 42 b3 e2 h2 j2 p3 q3 Log@e + f xD - 48 a b2 e f h2 i j p2 q2 Log@e + f xD2 + 24 a b2 e f g h j2 p2 q2 Log@e + f xD2 + 12 a b2 e2 h2 j2 p2 q2 Log@e + f xD2 - 18 b3 e2 h2 j2 p3 q3 Log@e + f xD2 + 16 b3 e f h2 i j p3 q3 Log@e + f xD3 8 b3 e f g h j2 p3 q3 Log@e + f xD3 - 4 b3 e2 h2 j2 p3 q3 Log@e + f xD3 - 96 a b2 e f h2 i j p q Log@c Hd He + f xLp Lq D + 48 a b2 e f g h j2 p q Log@c Hd He + f xLp Lq D + 96 b3 e f h2 i j p2 q2 Log@c Hd He + f xLp Lq D - 48 b3 e f g h j2 p2 q2 Log@c Hd He + f xLp Lq D + 48 a2 b f2 h2 i j x Log@c Hd He + f xLp Lq D - 24 a2 b f2 g h j2 x Log@c Hd He + f xLp Lq D - 96 a b2 f2 h2 i j p q x Log@c Hd He + f xLp Lq D + 48 a b2 f2 g h j2 p q x Log@c Hd He + f xLp Lq D + 24 a b2 e f h2 j2 p q x Log@c Hd He + f xLp Lq D + 96 b3 f2 h2 i j p2 q2 x Log@c Hd He + f xLp Lq D 48 b3 f2 g h j2 p2 q2 x Log@c Hd He + f xLp Lq D - 36 b3 e f h2 j2 p2 q2 x Log@c Hd He + f xLp Lq D + 12 a2 b f2 h2 j2 x2 Log@c Hd He + f xLp Lq D + + + - + 2.2 Logarithm Functions.nb 59 48 b3 f2 g h j2 p2 q2 x Log@c Hd He + f xLp Lq D - 36 b3 e f h2 j2 p2 q2 x Log@c Hd He + f xLp Lq D + 12 a2 b f2 h2 j2 x2 Log@c Hd He + f xLp Lq D 12 a b2 f2 h2 j2 p q x2 Log@c Hd He + f xLp Lq D + 6 b3 f2 h2 j2 p2 q2 x2 Log@c Hd He + f xLp Lq D + 96 a b2 e f h2 i j p q Log@e + f xD Log@c Hd He + f xLp Lq D 48 a b2 e f g h j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D - 24 a b2 e2 h2 j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D + 36 b3 e2 h2 j2 p2 q2 Log@e + f xD Log@c Hd He + f xLp Lq D - 48 b3 e f h2 i j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D + 24 b3 e f g h j2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D + 12 b3 e2 h2 j2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D 48 b3 e f h2 i j p q Log@c Hd He + f xLp Lq D2 + 24 b3 e f g h j2 p q Log@c Hd He + f xLp Lq D2 + 48 a b2 f2 h2 i j x Log@c Hd He + f xLp Lq D2 24 a b2 f2 g h j2 x Log@c Hd He + f xLp Lq D2 - 48 b3 f2 h2 i j p q x Log@c Hd He + f xLp Lq D2 + 24 b3 f2 g h j2 p q x Log@c Hd He + f xLp Lq D2 + 12 b3 e f h2 j2 p q x Log@c Hd He + f xLp Lq D2 + 12 a b2 f2 h2 j2 x2 Log@c Hd He + f xLp Lq D2 - 6 b3 f2 h2 j2 p q x2 Log@c Hd He + f xLp Lq D2 + 48 b3 e f h2 i j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 - 24 b3 e f g h j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 12 b3 e2 h2 j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 + 16 b3 f2 h2 i j x Log@c Hd He + f xLp Lq D3 - 8 b3 f2 g h j2 x Log@c Hd He + f xLp Lq D3 + 4 b3 f2 h2 j2 x2 Log@c Hd He + f xLp Lq D3 + 8 a3 f2 h2 i2 Log@g + h xD - 16 a3 f2 g h i j Log@g + h xD + 8 a3 f2 g2 j2 Log@g + h xD 24 a2 b f2 h2 i2 p q Log@e + f xD Log@g + h xD + 48 a2 b f2 g h i j p q Log@e + f xD Log@g + h xD - 24 a2 b f2 g2 j2 p q Log@e + f xD Log@g + h xD + 24 a b2 f2 h2 i2 p2 q2 Log@e + f xD2 Log@g + h xD - 48 a b2 f2 g h i j p2 q2 Log@e + f xD2 Log@g + h xD + 24 a b2 f2 g2 j2 p2 q2 Log@e + f xD2 Log@g + h xD 8 b3 f2 h2 i2 p3 q3 Log@e + f xD3 Log@g + h xD + 16 b3 f2 g h i j p3 q3 Log@e + f xD3 Log@g + h xD - 8 b3 f2 g2 j2 p3 q3 Log@e + f xD3 Log@g + h xD + 24 a2 b f2 h2 i2 Log@c Hd He + f xLp Lq D Log@g + h xD - 48 a2 b f2 g h i j Log@c Hd He + f xLp Lq D Log@g + h xD + 24 a2 b f2 g2 j2 Log@c Hd He + f xLp Lq D Log@g + h xD - 48 a b2 f2 h2 i2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 96 a b2 f2 g h i j p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD - 48 a b2 f2 g2 j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 24 b3 f2 h2 i2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD - 48 b3 f2 g h i j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 24 b3 f2 g2 j2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 24 a b2 f2 h2 i2 Log@c Hd He + f xLp Lq D2 Log@g + h xD 48 a b2 f2 g h i j Log@c Hd He + f xLp Lq D2 Log@g + h xD + 24 a b2 f2 g2 j2 Log@c Hd He + f xLp Lq D2 Log@g + h xD 24 b3 f2 h2 i2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + 48 b3 f2 g h i j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD 24 b3 f2 g2 j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + 8 b3 f2 h2 i2 Log@c Hd He + f xLp Lq D3 Log@g + h xD 16 b3 f2 g h i j Log@c Hd He + f xLp Lq D3 Log@g + h xD + 8 b3 f2 g2 j2 Log@c Hd He + f xLp Lq D3 Log@g + h xD + f Hg + h xL f Hg + h xL f Hg + h xL 24 a2 b f2 h2 i2 p q Log@e + f xD LogB F - 48 a2 b f2 g h i j p q Log@e + f xD LogB F + 24 a2 b f2 g2 j2 p q Log@e + f xD LogB Ffg-eh fg-eh fg-eh 24 a b2 f2 h2 i2 p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh 24 a b2 f2 g2 j2 p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh 16 b3 f2 g h i j p3 q3 Log@e + f xD3 LogB f Hg + h xL fg-eh F + 48 a b2 f2 g h i j p2 q2 Log@e + f xD2 LogB F + 8 b3 f2 h2 i2 p3 q3 Log@e + f xD3 LogB F + 8 b3 f2 g2 j2 p3 q3 Log@e + f xD3 LogB 48 a b2 f2 h2 i2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB 48 a b2 f2 g2 j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh f Hg + h xL fg-eh 48 b3 f2 g h i j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB 24 b3 f2 h2 i2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB 24 b3 f2 g2 j2 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB f Hg + h xL fg-eh f Hg + h xL fg-eh f Hg + h xL fg-eh F- F- F+ F - 24 b3 f2 h2 i2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB fg-eh fg-eh fg-eh F - 96 a b2 f2 g h i j p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB f Hg + h xL f Hg + h xL f Hg + h xL f Hg + h xL fg-eh f Hg + h xL fg-eh F - 24 b3 f2 g2 j2 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB F - 48 b3 f2 g h i j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB - F+ f Hg + h xL fg-eh f Hg + h xL fg-eh F + 24 b f2 Hh i - g jL2 p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, + F+ F+ F+ h He + f xL -f g + e h F- 60 2.2 Logarithm Functions.nb 48 b2 f2 Hh i - g jL2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, 96 b3 f2 g h i j p3 q3 PolyLogB4, h He + f xL -f g + e h h He + f xL -f g + e h F + 48 b3 f2 g2 j2 p3 q3 PolyLogB4, F + 48 b3 f2 h2 i2 p3 q3 PolyLogB4, h He + f xL -f g + e h F h He + f xL -f g + e h F- Problem ð192: Valid but suboptimal antiderivative: : Hi + j xL Ha + b Log@c Hd He + f xLp Lq DL3 , x, 10, 0> g+hx 6 a b2 j p2 q2 x 6 b3 j p3 q3 x - h + h 6 b3 j p2 q2 He + f xL Log@c Hd He + f xLp Lq D 3 b j p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 fh Hh i - g jL Ha + b Log@c Hd He + f xLp Lq DL3 LogB h2 + j He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 f Hg+h xL f g-e h F + fh + 3 b Hh i - g jL p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - 6 b2 Hh i - g jL p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, h2 - fh h2 h He+f xL F f g-e h + 6 b3 Hh i - g jL p3 q3 PolyLogB4, h2 h He+f xL F f g-e h h He+f xL F f g-e h - 2.2 Logarithm Functions.nb 1 - 3 a2 b e h j p q + 6 a b2 e h j p2 q2 - 6 b3 e h j p3 q3 + a3 f h j x - 3 a2 b f h j p q x + 6 a b2 f h j p2 q2 x - 6 b3 f h j p3 q3 x + 3 a2 b e h j p q Log@e + f xD f h2 3 a b2 e h j p2 q2 Log@e + f xD2 + b3 e h j p3 q3 Log@e + f xD3 - 6 a b2 e h j p q Log@c Hd He + f xLp Lq D + 6 b3 e h j p2 q2 Log@c Hd He + f xLp Lq D + 3 a2 b f h j x Log@c Hd He + f xLp Lq D - 6 a b2 f h j p q x Log@c Hd He + f xLp Lq D + 6 b3 f h j p2 q2 x Log@c Hd He + f xLp Lq D + 6 a b2 e h j p q Log@e + f xD Log@c Hd He + f xLp Lq D - 3 b3 e h j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D - 3 b3 e h j p q Log@c Hd He + f xLp Lq D2 + 3 a b2 f h j x Log@c Hd He + f xLp Lq D2 - 3 b3 f h j p q x Log@c Hd He + f xLp Lq D2 + 3 b3 e h j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 + b3 f h j x Log@c Hd He + f xLp Lq D3 + a3 f h i Log@g + h xD - a3 f g j Log@g + h xD - 3 a2 b f h i p q Log@e + f xD Log@g + h xD + 3 a2 b f g j p q Log@e + f xD Log@g + h xD + 3 a b2 f h i p2 q2 Log@e + f xD2 Log@g + h xD - 3 a b2 f g j p2 q2 Log@e + f xD2 Log@g + h xD b3 f h i p3 q3 Log@e + f xD3 Log@g + h xD + b3 f g j p3 q3 Log@e + f xD3 Log@g + h xD + 3 a2 b f h i Log@c Hd He + f xLp Lq D Log@g + h xD 3 a2 b f g j Log@c Hd He + f xLp Lq D Log@g + h xD - 6 a b2 f h i p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 6 a b2 f g j p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 3 b3 f h i p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD 3 b3 f g j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 3 a b2 f h i Log@c Hd He + f xLp Lq D2 Log@g + h xD 3 a b2 f g j Log@c Hd He + f xLp Lq D2 Log@g + h xD - 3 b3 f h i p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + 3 b3 f g j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + b3 f h i Log@c Hd He + f xLp Lq D3 Log@g + h xD f Hg + h xL f Hg + h xL b3 f g j Log@c Hd He + f xLp Lq D3 Log@g + h xD + 3 a2 b f h i p q Log@e + f xD LogB F - 3 a2 b f g j p q Log@e + f xD LogB Ffg-eh fg-eh 3 a b2 f h i p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh b3 f g j p3 q3 Log@e + f xD3 LogB f Hg + h xL fg-eh F + 3 a b2 f g j p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh 3 b3 f g j p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB 3 b3 f g j p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB fg-eh f Hg + h xL fg-eh -f g + e h : Ha + b Log@c Hd He + f xLp Lq DL3 g+hx , x, 4, 0> fg-eh F- F + 3 b3 f h i p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB f Hg + h xL h He + f xL -f g + e h F+ h He + f xL -f g + e h F fg-eh f Hg + h xL fg-eh f Hg + h xL fg-eh F + 3 b f Hh i - g jL p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, F - 6 b3 f g j p3 q3 PolyLogB4, Problem ð193: Valid but suboptimal antiderivative: f Hg + h xL F - 3 b3 f h i p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB f Hg + h xL 6 b2 f Hh i - g jL p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, h He + f xL fg-eh F + b3 f h i p3 q3 Log@e + f xD3 LogB F + 6 a b2 f h i p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB 6 a b2 f g j p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB 6 b3 f h i p3 q3 PolyLogB4, f Hg + h xL F- F+ F- h He + f xL -f g + e h F- 61 62 2.2 Logarithm Functions.nb Ha + b Log@c Hd He + f xLp Lq DL3 LogB f Hg+h xL f g-e h h F + 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - 6 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, h 1 h h He+f xL F f g-e h h 6 b3 p3 q3 PolyLogB4, + h h He+f xL F f g-e h h He+f xL F f g-e h - a3 Log@g + h xD - 3 a2 b p q Log@e + f xD Log@g + h xD + 3 a b2 p2 q2 Log@e + f xD2 Log@g + h xD b3 p3 q3 Log@e + f xD3 Log@g + h xD + 3 a2 b Log@c Hd He + f xLp Lq D Log@g + h xD - 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 3 a b2 Log@c Hd He + f xLp Lq D2 Log@g + h xD 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + b3 Log@c Hd He + f xLp Lq D3 Log@g + h xD + f Hg + h xL f Hg + h xL f Hg + h xL 3 a2 b p q Log@e + f xD LogB F - 3 a b2 p2 q2 Log@e + f xD2 LogB F + b3 p3 q3 Log@e + f xD3 LogB F+ fg-eh fg-eh fg-eh 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB f Hg + h xL fg-eh f Hg + h xL fg-eh 6 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, F - 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB F + 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, h He + f xL -f g + e h F + 6 b3 p3 q3 PolyLogB4, h He + f xL -f g + e h F f Hg + h xL fg-eh h He + f xL -f g + e h F+ F- Problem ð194: Valid but suboptimal antiderivative: : Ha + b Log@c Hd He + f xLp Lq DL3 Hg + h xL Hi + j xL , x, 10, 0> Ha + b Log@c Hd He + f xLp Lq DL3 LogB f Hg+h xL hi-gj f g-e h F - Ha + b Log@c Hd He + f xLp Lq DL3 LogB 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - f i-e j hi-gj j He+f xL f i-e j hi-gj 6 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, hi-gj f Hi+j xL F j He+f xL f i-e j - F + 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, hi-gj 6 b2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, hi-gj F 6 b3 p3 q3 PolyLogB4, + hi-gj h He+f xL F f g-e h h He+f xL F f g-e h 6 b3 p3 q3 PolyLogB4, hi-gj + j He+f xL f i-e j F h He+f xL F f g-e h - 2.2 Logarithm Functions.nb 1 a3 Log@g + h xD - 3 a2 b p q Log@e + f xD Log@g + h xD + 3 a b2 p2 q2 Log@e + f xD2 Log@g + h xD - hi-gj b3 p3 q3 Log@e + f xD3 Log@g + h xD + 3 a2 b Log@c Hd He + f xLp Lq D Log@g + h xD - 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@g + h xD + 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@g + h xD + 3 a b2 Log@c Hd He + f xLp Lq D2 Log@g + h xD f Hg + h xL 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@g + h xD + b3 Log@c Hd He + f xLp Lq D3 Log@g + h xD + 3 a2 b p q Log@e + f xD LogB Ffg-eh 3 a b2 p2 q2 Log@e + f xD2 LogB f Hg + h xL fg-eh F + b3 p3 q3 Log@e + f xD3 LogB 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB f Hg + h xL fg-eh f Hg + h xL fg-eh F + 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB F + 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB f Hg + h xL fg-eh F- f Hg + h xL fg-eh a3 Log@i + j xD + 3 a2 b p q Log@e + f xD Log@i + j xD - 3 a b2 p2 q2 Log@e + f xD2 Log@i + j xD + b3 p3 q3 Log@e + f xD3 Log@i + j xD 3 a2 b Log@c Hd He + f xLp Lq D Log@i + j xD + 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D Log@i + j xD 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D Log@i + j xD - 3 a b2 Log@c Hd He + f xLp Lq D2 Log@i + j xD + f Hi + j xL 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 Log@i + j xD - b3 Log@c Hd He + f xLp Lq D3 Log@i + j xD - 3 a2 b p q Log@e + f xD LogB F+ fi-ej 3 a b2 p2 q2 Log@e + f xD2 LogB f Hi + j xL fi-ej F - b3 p3 q3 Log@e + f xD3 LogB 3 b3 p2 q2 Log@e + f xD2 Log@c Hd He + f xLp Lq D LogB 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, 6 a b2 p2 q2 PolyLogB3, h He + f xL -f g + e h f Hi + j xL fi-ej h He + f xL -f g + e h j He + f xL -f i + e j Problem ð195: Valid but suboptimal antiderivative: Ha + b Log@c Hd He + f xLp Lq DL3 Hg + h xL Hi + j xL2 fi-ej F - 6 a b2 p q Log@e + f xD Log@c Hd He + f xLp Lq D LogB F - 3 b3 p q Log@e + f xD Log@c Hd He + f xLp Lq D2 LogB , x, 14, 0> f Hi + j xL F - 3 b p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, F - 6 b3 p2 q2 Log@c Hd He + f xLp Lq D PolyLogB3, 6 b3 p2 q2 Log@c Hd He + f xLp Lq D PolyLogB3, : f Hi + j xL F + 6 b3 p3 q3 PolyLogB4, h He + f xL -f g + e h h He + f xL -f g + e h fi-ej F+ j He + f xL -f i + e j F + 6 a b2 p2 q2 PolyLogB3, F - 6 b3 p3 q3 PolyLogB4, F- j He + f xL -f i + e j j He + f xL -f i + e j F f Hi + j xL F+ fi-ej F- F+ 63 64 2.2 Logarithm Functions.nb - j He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 Hf i - e jL Hh i - g jL Hi + j xL h Ha + b Log@c Hd He + f xLp Lq DL3 LogB Hh i - g jL2 + h Ha + b Log@c Hd He + f xLp Lq DL3 LogB f Hi+j xL f i-e j F Hh i - g jL2 + Hf i - e jL Hh i - g jL 6 b2 h p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, Hh i - g jL2 6 b2 h p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, H- h i + g jL Hi + j xL f g-e h F + 3 b f p q Ha + b Log@c Hd He + f xLp Lq DL2 LogB 3 b h p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - 6 b2 f p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, - 1 f Hg+h xL Hh i - g jL2 j He+f xL f i-e j F h He+f xL F f g-e h j He+f xL f i-e j F - Hh i - g jL2 Hf i - e jL Hh i - g jL h He+f xL F f g-e h Hh i - g jL2 j He+f xL f i-e j Hf i - e jL Hh i - g jL - 6 b3 h p3 q3 PolyLogB4, Hh i - g jL2 + F h He+f xL F f g-e h f i-e j F - + 3 b h p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, 6 b3 f p3 q3 PolyLogB3, - f Hi+j xL j He+f xL f i-e j F - + 6 b3 h p3 q3 PolyLogB4, Hh i - g jL2 - j He+f xL f i-e j F - a3 - 3 a2 b q H- p Log@e + f xD + Log@d He + f xLp DL - 3 a b2 q2 H- p Log@e + f xD + Log@d He + f xLp DL2 - b3 q3 H- p Log@e + f xD + Log@d He + f xLp DL3 3 a2 b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D F - 6 a b2 q H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F - q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F - Log@d He + f xLp D 3 b3 q2 H- p Log@e + f xD + Log@d He + f xLp DL2 - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - + Log@d He + f xLp D 3 a b2 - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- q- q H- p Log@e + f xD + Log@d He + f xLp DL q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D q- F 2 - Log@d He + f xLp D 3 b3 q H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q 2 - + Log@d He+f xLp D Log@d He+f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + 2.2 Logarithm Functions.nb LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL q- Log@d He+f xLp D b3 - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL 1 Hh i - g jL2 p DL q- F 2 F 3 - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He+f xLp D Log@d He + f xLp D + h a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q 1 Log@g + h xD - Hh i - g jL2 q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D + p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F 3 Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F 3 q- h a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - Log@d He + f xLp D p DL q- Log@d He+f xLp D Log@d He+f xLp D Log@i + j xD + 3 a2 b p q + 2 a b2 p q2 H- p Log@e + f xD + Log@d He + f xLp DL + b3 p q3 H- p Log@e + f xD + Log@d He + f xLp DL2 + 2 a b2 p q - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D F + 2 b3 p q2 H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL b3 p q - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- - j He + f xL Log@e + f xD + f Hi + j xL Log@i + j xD H- f i + e jL H- h i + g jL Hi + j xL h JLog@e + f xD LogB 3 f Hi+j xL f i-e j F + PolyLogB2, Hh i - g jL2 + j He+f xL -f i+e j h JLog@e + f xD LogB FN + Hd He + f xLp L q- q H- p Log@e + f xD + Log@d He + f xLp DL q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D p DL F 2 f Hg+h xL f g-e h Log@d He + f xLp D F + PolyLogB2, Hh i - g jL2 + h He+f xL FN -f g+e h - + q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D F + 65 66 2.2 Logarithm Functions.nb 3 a b2 p2 q2 + b3 p2 q3 H- p Log@e + f xD + Log@d He + f xLp DL + b3 p2 q2 - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D - Log@e + f xD Jj He + f xL Log@e + f xD - 2 f Hi + j xL LogB h JLog@e + f xD2 LogB h JLog@e + f xD2 LogB b3 p3 q3 f Hg+h xL f g-e h f Hi+j xL f i-e j f i-e j FN + 2 f Hi + j xL PolyLogB2, H- f i + e jL H- h i + g jL Hi + j xL F + 2 Log@e + f xD PolyLogB2, Hh i - g jL2 F + 2 Log@e + f xD PolyLogB2, Hh i - g jL2 h He+f xL F -f g+e h j He+f xL -f i+e j - Log@e + f xD2 j He + f xL Log@e + f xD - 3 f Hi + j xL LogB 6 f Hi + j xL PolyLogB3, h Log@e + f xD3 LogB j He + f xL -f i + e j f Hg + h xL fg-eh 1 Hh i - g jL2 h Log@e + f xD3 LogB f Hi + j xL fi-ej Ha + b Log@c Hd He + f xLp Lq DL3 Hg + h xL Hi + j xL3 F + 3 Log@e + f xD2 PolyLogB2, F + 3 Log@e + f xD2 PolyLogB2, , x, 24, 0> - 2 PolyLogB3, F - 2 PolyLogB3, f Hi + j xL fi-ej h He + f xL -f g + e h j He + f xL -f i + e j j He+f xL h He+f xL FN -f g+e h j He+f xL -f i+e j p DL FN -f i+e j F Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D F + - + F + 6 f Hi + j xL Log@e + f xD PolyLogB2, F HH- f i + e jL H- h i + g jL Hi + j xLL + Problem ð196: Valid but suboptimal antiderivative: : f Hi+j xL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL 1 Hh i - g jL2 F - 6 Log@e + f xD PolyLogB3, F - 6 Log@e + f xD PolyLogB3, h He + f xL -f g + e h j He + f xL -f i + e j j He + f xL -f i + e j F- F + 6 PolyLogB4, F + 6 PolyLogB4, h He + f xL -f g + e h j He + f xL -f i + e j F F 2.2 Logarithm Functions.nb 3 b f j p q He + f xL Ha + b Log@c Hd He + f xLp Lq DL2 2 Hf i - e jL2 Hh i - g jL Hi + j xL h j He + f xL Ha + b Log@c Hd He + f xLp Lq DL3 Hf i - e jL Hh i - g jL2 Hi + j xL + h2 Ha + b Log@c Hd He + f xLp Lq DL3 LogB Hh i - g jL3 3 b3 f2 p3 q3 PolyLogB2, - j He+f xL f i-e j Hf i - e jL2 Hh i - g jL F + f Hi+j xL f i-e j f2 Ha + b Log@c Hd He + f xLp Lq DL3 2 Hf i - e jL2 Hh i - g jL h2 Ha + b Log@c Hd He + f xLp Lq DL3 LogB 3 b f h p q Ha + b Log@c Hd He + f xLp Lq DL2 LogB Hf i - e jL Hh i - g jL2 - f Hi+j xL F f i-e j + Hh i - g jL3 F + f Hg+h xL f g-e h 2 Hh i - g jL Hi + j xL2 F - 2 Hf i - e jL2 Hh i - g jL 6 b2 f h p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, - 6 b2 h2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, Hh i - g jL3 6 b2 h2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB3, Hh i - g jL3 Hf i - e jL Hh i - g jL2 j He+f xL f i-e j F h He+f xL F f g-e h j He+f xL f i-e j F - - 3 b2 f2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL LogB 3 b f2 p q Ha + b Log@c Hd He + f xLp Lq DL2 LogB Hh i - g jL3 Hf i - e jL2 Hh i - g jL 2 H- h i + g jL Hi + j xL2 Ha + b Log@c Hd He + f xLp Lq DL3 f Hi+j xL f i-e j 3 b h2 p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, - 3 b2 f2 p2 q2 Ha + b Log@c Hd He + f xLp Lq DL PolyLogB2, - 1 + j He+f xL f i-e j F F Hf i - e jL2 Hh i - g jL + Hh i - g jL3 j He+f xL f i-e j Hf i - e jL Hh i - g jL2 - 6 b3 h2 p3 q3 PolyLogB4, Hh i - g jL3 + F h He+f xL F f g-e h f i-e j - 3 b h2 p q Ha + b Log@c Hd He + f xLp Lq DL2 PolyLogB2, 6 b3 f h p3 q3 PolyLogB3, - f Hi+j xL - h He+f xL F f g-e h - j He+f xL f i-e j F - 3 b3 f2 p3 q3 PolyLogB3, - j He+f xL f i-e j Hf i - e jL2 Hh i - g jL 6 b3 h2 p3 q3 PolyLogB4, Hh i - g jL3 - j He+f xL f i-e j F + F - a3 - 3 a2 b q H- p Log@e + f xD + Log@d He + f xLp DL - 3 a b2 q2 H- p Log@e + f xD + Log@d He + f xLp DL2 - b3 q3 H- p Log@e + f xD + Log@d He + f xLp DL3 3 a2 b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL F - Log@d He + f xLp D 6 a b2 q H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL 3 b3 q2 H- p Log@e + f xD + Log@d He + f xLp DL2 - q H- p Log@e + f xD + Log@d He + f xLp DL + p DL 67 + Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D F - - F + 68 2.2 Logarithm Functions.nb Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL 3 a b2 - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- q H- p Log@e + f xD + Log@d He + f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL q- p DL Log@d He+f xLp D F 2 F 2 F 3 Log@d He + f xLp D - 3 b3 q H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL q- Log@d He+f xLp D b3 - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL 1 Hh i - g jL2 Hi + j xL p DL q- Log@d He+f xLp D + Log@d He + f xLp D F - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He+f xLp D q H-p Log@e+f xD+Log@d He+f xLp DL + + h a3 + 3 a2 b q H- p Log@e + f xD + Log@d He + f xLp DL + 3 a b2 q2 H- p Log@e + f xD + Log@d He + f xLp DL2 + b3 q3 H- p Log@e + f xD + Log@d He + f xLp DL3 + 3 a2 b - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F + Log@d He + f xLp D 6 a b2 q H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - Log@d He + f xLp D 3 b3 q2 H- p Log@e + f xD + Log@d He + f xLp DL2 - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - Log@d He + f xLp D 3 a b2 - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - q- q- q H- p Log@e + f xD + Log@d He + f xLp DL q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D q- F 2 Log@d He + f xLp D Log@d He+f xLp D + Log@d He+f xLp D + 3 b3 q H- p Log@e + f xD + Log@d He + f xLp DL q H- p Log@e + f xD + Log@d He + f xLp DL 2 Log@d He+f xLp D Log@d He + f xLp D + + + + 2.2 Logarithm Functions.nb LogBc ãq H-p Log@e+f xD+Log@d He+f xL 1 Log@d He + f xLp D q - Hh i - g jL3 p DL Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D F 2 + b3 - q H- p Log@e + f xD + Log@d He + f xLp DL - + LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL h2 a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q 1 Log@g + h xD - Hh i - g jL3 q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL q H- p Log@e + f xD + Log@d He + f xLp DL + LogBc ãq H-p Log@e+f xD+Log@d He+f xL Log@d He + f xLp D p DL Hd He + f xLp L q- Log@d He + f xLp D p DL Log@d He+f xLp D LogBc ãq H-p Log@e+f xD+Log@d He+f xL p DL Hd He + f xLp L q- q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D 3 Hd He + f xLp L q H-p Log@e+f xD+Log@d He+f xLp DL F 3 q- Log@d He + f xLp D 2 b3 p q2 H- p Log@e + f xD + Log@d He + f xLp DL - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D + LogBc ãq H-p Log@e+f xD+Log@d He+f xL b3 p q - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q LogBc ãq H-p Log@e+f xD+Log@d He+f xL - Hd He + f xLp L q- h H- j He + f xL Log@e + f xD + f Hi + j xL Log@i + j xDL H- f i + e jL Hh i - g jL2 Hi + j xL h2 JLog@e + f xD LogB h2 JLog@e + f xD LogB 3 p DL Log@d He+f xLp D Log@d He+f xLp D f Hg+h xL f g-e h F + PolyLogB2, Hh i - g jL3 f Hi+j xL f i-e j F + PolyLogB2, Hh i - g jL3 - h He+f xL FN -f g+e h j He+f xL -f i+e j FN F 2 Log@d He + f xLp D + Hd He + f xLp L q H- p Log@e + f xD + Log@d He + f xLp DL q H-p Log@e+f xD+Log@d He+f xLp DL Log@d He+f xLp D p DL + F q H- p Log@e + f xD + Log@d He + f xLp DL F + 3 q H-p Log@e+f xD+Log@d He+f xLp DL q- Log@i + j xD + 3 a2 b p q + 2 a b2 p q2 H- p Log@e + f xD + Log@d He + f xLp DL + b3 p q3 H- p Log@e + f xD + Log@d He + f xLp DL2 + 2 a b2 p q - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - F Hd He + f xLp L h2 a + b q H- p Log@e + f xD + Log@d He + f xLp DL + b - q H- p Log@e + f xD + Log@d He + f xLp DL - Log@d He + f xLp D q - q H-p Log@e+f xD+Log@d He+f xLp DL q- q H-p Log@e+f xD+Log@d He+f xLp DL + Log@d He+f xLp D F + j He + f xL He j - f H2 i + j xLL Log@e + f xD + f Hi + j xL H- f i + e j + f Hi + j xL Log@i + j xDL 2 Hf i - e jL2 H- h i + g jL Hi + j xL2 - + 69 + 70 2.2 Logarithm Functions.nb 3 a b2 p2 q2 + b3 p2 q3 H- p Log@e + f xD + Log@d He + f xLp DL + b3 p2 q2 - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D q - q H- p Log@e + f xD + Log@d He + f xLp DL Log@d He + f xLp D - h - Log@e + f xD j He + f xL Log@e + f xD - 2 f Hi + j xL LogB + LogBc ãq H-p Log@e+f xD+Log@d He+f xL f Hi + j xL fi-ej p DL F + 2 f Hi + j xL PolyLogB2, Hd He + f xLp L j He + f xL -f i + e j IH- f i + e jL Hh i - g jL2 Hi + j xLM - j He + f xL He j - f H2 i + j xLL Log@e + f xD2 - 2 f2 Hi + j xL2 LogB 2 f Hi + j xL Log@e + f xD j He + f xL + f Hi + j xL LogB I2 Hf i - e jL H- h i + g jL Hi + j xL M + 2 2 h2 JLog@e + f xD2 LogB f Hi+j xL f i-e j f Hi + j xL fi-ej h2 JLog@e + f xD2 LogB F + 2 Log@e + f xD PolyLogB2, Hh i - g jL3 F + 2 f2 Hi + j xL2 PolyLogB2, f Hg+h xL f g-e h j He+f xL -f i+e j j He + f xL -f i + e j F Hh i - g jL3 F - 2 PolyLogB3, b3 p3 q3 - h - Log@e + f xD2 j He + f xL Log@e + f xD - 3 f Hi + j xL LogB 6 f Hi + j xL PolyLogB3, F + 2 Log@e + f xD PolyLogB2, f Hi + j xL fi-ej 6 f2 Hi + j xL2 PolyLogB3, h2 Log@e + f xD3 LogB j He + f xL -f i + e j f Hg + h xL fg-eh 1 Hh i - g jL3 h2 Log@e + f xD3 LogB f Hi + j xL fi-ej f Hi + j xL fi-ej -f i+e j F FN f Hi + j xL fi-ej F I2 Hf i - e jL2 H- h i + g jL Hi + j xL2 M + F + 3 Log@e + f xD2 PolyLogB2, Problem ð237: Valid but suboptimal antiderivative: h He + f xL -f g + e h j He + f xL -f i + e j f Hi + j xL fi-ej j He + f xL -f i + e j h He+f xL F -f g+e h F Log@d He+f xLp D F F+ - 2 PolyLogB3, h He+f xL FN -f g+e h - + j He + f xL -f i + e j F- F + 3 f Hi + j xL Log@e + f xD + 6 f2 Hi + j xL2 H- 1 + Log@e + f xDL PolyLogB2, F + 3 Log@e + f xD2 PolyLogB2, F q H-p Log@e+f xD+Log@d He+f xLp DL F + 6 f Hi + j xL Log@e + f xD PolyLogB2, IH- f i + e jL Hh i - g jL2 Hi + j xLM - Log@e + f xD j He + f xL He j - f H2 i + j xLL Log@e + f xD2 - 6 f2 Hi + j xL2 LogB j He + f xL + f Hi + j xL LogB j He+f xL q- 1 j He + f xL -f i + e j Hh i - g jL3 F - 6 Log@e + f xD PolyLogB3, F - 6 Log@e + f xD PolyLogB3, F- h He + f xL -f g + e h j He + f xL -f i + e j F + 6 PolyLogB4, F + 6 PolyLogB4, h He + f xL -f g + e h j He + f xL -f i + e j F F 2.2 Logarithm Functions.nb : Log@c Ha + b xLn D3 , x, 11, 0> d x + e x2 LogA- bx E a Log@c Ha + b xLn D3 d - Log@c Ha + b xLn D3 LogA 6 n2 Log@c Ha + b xLn D PolyLogA3, d 1 d a+b x E a d + b Hd+e xL E b d-a e + 3 n Log@c Ha + b xLn D2 PolyLogA2, 6 n2 Log@c Ha + b xLn D PolyLogA3, d bx Log@xD Log@a + b xD - Log@xD LogB1 + a F - Log@a + b xD LogB 3 n2 Hn Log@a + b xD - Log@c Ha + b xLn DL LogB2 Log@a + b xD PolyLogB2, e Ha + b xL -b d + a e bx a bx e Ha + b xL -b d + a e a b Hd + e xL bd-ae b Hd + e xL e Ha+b xL E b d-a e - 6 n3 PolyLogA4, + d bd-ae F - 6 Log@a + b xD PolyLogB3, 1 + IntegrationTest@"2 Exponential Functions\\Logarithm Functions"D; Testing Mathematica on 807 integration problems... Problem ð179: Valid but suboptimal antiderivative: LogA E x a , x, 1, 0> a-x a-x PolyLogB2, a F x x x - LogB F LogB1 - F - PolyLogB2, F a a a Problem ð181: Valid but suboptimal antiderivative: x LogB x2 a a - x2 F , x, 2, 0> F - PolyLogB2, bx a bx a bx 3 n Log@c Ha + b xLn D2 PolyLogA2, - a+b x E a a F - PolyLogB2, b Hd + e xL bd-ae F + 2 PolyLogB3, F - 3 Log@a + b xD2 PolyLogB2, Test complete! : d F Log@a + b xD2 - Log@a + b xD2 LogB F + 2 Log@a + b xD PolyLogB2, 1 + F Log@a + b xD3 - Log@a + b xD3 LogB 6 Log@a + b xD PolyLogB3, : a+b x E a d 6 n3 PolyLogA4, d e Ha+b xL E b d-a e e Ha+b xL E b d-a e Log@xD H- n Log@a + b xD + Log@c Ha + b xLn DL3 + Hn Log@a + b xD - Log@c Ha + b xLn DL3 Log@d + e xD + 3 n H- n Log@a + b xD + Log@c Ha + b xLn DL2 n3 LogB- 71 F- e Ha + b xL -b d + a e e Ha + b xL -b d + a e F - 6 PolyLogB4, e Ha + b xL -b d + a e F - F - 2 PolyLogB3, 1 + bx a F + F + 3 Log@a + b xD2 PolyLogB2, 1 + e Ha + b xL -b d + a e F + 6 PolyLogB4, 1 + bx a F bx a F+ - 72 2.2 Logarithm Functions.nb a - x2 1 PolyLogB2, 2 a x2 1 LogB 2 a F LogB1 - F x2 a F- x2 1 PolyLogB2, 2 a F Problem ð184: Valid but suboptimal antiderivative: : LogA E a x , x, 2, 0> a x - x2 PolyLogA2, 1 a 2 a LogA E x a E x HLog@xD - Log@- a + xDL + Log@xD ILog@xD - 2 Log@- a + xD + 2 LogA1 2a x EM a + 2 PolyLogA2, x E a Problem ð185: Valid but suboptimal antiderivative: : LogA a x2 E , x, 2, 0> a x - x3 PolyLogB2, - a-x2 x2 2a 1 a 2 LogB x2 2a a LogB x2 F F Log@xD + 2 Log@xD2 + 2 Log@xD LogB1 - x a F + 2 Log@xD LogB1 + F LogA- a + x2 E - 2 Log@xD LogA- a + x2 E + 2 PolyLogB2, - x a x a F- F + 2 PolyLogB2, x a F Problem ð186: Valid but suboptimal antiderivative: : - LogAa x1-n E , x, 3, 0> a x - xn PolyLogA2, 1 - a x1-n E 1 a H1 - nL 2 a H- 1 + nL I- 1 + n2 M Log@xD2 + 2 Log@xD n LogAa x1-n E + H- 1 + nL LogB1 - x-1+n a F - Log@- a x + xn D - 2 LogAa x1-n E Log@- a x + xn D + 2 PolyLogB2, x-1+n a F 2.2 Logarithm Functions.nb Problem ð187: Unable to integrate: : LogAc - a H1-cL x-m b x Ha + b PolyLogA2, xm L E H1-cL x-m Ha+b xm L E b am á LogAc - , x, 1, 0> a H1-cL x-m b x Ha + b xm L E âx Problem ð188: Unable to integrate: : LogA x-m H-a+a c+b c xm L E b x Ha + b xm L PolyLogA2, , x, 1, 0> H1-cL x-m Ha+b xm L E b am á x-m H-a+a c+b c xm L LogA E b x Ha + b xm L âx Problem ð189: Unable to integrate: : LogAc Ia - Hd-a c dL x-m ce x Hd + e xm L PolyLogA2, á ME , x, 1, 0> H1-a cL x-m Hd+e xm L E e LogAc Ia - dm Hd-a c dL x-m ce x Hd + e xm L ME âx Problem ð190: Unable to integrate: : LogA x-m H-d+a c d+a c e xm L E e x Hd + e xm L PolyLogA2, , x, 1, 0> H1-a cL x-m Hd+e xm L E e dm 73 74 2.2 Logarithm Functions.nb á LogA x-m H-d+a c d+a c e xm L E e x Hd + e xm L âx Problem ð191: Valid but suboptimal antiderivative: : LogA 2a E a+b x Ha - b xL Ha + b xL PolyLogA2, 2ab , x, 1, 0> a-b x E a+b x 1 bx 4 ArcTanhB 4ab a F LogB a 2a + xF + LogB b a+bx F - LogB a a + xF Log@4D + LogB b bx + xF - 2 LogB1 - b a F + 2 PolyLogB2, a+bx F + 2 PolyLogB2, a+bx 2a F Problem ð192: Valid but suboptimal antiderivative: : LogA 2a E a+b x , x, 2, 0> a2 - b2 x2 PolyLogA2, 2ab a-b x E a+b x 1 bx 4 ArcTanhB 4ab a F LogB a 2a + xF + LogB b a+bx F - LogB a a + xF Log@4D + LogB b bx + xF - 2 LogB1 - b a 2a F Problem ð193: Valid but suboptimal antiderivative: : LogA a H1-cL+b H1+cL x E a+b x Ha - b xL Ha + b xL PolyLogA2, 2ab , x, 1, 0> c Ha-b xL E a+b x 1 bx 4 ArcTanhB 4ab a a-ac 2 LogB F LogB + xF LogB b+bc a a + xF - LogB b b H1 + cL Ha - b xL 2a a+bx 2 PolyLogB2, 2a bx 2 + xF - 4 ArcTanhB F - 2 PolyLogB2, a F + 2 LogB a-ac + xF LogB b+bc a - a c + b H1 + cL x 2a Problem ð194: Valid but suboptimal antiderivative: F LogB a-ac a + xF + 2 LogB b+bc H1 + cL Ha + b xL 2ac F + 2 PolyLogB2, - a-bx + xF LogB b 2a F + 4 ArcTanhB a - a c + b H1 + cL x 2ac F bx a F LogB F- a - a c + b H1 + cL x a+bx F+ 2.2 Logarithm Functions.nb : LogA a H1-cL+b H1+cL x E a+b x , x, 2, 0> a2 - b2 x2 PolyLogA2, 2ab c Ha-b xL E a+b x 1 bx 4 ArcTanhB 4ab a a-ac 2 LogB F LogB + xF LogB b+bc a a + xF - LogB b b H1 + cL Ha - b xL 2a a+bx 2 PolyLogB2, 2a bx 2 + xF - 4 ArcTanhB F - 2 PolyLogB2, a F + 2 LogB a-ac F LogB + xF LogB b+bc a - a c + b H1 + cL x 2a a-ac a + xF + 2 LogB b+bc H1 + cL Ha + b xL 2ac F + 2 PolyLogB2, - a-bx + xF LogB b 2a F + 4 ArcTanhB a - a c + b H1 + cL x 2ac bx a F F LogB F- a - a c + b H1 + cL x a+bx F+ Problem ð195: Valid but suboptimal antiderivative: : LogA1 - c Ha-b xL E a+b x Ha - b xL Ha + b xL PolyLogA2, 2ab , x, 2, 0> c Ha-b xL E a+b x 1 bx 4 ArcTanhB 4ab a a-ac 2 LogB F LogB + xF LogB b+bc a a + xF - LogB b b H1 + cL Ha - b xL 2a a+bx 2 PolyLogB2, 2a bx 2 + xF - 4 ArcTanhB F - 2 PolyLogB2, a F + 2 LogB a-ac Problem ð196: Valid but suboptimal antiderivative: : LogA1 - c Ha-b xL E a+b x a2 - b2 x2 PolyLogA2, 2ab , x, 3, 0> c Ha-b xL E a+b x + xF LogB b+bc a - a c + b H1 + cL x 2a F LogB a-ac a + xF + 2 LogB b+bc H1 + cL Ha + b xL 2ac F + 2 PolyLogB2, - a-bx + xF LogB b 2a F + 4 ArcTanhB a - a c + b H1 + cL x 2ac F bx a F LogB F- a - a c + b H1 + cL x a+bx F+ 75 76 2.2 Logarithm Functions.nb 1 bx 4 ArcTanhB 4ab a a-ac 2 LogB F LogB + xF LogB b+bc a a + xF - LogB b b H1 + cL Ha - b xL 2a a+bx 2 PolyLogB2, 2a bx 2 + xF - 4 ArcTanhB a F + 2 LogB a-ac + xF LogB b+bc a - a c + b H1 + cL x F - 2 PolyLogB2, F LogB 2a a-ac a + xF + 2 LogB b+bc H1 + cL Ha + b xL 2ac F + 2 PolyLogB2, - a-bx + xF LogB b 2a F + 4 ArcTanhB a - a c + b H1 + cL x 2ac bx a F F LogB F- a - a c + b H1 + cL x Problem ð304: Timed out after 120 seconds: : Log@a + b xn D , x, 1, 0> c+dx Log@c + d xD Log@a + b xn D b n IntB x-1+n Log@c+d xD a+b xn - d , xF d ??? Problem ð358: Valid but suboptimal antiderivative: : LogAc Ia + b x2 M E n 2 , x, 4, 0> x b x2 1 LogB- 2 a F LogAc Ia + b x2 M E + n LogAc Ia + b x2 M E PolyLogB2, n 2 n a + b x2 a Log@xD I- n LogAa + b x2 E + LogAc Ia + b x2 M EM + n 2 F - n2 PolyLogB3, 2 n I- n LogAa + b x2 E + LogAc Ia + b x2 M EM Log@xD LogAa + b x2 E - LogB1 + n 1 n2 LogB- 2 b x2 a F LogAa + b x2 E + 2 LogAa + b x2 E PolyLogB2, 1 + 2 b x2 a b x2 a F - : b n LogB- b x2 2 a F - 2 PolyLogB3, 1 + , x, 4, 0> x3 F LogAc Ia + b x2 M E n a Ia + b x2 M LogAc Ia + b x2 M E n 2 - 2 a x2 b n2 PolyLogB2, + a F PolyLogB2, - n 2 b x2 a a 1 Problem ð359: Valid but suboptimal antiderivative: LogAc Ia + b x2 M E a + b x2 a+b x2 a F b x2 a F F + a+bx F+ 2.2 Logarithm Functions.nb n I2 b x2 Log@xD - Ia + b x2 M LogAa + b x2 EM In LogAa + b x2 E - LogAc Ia + b x2 M EM n - a x2 LogAa + b x2 E 2 2 n - 2 1 b x LogB1 a a ä 2 + xF LogB b F + 4 Log@xD LogB1 + 2 LogAa + b x2 E + 4 PolyLogB2, 2 b x ä a b x ä a 1 + xF + 2 LogB- b F - 4 Log@xD LogAa + b x2 E - 2 LogBb x ä a F + 2 LogB b x ä - 2 b F + 4 PolyLogB2, 2 + 2 x2 a ä 2 + xF + LogB 2a ä a ä b LogB- - x2 I- n LogAa + b x2 E + LogAc Ia + b x2 M EM n - ä 2 a 1 + xF LogB 1 b x ä 2 2 a F + 2 PolyLogB2, b 1 2 b x ä + 2 a 2 a F Problem ð360: Valid but suboptimal antiderivative: : LogAc Ia + b x2 M E n 2 x5 b2 n2 Log@xD a2 , x, 10, 0> b2 n2 LogAa + b x2 E 2 a2 b2 n LogB- b x2 a b n LogAc Ia + b x2 M E n - F LogAc Ia + b x2 M E n 4 b2 n2 x4 + 2 b2 n2 x4 LogB- b2 LogAc Ia + b x2 M E + b x ä a a ä 2 + xF - 2 b2 n2 x4 LogB 1 + xF LogB ä ä a b b x + 2 b 2 b2 n2 x4 LogB- ä a b x ä a 2 a 2 a2 F - 2 b2 n2 x4 LogB- + xF + b2 n2 x4 LogB ä a a+b x2 a b2 n2 PolyLogB2, 4 x4 b a ä - F + 2 b2 n2 x4 LogB b 2 b2 n2 x4 LogB n 2 4 a2 4 a2 x4 b2 n2 x4 LogB- LogAc Ia + b x2 M E n 2 2 a2 1 - 2 a x2 ä + xF + b 2 + xF LogAa + b x2 E - 2 b2 n2 x4 LogB ä a b ä a ä 1 + xF LogB b F + 4 b2 n2 x4 Log@xD LogB1 - F a + xF + 2 b2 n2 x4 LogB- 2 b b x a ä F + 4 b2 n2 x4 Log@xD LogB1 + ä b x a + xF LogAa + b x2 E - 2 a b n x2 LogAc Ia + b x2 M E - b x 2 F- a F+ n 4 b2 n x4 Log@xD LogAc Ia + b x2 M E + 2 b2 n x4 LogAa + b x2 E LogAc Ia + b x2 M E - a2 LogAc Ia + b x2 M E + 4 b2 n2 x4 PolyLogB2, n 4 b2 n2 x4 PolyLogB2, ä b x a F + 2 b2 n2 x4 PolyLogB2, Problem ð361: Valid but suboptimal antiderivative: n 2 n 1 ä b x 2 2 a F + 2 b2 n2 x4 PolyLogB2, 1 ä b x + 2 2 a F F + 4 Log@xD + xF LogAa + b x2 E + a ä b x ä + 2 b + xF LogAa + b x2 E - 2 LogB a b F + 2 PolyLogB2, a ä ä b x a F+ 77 78 2.2 Logarithm Functions.nb : LogAc Ia + b x2 M E n 2 , x, 13, 0> x7 b2 n2 b3 n2 LogAa + b x2 E b3 n2 Log@xD - - + 6 a2 x2 a3 b3 n LogB- b x2 a 18 - F LogAc Ia + b x2 M E n b3 LogAc Ia + b x2 M E n 2 - ä b x a a ä 3 b3 n2 x6 LogB- 2 + xF - 9 b3 n2 x6 LogB a ä b 6 b3 n2 x6 LogB 1 + xF LogB ä LogAc Ia + b x2 M E 2 a b b3 n2 PolyLogB2, + 6 x6 F + 9 b3 n2 x6 LogB + xF + 3 b3 n2 x6 LogB 3 a3 b x ä a a ä a+b x2 a F - 9 b3 n2 x6 LogB- F a ä 2 + xF + 6 b3 n2 x6 LogB- + xF + b a ä F + 12 b3 n2 x6 Log@xD LogB1 ä a b ä 1 + xF LogB b + xF LogAa + b x2 E - 6 b3 n2 x6 LogB a ä b x + 2 b 6 b3 n2 x6 LogB- - + 3 a2 x2 b a ä + n 2 6 a3 x6 b2 n LogAc Ia + b x2 M E n 6 a x4 3 a b2 n2 x4 + 23 b3 n2 x6 + 9 b3 n2 x6 LogB- a3 n 2 a3 3 a3 1 b n LogAc Ia + b x2 M E 2 b b x a ä F + 12 b3 n2 x6 Log@xD LogB1 + ä b x - b x a + xF LogAa + b x2 E + 3 a2 b n x2 LogAc Ia + b x2 M E - 2 F- a F+ n 6 a b n x LogAc Ia + b x M E - 12 b n x Log@xD LogAc Ia + b x2 M E + 6 b3 n x6 LogAa + b x2 E LogAc Ia + b x2 M E + 3 a3 LogAc Ia + b x2 M E + 2 2 n 4 12 b3 n2 x6 PolyLogB2, - ä 3 b x a n 6 F + 12 b3 n2 x6 PolyLogB2, b x ä n 2 n a F + 6 b3 n2 x6 PolyLogB2, 1 ä b x - 2 2 a F + 6 b3 n2 x6 PolyLogB2, Problem ð366: Valid but suboptimal antiderivative: : LogAc Ia + b x2 M E n 2 , x, 11, 0> x4 8 b32 n2 ArcTanB b x a 3 a32 F 4 ä b32 n2 ArcTanB a - 4 b n LogAc Ia + b x2 M E 3 a32 4 b32 n ArcTanB n 3ax b x - b x a F 2 8 b32 n2 ArcTanB a - F LogB 2ä ä a a - b x 3 a32 F LogAc Ia + b x2 M E 3 a32 b x n LogAc Ia + b x2 M E 3 x3 - 4 ä b32 n2 PolyLogB2, - n 2 - F 3 a32 a -ä b x a +ä b x F 1 ä b x + 2 2 a F 2.2 Logarithm Functions.nb 1 3 b x ä ä 4 b32 n2 x3 LogB- a32 x3 a 4 ä b32 n2 x3 ArcTanB b x a 4 ä b32 n2 x3 ArcTanB b x a 2 b32 n2 x3 LogB ä a F - 4 b32 n2 x3 LogB F LogBF LogB a F - 4 b32 n2 x3 LogBä + xF + b32 n2 x3 LogB- a + xF - b32 n2 x3 LogB + 2 a + xF b 2 + xF + 4 b32 n2 x3 LogB a ä + xF - a ä b 2 + xF - 2 b32 n2 x3 LogB- a F+4ä ä a32 LogAc Ia + b x2 M E + 2 b32 n2 x3 PolyLogB2, n 2 a ä 1 + xF LogB b b x ä a ä b b 2 b b x b ä 1 + xF LogB a ä ä a b n x2 LogAc Ia + b x2 M E + 4 ä b32 n x3 ArcTanB ä b x 2 2 a F - 2 b32 n2 x3 PolyLogB2, 1 ä b x + 2 2 ä b x a a F b x 2 b n 1 79 2 a F+ F LogAc Ia + b x2 M E + n Problem ð372: Valid but suboptimal antiderivative: : LogAc Ia + b x2 M E n 3 , x, 5, 0> x b x2 1 LogB- 2 a F LogAc Ia + b x2 M E + n 3 3 2 n LogAc Ia + b x2 M E PolyLogB2, n 2 Log@xD I- n LogAa + b x2 E + LogAc Ia + b x2 M EM + n 3 n I- n LogAa + b x2 E + LogAc Ia + b x2 M EM n 3 2 1 2 n3 LogB- b x2 Log@xD LogAa + b x2 E - LogB1 + 2 a b x2 a 2 F - 3 n2 LogAc Ia + b x2 M E PolyLogB3, n b x2 a F - b x2 1 PolyLogB2, 2 a F LogAa + b x2 E + 2 LogAa + b x2 E PolyLogB2, 1 + 2 F LogAa + b x2 E + 3 LogAa + b x2 E PolyLogB2, 1 + 3 a 3 n2 In LogAa + b x2 E - LogAc Ia + b x2 M EM LogBn a + b x2 b x2 a F - 6 LogAa + b x2 E PolyLogB3, 1 + a F + 3 n3 PolyLogB4, F - b x2 a a + b x2 F - 2 PolyLogB3, 1 + b x2 a b x2 F + 6 PolyLogB4, 1 + a F + b x2 a F Problem ð373: Valid but suboptimal antiderivative: : LogAc Ia + b x2 M E n 3 , x, 5, 0> x3 3 b n LogB- b x2 a F LogAc Ia + b x2 M E n 2 2a Ia + b x2 M LogAc Ia + b x2 M E - 2 a x2 3 b n2 LogAc Ia + b x2 M E PolyLogB2, n n 3 + a a+b x2 a F 3 b n3 PolyLogB3, a a+b x2 a F a + b x2 a F 80 2.2 Logarithm Functions.nb 1 2 In LogAa + b x2 E - LogAc Ia + b x2 M EM 3 n + x2 6 b n Log@xD I- n LogAa + b x2 E + LogAc Ia + b x2 M EM 3 b n LogAa + b x2 E I- n LogAa + b x2 E + LogAc Ia + b x2 M EM 2 n n - a LogAa + b x2 E 2 2 1 x2 a ä b LogB- - + xF + LogB 2a ä b x LogB1 a b F + 4 Log@xD LogB1 + 2 2 - 3 b x2 LogB- b x2 a ä 2 b x ä a b x ä a a + xF + 2 LogB- b 2 LogAa + b x2 E + 4 PolyLogB2, - n3 LogAa + b x2 E a ä 2 + 6 n2 I- n LogAa + b x2 E + LogAc Ia + b x2 M EM n x2 2 - - a 3 n LogAa + b x2 E I- n LogAa + b x2 E + LogAc Ia + b x2 M EM n 2 1 + xF LogB 2 b F - 4 Log@xD LogAa + b x2 E - 2 LogB- F + 4 PolyLogB2, b x ä a b x ä - ä 2 1 ä b x 2 ä a 1 + xF LogB 2 F + Ia + b x2 M LogAa + b x2 E - 6 b x2 LogAa + b x2 E PolyLogB2, 1 + a F + 2 PolyLogB2, b x2 a a ä b 1 ä 2 2 a b x a F + 6 b x2 PolyLogB3, 1 + F b x2 a Problem ð374: Valid but suboptimal antiderivative: : LogAc Ia + b x2 M E n 3 , x, 11, 0> x5 3 b2 n2 LogB- b x2 a F LogAc Ia + b x2 M E n n 2 - 2 a2 LogAc Ia + b x2 M E n 3 4 x4 3 b n Ia + b x2 M LogAc Ia + b x2 M E 3 b2 n3 PolyLogB2, + 2 a2 3 b2 n LogB- b x2 a 4 a2 x2 a+b x2 a F n 2 3 b2 n2 LogAc Ia + b x2 M E PolyLogB2, 2 a2 a+b x2 a F b2 LogAc Ia + b x2 M E n 3 + 4 a2 n - F LogAc Ia + b x2 M E 3 b2 n3 PolyLogB3, + 2 a2 4 a2 a+b x2 a F F + 4 Log@xD + xF LogAa + b x2 E + + 2 b x ä + 2 b + xF LogAa + b x2 E - 2 LogB a b F + 2 PolyLogB2, a F + 2 LogB - 1 a x2 F 2.2 Logarithm Functions.nb 1 4 In LogAa + b x2 E - LogAc Ia + b x2 M EM 3 n 3 b n I- n LogAa + b x2 E + LogAc Ia + b x2 M EM x4 2 n 3 b2 n LogAa + b x2 E I- n LogAa + b x2 E + LogAc Ia + b x2 M EM n + a2 3 n LogAa + b x2 E I- n LogAa + b x2 E + LogAc Ia + b x2 M EM 1 + x4 1 a2 2 b x2 LogB a ä + xF + b x2 LogB ä b x a ä a b 4 b x2 PolyLogB2, n F + 2 b x2 LogB b x ä F + 4 b x2 Log@xD LogB1 + b x2 a a a F - 2 b x2 LogB1 + xF LogB ä a b 1 ä b x ä a 2 ä a a b F + 2 b x2 LogB ä a 1 + xF LogB F - 2 a LogAa + b x2 E - 4 b x2 Log@xD LogAa + b x2 E 2 b x a F + 2 b x2 PolyLogB2, 1 ä b x + 2 2 a F b x2 a F + 6 b2 x4 PolyLogB3, 1 + b x2 a + F Problem ð375: Valid but suboptimal antiderivative: : n 3 , x, 20, 0> x7 b3 n3 Log@xD a3 b3 n3 LogAa + b x2 E 2 a3 b n LogAc Ia + b x2 M E n 2 4 a x4 LogAc Ia + b x2 M E n 3 6 x6 b2 n2 LogAc Ia + b x2 M E n - 3 b3 n2 LogB- 2 a2 x2 b2 n Ia + b x2 M LogAc Ia + b x2 M E b3 n LogB+ b x2 a 2 a3 x2 3 b3 n3 PolyLogB2, 2 a3 a+b x2 a F n F LogAc Ia + b x2 M E n 2 2 a3 b3 n2 LogAc Ia + b x2 M E PolyLogB2, n + F LogAc Ia + b x2 M E 2 a3 n 2 + b x2 a a3 a+b x2 a F b3 n LogAc Ia + b x2 M E n 2 + 4 a3 b3 LogAc Ia + b x2 M E n 3 - 6 a3 b3 n3 PolyLogB3, a3 a+b x2 a ä F - - b x + 2 b F I- 2 + LogAa + b x2 EM - Ia + b x2 M LogAa + b x2 E I3 b x2 + Ia - b x2 M LogAa + b x2 EM - 6 b2 x4 I- 1 + LogAa + b x2 EM PolyLogB2, 1 + LogAc Ia + b x2 M E 2 + xF - + xF LogAa + b x2 E + 2 b x2 LogAa + b x2 E + 4 b x2 PolyLogB2, - 2 2 + xF + b x2 LogB- b b x ä a ä 2 b F + 2 b x2 PolyLogB2, n3 LogAa + b x2 E - 3 b2 x4 LogB- b x ä ä + xF + 2 b x2 LogB- + xF LogAa + b x2 E - 2 b x2 LogB a a2 x4 2 - 3 n2 I- n LogAa + b x2 E + LogAc Ia + b x2 M EM b 4 b x2 Log@xD LogB1 - 2 b x2 LogB- a ä x4 b x a b 1 ä b x2 4 b x2 + 2 b x2 LogB- 2 a2 2 n 2 - a x2 6 b2 n Log@xD I- n LogAa + b x2 E + LogAc Ia + b x2 M EM - LogAa + b x2 E + 2 n - 2 ä a b x a F+ F+ 81 82 2.2 Logarithm Functions.nb 1 12 2 In LogAa + b x2 E - LogAc Ia + b x2 M EM n 3 3 b n I- n LogAa + b x2 E + LogAc Ia + b x2 M EM x6 2 n - 6 n LogAa + b x2 E I- n LogAa + b x2 E + LogAc Ia + b x2 M EM n 6 b3 n LogAa + b x2 E I- n LogAa + b x2 E + LogAc Ia + b x2 M EM b x a 3 b3 x6 LogB a ä 2 + xF + 6 b3 x6 LogB- 2 b b x ä a 6 b3 x6 LogB- a ä b ä b x a 1 a n3 - 6 b3 x6 LogB- a3 x6 b x2 a b x ä a 1 + xF LogB b x ä 2 a ä F + 12 b3 x6 PolyLogB2, 2 Problem ð412: Valid but suboptimal antiderivative: - , x, 2, 0> x PolyLog@2, - b xm D m Log@- b xm D Log@1 + b xm D + PolyLog@2, 1 + b xm D m + xF + 3 b3 x6 LogB- ä ä a 2 + xF - 9 b3 x6 LogB ä a 1 + xF LogB ä 2 b b x + 2 a a + xF + b a b F + 12 b3 x6 Log@xD LogB1 - b ä + xF LogAa + b x2 E + 3 a3 LogAa + b x2 E + 6 b3 x6 LogAa + b x2 E + 2 b x a F + 6 b3 x6 PolyLogB2, 1 b x2 a ä b x 2 F + 6 a b2 x4 LogAa + b x2 E + 6 b3 x6 LogAa + b x2 E + 18 b3 x6 LogB- 3 Log@1 + b xm D a b F + 6 b3 x6 LogB 2 b x2 a a 2 F + 6 b3 x6 PolyLogB2, 2 b x2 a 1 ä b x + 2 2 a F - F LogAa + b x2 E + 3 a2 b x2 LogAa + b x2 E - F LogAa + b x2 E + 2 a3 LogAa + b x2 E + 2 b3 x6 LogAa + b x2 E - 6 b3 x6 I- 3 + 2 LogAa + b x2 EM PolyLogB2, 1 + : ä F + 3 a2 b x2 LogAa + b x2 E - 6 a b2 x4 LogAa + b x2 E - 12 b3 x6 Log@xD LogAa + b x2 E - 6 a b2 x4 LogAa + b x2 E - 9 b3 x6 LogAa + b x2 E - 6 b3 x6 LogB2 F - 9 b3 x6 LogB- 2 + xF LogAa + b x2 E - 6 b3 x6 LogB 12 b3 x6 PolyLogB2, - - n a3 x6 b 12 b3 x6 Log@xD LogB1 + 2 n2 In LogAa + b x2 E - LogAc Ia + b x2 M EM 1 + F + 9 b3 x6 LogB ä 2 a3 x6 ä + a2 x2 n a3 2 n + a x4 12 b3 n Log@xD I- n LogAa + b x2 E + LogAc Ia + b x2 M EM 3 a b2 x4 + 23 b3 x6 + 9 b3 x6 LogB- 6 b2 n I- n LogAa + b x2 E + LogAc Ia + b x2 M EM 2 n - F + 12 b3 x6 PolyLogB3, 1 + 3 b x2 a F 2 ä b x a F+ 2.2 Logarithm Functions.nb 83 Problem ð417: Valid but suboptimal antiderivative: : Log@c Ha + b xm Ln D2 , x, 4, 0> x LogA- b xm a E Log@c Ha + b xm Ln D2 + m 2 n Log@c Ha + b xm Ln D PolyLogA2, a+b xm a m E 2 n2 PolyLogA3, - a+b xm a m E Log@xD H- n Log@a + b x D + Log@c Ha + b x L DL + 2 n H- n Log@a + b x D + Log@c Ha + b x L DL Log@xD Log@a + b x D - LogB1 + n2 ILogA- b xm a 2 m n m m n m E Log@a + b xm D2 + 2 Log@a + b xm D PolyLogA2, 1 + m b xm a E - 2 PolyLogA3, 1 + b xm a m EM b xm a F - PolyLogA2, - b xm a m E Problem ð418: Valid but suboptimal antiderivative: : Log@c Ha + b xm Ln D3 , x, 5, 0> x LogA- b xm a E Log@c Ha + b xm Ln D3 m 1 m + 3 n Log@c Ha + b xm Ln D2 PolyLogA2, m - m n3 Log@xD Log@a + b xm D3 + n3 LogB3 n2 LogB- b xm a a+b xm a b xm a E - 6 n2 Log@c Ha + b xm Ln D PolyLogA3, m F Log@a + b xm D3 + 3 m n2 Log@xD Log@a + b xm D2 Log@c Ha + b xm Ln D - F Log@a + b xm D2 Log@c Ha + b xm Ln D - 3 m n Log@xD Log@a + b xm D Log@c Ha + b xm Ln D2 + 3 n LogB- m Log@xD Log@c Ha + b xm Ln D3 + 3 n Log@c Ha + b xm Ln D2 PolyLogB2, 1 + b xm a a+b xm a b xm a E 6 n3 PolyLogA4, + a+b xm a m E F Log@a + b xm D Log@c Ha + b xm Ln D2 + F - 6 n2 Log@c Ha + b xm Ln D PolyLogB3, 1 + b xm a F + 6 n3 PolyLogB4, 1 + b xm a Problem ð419: Valid but suboptimal antiderivative: : a + b LogBc d + 4 b e p LogB- - p e f+gx e F d Hf+g xL F 4 , x, 5, 0> Ja + b LogBc Jd + dg 12 b2 e p2 Ja + b LogBc Jd + 2 p e N FN f+g x dg 3 p e N FN f+g x + dg d+ PolyLogB2, He + d Hf + g xLL Ja + b LogBc Jd + e f+g x d F + 4 p e N FN f+g x 24 b3 e p3 Ja + b LogBc Jd + p e N FN f+g x dg d+ PolyLogB3, e f+g x d F 24 b4 e p4 PolyLogB4, dg d+ e f+g x d F F + 84 2.2 Logarithm Functions.nb 4bp - d f Log@f + g xD + He + d fL Log@e + d f + d g xD e+df+dgx + x LogB dg e x a - b p LogBd + f+gx f d f LogB F + b LogBc d + p e f+gx f 2 + xF - 2 d f LogB g 4 1 f d Hf + g xL e e+df+dgx e LogB dg e F LogB f+gx f F - 2 d f LogB e+df+dgx dg F + d f LogB 2 e+df+dgx dg e+df+dgx f+gx e dg f+gx e 6 e LogBd + f+gx F PolyLogB2, 1 + e b4 p4 - 4 e LogB- F - b LogBc d + df+dgx e f+gx df+dgx g e 2 f+gx e+df+dgx f+gx F df+dgx e+df+dgx dg e+df+dgx 2 f+gx F e e - 3 e LogB- df+dgx df+dgx F + d f LogBd + e f+gx F + e 4 F + 24 e LogBd + 3 b e p LogB- - p e f+gx e F d Hf+g xL F f+gx 1 4 F PolyLogB3, 1 + dg 6 b2 e p2 Ja + b LogBc Jd + p e N FN f+g x dg d+ PolyLogB2, + He + d Hf + g xLL Ja + b LogBc Jd + dg e f+g x d F 6 b3 e p3 PolyLogB3, + dg d+ e f+g x d F d Hf + g xL e + F- e+df+dgx dg e+df+dgx f+gx F - 2 d f PolyLogB2, F + He + d f + d g xL LogBd + F + d g x LogBd + e df+dgx , x, 4, 0> 2 p e N FN f+g x 3 F+ F+ e+df+dgx e e f+gx F - dg 3 Ja + b LogBc Jd + dg F + 2 e Log@e + d f + d g xD LogB Problem ð420: Valid but suboptimal antiderivative: : a + b LogBc d + 2 F - 2 d f Log@e + d f + d g xD LogB 2 f+gx e f+gx e+df+dgx F - 2 He + d fL PolyLogB2, - e LogBd + F + 6 e PolyLogB3, 1 + e F f+gx F + xF Log@e + d f + d g xD - F + 2 d f Log@f + g xD LogB F - 2 d f Log@f + g xD LogB F + e LogBd + 3 f+gx F PolyLogB2, 1 + 2 12 e LogBd + F LogBd + e e e+df+dgx p f+gx p p e g + xF LogB e e f f F + d g x LogB F + b LogBc d + F + b LogBc d + + xF Log@e + d f + d g xD + 2 d f LogB F - 2 e Log@e + d f + d g xD LogB 2 d f Log@e + d f + d g xD LogB 4 b3 p3 - a + b p LogBd + e g e+df+dgx 2 d f LogB- f+gx dg + xF Log@f + g xD + 2 e LogB + xF LogB g e a - b p LogBd + 6 b2 p2 a - b p LogBd + + g 2 e LogB 1 F f+gx F 3 p e N FN f+g x - e f+gx F 4 F - 24 e PolyLogB4, 1 + e df+dgx F F - 2.2 Logarithm Functions.nb 1 g 3 b p Hf + g xL LogBd + e f+gx Hf + g xL a - b p LogBd + 3 b2 p2 a - b p LogBd + F a - b p LogBd + e f+gx e f+gx e f+gx F + b LogBc d + F + b LogBc d + F + b LogBc d + p e f+gx p e f+gx e 2 Log@f + g xD - LogB e + f + g xF + LogBd + d 1 e b3 p3 LogBd + d F f+gx e 2 f+gx F F - 3 e LogBdf+dgx e 6 e PolyLogB3, 1 + df+dgx F 3 + e f+gx p F 2 + 3 b e p Ja - b p LogBd + Hf + g xL LogB e F f+g x e+df+dgx f+gx F + 2 + b LogBc Jd + d 1 e f+gx 2 p e N FN f+g x e e LogB d + 2 d F - 6 e LogBd + Log@e + d Hf + g xLD + f + g xF + F Log@e + d Hf + g xLD - 2 Log@f + g xD LogB1 + F + He + d f + d g xL LogBd + e f+gx d Hf + g xL e F + PolyLogB2, - F PolyLogB2, 1 + e df+dgx d Hf + g xL e F+ Problem ð421: Valid but suboptimal antiderivative: : a + b LogBc d + 2 b e p LogB- 1 g p e f+gx e F d Hf+g xL F 2 dg Hf + g xL a - b p LogBd + e f+gx e 2 b p a - b p LogBd + f+gx b2 p2 Hf + g xL LogB , x, 3, 0> Ja + b LogBc Jd + p e N FN f+g x F + b LogBc d + F + b LogBc d + e+df+dgx f+gx e 2 Log@f + g xD - LogB + He + d Hf + g xLL Ja + b LogBc Jd + dg f+gx p e f+gx F + 1 de LogB 2 p e e F F Problem ð426: Unable to integrate: 8Log@c Hd + e Hf + g xLp Lq D, x, 3, 0< 2 b2 e p2 PolyLogB2, dg 2 e+df+dgx f+gx F+ e Log@e + d Hf + g xLD f+gx e f+g x d F + d d e d+ + + f + g xF + + f + g xF + LogBd + d 2 p e N FN f+g x 2 Hf + g xL LogB 85 F Log@e + d Hf + g xLD - 2 Log@f + g xD LogB1 + d Hf + g xL e F + PolyLogB2, - d Hf + g xL e F F + 86 2.2 Logarithm Functions.nb -p q x + p q Hf + g xL Hypergeometric2F1B1, 1 , p 1+ 1 , p - e Hf+g xLp d g F + Hf + g xL Log@c Hd + e Hf + g xLp Lq D g p q à Log@c Hd + e Hf + g xL L D â x Problem ð444: Valid but suboptimal antiderivative: 9xm LogAd Ia + b x + c x2 M E, x, 5, 0= n 2cx 2 c n x2+m Hypergeometric2F1B1, 2 + m, 3 + m, b- b- b2 - 4 a c 1 c b2 - 4 a c m H1 + mL 2 -b + 4 a c + b 2 b -4ac b2 - 4 a c b2 -4 a c H1 + mL H2 + mL F x1+m LogAd Ia + b x + c x2 M E n + 1+m b2 - 4 a c + 2 c x H1 + mL n Hypergeometric2F1B- m, - m, 1 - m, b+ b2 - 4 a c + 2 c x b2 - 4 a c cx 2 b -4ac mx b+ b2 - 4 a c + 2 c x b- H1 + mL n Hypergeometric2F1B- m, - m, 1 - m, H- 2 n + H1 + mL Log@d Ha + x Hb + c xLLn DL Problem ð471: Valid but suboptimal antiderivative: :LogAd Ia + b x + c x2 M E , x, 24, 0> b2 - 4 a c b- cx b2 - 4 a c + b n 2 b+ -m b+ -4ac +2cx c - cx m 1+m b+ -m b2 - 4 a c + 2 c x 2cx 2 c n x2+m Hypergeometric2F1B1, 2 + m, 3 + m, - m 2 m b- F cx b- cx b2 H1 + mL H2 + mL 2-1-m xm 2 b2 -4 a c b2 - 4 a c + 2 c x b+ b+ b2 F+ b2 - 4 a c -4ac +2cx F+ 2.2 Logarithm Functions.nb 4 b2 - 4 a c n2 ArcTanhB b+2 c x b2 -4 8 n2 x - ac c b+ b2 - 4 a c b2 -4 a c +2 c x b- n2 LogB- b2 -4 2 ac c b- b2 - 4 a c n2 LogBb - F 2 b- b2 - 4 a c n2 LogBb - - 2c F LogBb + n2 LogBb + b2 - 4 a c + 2 c xF - 2c b2 - 4 a c + 2 c xF LogB b2 -4 a c +2 c x b+ b2 -4 a c F - 2 b n2 LogAa + b x + c x2 E c b2 - 4 a c + 2 c xF LogAd Ia + b x + c x2 M E n n LogBb - b2 - 4 a c b+ - c b2 - 4 a c 2 b2 - 4 a c + 2 c xF 2 b- b2 - 4 a c + 2 c xF b2 - 4 a c b+ - 4 n x LogAd Ia + b x + c x2 M E + n b2 - 4 a c + 2 c xF LogAd Ia + b x + c x2 M E n LogBb + + c x LogAd Ia + b x + c x M E - b2 - 4 a c b- n2 PolyLogB2, - 2 n x LogAa + b x + c x E + b2 -4 a c 2 2 n 2 c -4 c x + 2 - b2 + 4 a c ArcTanB b+2 c x -b2 +4 a c 2 2c F b+ b2 - 4 a c 1 2 32 a c - 16 c - Ib2 - 4 a cM 2 2c b2 - 4 a c ArcTanB b+2cx - b2 + 4 a c 4 b2 b2 - 4 a c ArcTanB b+2cx - b2 b - Ib2 - 4 a cM 2 b- +4ac b2 - 4 a c LogB F-8c F LogB b- 2 - Ib2 - 4 a cM 2 b+2cx b2 - 4 a c ArcTanB b- + xF 2c + xF + 16 a c b2 - 4 a c ArcTanB 2c b+2cx - b2 b- b2 - 4 a c +4ac F LogB 2 + xF - 4 a c 2c - F+ b2 - 4 a c x LogB b2 - 4 a c 2c F H- n Log@a + x Hb + c xLD + Log@d Ha + x Hb + c xLLn DL + - b2 + 4 a c - b2 + 4 a c LogB b2 -4 a c c x - 8 b2 2 + xF + b2 b2 -4 a c +2 c x b+ - F + b Log@a + x Hb + c xLD - Ib2 - 4 a cM n2 PolyLogB2, 2 x H- n Log@a + x Hb + c xLD + Log@d Ha + x Hb + c xLLn DL2 + n2 x LogAa + b x + c x2 E - + c b2 -4 a c +2 c x b- n - b2 + 4 a c LogB b- b2 - 4 a c + xF 2c b- b2 - 4 a c 2 + xF + 2c + 87 88 2.2 Logarithm Functions.nb 8c - Ib2 - 4 a cM x LogB 2c b+2cx ArcTanB - b2 + 4 a c 2 b2 - b2 + 4 a c LogB F LogB b2 - 4 a c + 2 c x LogB 2c - b2 + 4 a c LogB 4ac b+ 2c b2 F-b -4ac 2 b2 - 4 a c + 2 c x - b2 + 4 a c LogB b- - Ib2 - 4 a cM 16 a c - Ib2 - 4 a cM b+ b2 - 4 a c + 2 c x + xF LogB 2 b 2 b 2 b+2cx b2 - 4 a c ArcTanB - Ib2 - 4 a cM b+ 2 +4ac b2 - 4 a c + 2 c x 2c - Ib2 - 4 a cM 2 - Ib2 - 4 a cM 2 + b2 b2 -4ac - Ib2 - 4 a cM 2 - b2 + 4 a c - 4 a c F - b2 - b2 + 4 a c + 4 a c - b2 + 4 a c Problem ð474: Valid but suboptimal antiderivative: - b2 + 4 a c LogB b+ b+ F + 16 a c 2c b2 -4ac b2 - 4 a c + 2 c x 2c b2 - 4 a c b2 - 4 a c + 2 c x b+ + xF LogB 2c F+8ac 2 - b2 + 4 a c LogB b- b2 - 4 a c b2 - 4 a c 2 b2 - 4 a c 2c 2 b2 - 4 a c - 2 c x -b + b2 -4ac b2 - 4 a c + 2 c x b+ PolyLogB2, 2 b2 - 4 a c F 2 b2 - 4 a c + 2 c x 2 b+2cx F- +4ac b2 -4ac F+ F Log@a + x Hb + c xLD - + xF Log@a + x Hb + c xLD + Log@a + x Hb + c xLD2 - PolyLogB2, F + b+ - b2 LogB - Ib2 - 4 a cM F + xF LogB b2 - 4 a c ArcTanB b- F+ F- 2c - Ib2 - 4 a cM b2 - 4 a c b2 - 4 a c + 2 c x b2 - 4 a c - 2 c x -b + 2 2 - b2 F LogB 2 LogB F Log@a + x Hb + c xLD - 2 b - b2 + 4 a c - b2 + 4 a c LogB x Log@a + x Hb + c xLD + 4 b2 F Log@a + x Hb + c xLD + 2 b LogB F-8ac b- 2 2c b2 - 4 a c 2 b2 - 4 a c + 2 c x b+ b2 - 4 a c + 2 c x b+ LogB LogB 2 b2 - 4 a c - b2 2b 2c F LogB b2 - 4 a c - 2 c x -b + 2 F -2b Log@a + x Hb + c xLD + 8 c 2 - Ib2 - 4 a cM 2c 2c 4b - b2 + 4 a c b2 - 4 a c + 2 c x b+ b+2cx b2 - 4 a c ArcTanB F-2b F LogB - Ib2 - 4 a cM 2c 2 b2 F - 4 b2 b2 - 4 a c + 2 c x b+ b2 - 4 a c - 2 c x -b + 2 b+ b2 - 4 a c + 2 c x b- 2 2.2 Logarithm Functions.nb : LogA- 1 + x + x2 E 2 , x, 30, 0> x3 1 Log@xD 2 1 4 1 2 1 2 J3 + J3 + J3 + J1 + 5 N LogB1 - 5 N LogB1 5 N LogB1 5 N LogB1 - 2x 2 2 J1 - 1 5 + 2 xF 2 1+ J3 - 2 5 N LogB2 5 N LogB1 + 5 +2x 5 + 2 xF LogB 5 5 F+ 1 2 J3 - 5 F LogB1 1 5 + 2 xF - 4 F + 3 Log@xD LogB 5 + 2 xF LogA- 1 + x + x E + 2 3 PolyLogB2, 1+ 1 5 + 2 xF - 1 2 J3 - 5 N PolyLogB2, - 1- J3 - 1+ 5 +2x 2 5 F- 2 5 N LogB1 + 5 +2x 1+ 5 N LogB1 + 1 5 + 2 xF + 3 LogB 5 F+ J- 1 + 2 5 + 2 xF - LogA- 1 + x + x2 E x 5 + 2 xF LogA- 1 + x + x E 2 1 2 J3 + 5 N PolyLogB2, - 5 NF LogB1 - 5 + 2 xF - - 3 Log@xD LogA- 1 + x + x2 E + LogA- 1 + x + x2 E 2 + 2 x2 1- 5 +2x 2 5 F - 3 PolyLogB2, 1- 5 +2x 1- 5 F 89 90 2.2 Logarithm Functions.nb 1 20 - 10 LogA- 1 + x + x2 E + 2 x2 x - 10 x LogB- 1 + 1 5 - 2 xF - 10 5 x LogB- 1 + 5 - 2 xF + 20 x Log@xD + 2 5 x Log@100D LogB 2 1 LogB 5 1 + xF - 10 2 5 x LogB- 1 + 2 5 1 2 + xF + 5 2 5 x LogB 5 2 5 x LogB- 1 + 5 - 2 xF LogB 2 J1 + 5 + 2 xF - 30 x LogB 1 2 J1 + 5 N + xF LogB1 + 1 6 5 x LogB 5 2 + xF LogB1 + 1 10 5 x LogB 2 J1 + 5 +2x 20 LogA- 1 + x + x2 E + 30 x LogB- 1 + 30 x LogB1 + 5 1 10 J5 - -1 + 5 -2x -1 + 5 5 x LogB 5 -2 1 2 J1 + Log@x-n Ha + xn LD J1 + 5 xNF - 4 1 5 x LogB 5 - 2 xF LogA- 1 + x + x2 E + 10 2x 1+ 5 , x, 2, 0> n 2 a + xn a F - 1 10 2 1 5 5 PolyLogA2, n xn a E J1 + 5 1 5 - 2 xF LogB 2 J1 + 5 N + xF - 10 x LogB1 + 2 + xF LogB1 + - 1 5 + xF + 2 2 5 N + xF - 5 + 2 xF + 5 + 2 xF - 2 1 5 + 2 xF + 30 x LogB 5 -2 5 1+ 5 +2x + xF LogB 2 J5 - 2 2 5 2x 5 +2 5 xF + 60 x Log@xD LogB1 + 2 5 x LogB- 1 + F- 5 xNF + + xF LogB5 + - F + 30 x PolyLogB2, x Log@xD n Log@xD + 2 Log@1 + a x-n D - 2 LogB 1 1+ 5 F+ 5 - 2 xF LogA- 1 + x + x2 E - 60 x Log@xD LogA- 1 + x + x2 E + 5 + 2 xF LogA- 1 + x + x2 E - 10 J- 3 + 5 x LogB1 + F + 60 x PolyLogB2, - -1 + 5 x LogB 5 x LogB 5 N + xF LogB 2 PolyLog@2, - a x-n D 1 2 5 N + xF LogB1 + Problem ð477: Valid but suboptimal antiderivative: : 2 2 F + 30 x LogB 5 + 2 xF LogA- 1 + x + x2 E - 10 60 x PolyLogB2, 5 N + xF - 2 5 - 2 xF F LogB 2x + xF - 60 x LogB 5 N + xF - 30 x LogB- 1 + 2 2 5 N + xF LogB 2 J1 + 5 + 2 xF + 10 1 2 1 5 5 + 2 xF + 7 1+ 2 2 - + xF LogB - J1 + 1 + xF - 30 x LogB- 1 + 2 5 - 2 5 x Log@8D LogB 5 N + xF + 15 x LogB 2 30 x LogB 2 2 1 5 x LogB1 + 1 + xF + 60 x Log@xD LogB 2 + xF + - 2 1 10 5 - 2 1 15 x LogB 10 5 - 2 xF LogB 5 - 1+ 5 +2x 2 5 5 N x PolyLogB2, F + 10 -1 + 2 5 -2x 5 1+ F- 5 +2x 5 x PolyLogB2, 2 5 F 2.2 Logarithm Functions.nb Problem ð484: Valid but suboptimal antiderivative: :LogBe a+bx F , x, 6, 0> n 4 c+dx Ha + b xL LogAe I b a+b x n 4 M E c+d x 24 Hb c - a dL n3 LogAe I n a+bx x LogBe c+dx 4 n I- LogAe I a+b x n M E c+d x c+dx c 2 F + LogB b c-a d F b Hc+d xL + 12 Hb c - a dL n2 LogAe I bd Ia d Log@a + b xD + b d x LogA bd a+b x E c+d x - b c Log@c + d xDM d bd 6 n2 LogBe bd c + xF Log@a + b xD - 2 a d LogB d a+bx F + 2 b c LogB 2 a+bx c+dx c + xF Log@a + b xD + 2 a d LogB b F + b d x LogB 1 + a 2 a+bx d Ha+b xL F b Hc+d xL PolyLogB2, - - + xF - 2 a d LogB 2 a d Log@a + b xD LogB d Ha+b xL F b Hc+d xL 24 Hb c - a dL n4 PolyLogB4, a+b x n 2 M E c+d x 4 3 a+b x EM c+d x + xF + b c LogB b n F - n LogB + xF LogB d a a+bx c+dx d Ha + b xL -b c + a d c a+bx 4 n3 - LogBe bd c+dx 6 Hb c - a dL LogB n4 a d LogB a+bx c+dx PolyLogB2, n F + n LogB a+bx c+dx c+dx F PolyLogB2, F + b d x LogB 4 a+bx d Ha + b xL b Hc + d xL a+bx c+dx F a+bx LogB d Ha + b xL b Hc + d xL c+dx F - 24 Hb c - a dL LogB a+bx c+dx a+bx c+dx a+bx c+dx F , x, 4, 0> n 3 F LogB 3 bc-ad bc+bdx F PolyLogB3, Problem ð485: Valid but suboptimal antiderivative: :LogBe d Ha + b xL LogB 2 a+bx c+dx F + H- 6 b c + 6 a dL PolyLogB3, F + 4 b c LogB 4 F + xF Log@c + d xD - 2 b c LogB F+ F 2 + xF Log@c + d xD c+dx b d a+bx a b Hc + d xL d Ha + b xL b Hc + d xL 2 b c LogB F Log@c + d xD - 2 b c LogB + xF LogB F - 2 b c PolyLogB2, F - 2 a d PolyLogB2, F c+dx b bc-ad -b c + a d bc-ad c+dx 1 d Ha+b xL F b Hc+d xL PolyLogB3, a+bx + n LogA a+b x n 3 M E c+d x bd bd F - n LogB a+b x n M E c+d x a a d LogB + 4 Hb c - a dL n LogAe I F + 3 Hb c - a dL LogB d Ha + b xL b Hc + d xL F - 4 a d LogB d Ha + b xL b Hc + d xL F + a+bx c+dx bc-ad bc+bdx 1 F + bd F LogB 3 F + 24 b c PolyLogB4, bc-ad bc+bdx d Ha + b xL b Hc + d xL F + 12 Hb c - a dL LogB F - 24 a d PolyLogB4, a+bx c+dx F 2 d Ha + b xL b Hc + d xL F 91 92 2.2 Logarithm Functions.nb Ha + b xL LogAe I b a+b x n 3 M E c+d x 6 Hb c - a dL n2 LogAe I + 3 Hb c - a dL n LogAe I a+b x n M E c+d x bd PolyLogB2, bd 3 a n Log@a + b xD ILogAe a+b x n I M E c+d x -n b n a+bx x LogBe c+dx 1 F - n LogB bd c+dx c LogB + xF LogB d a+bx c+dx n3 LogB bd a+bx c+dx d Ha + b xL a+bx c+dx F 2 d Ha+b xL F b Hc+d xL 2 a+b x LogA EM c+d x F 3 - F + n LogB -b c + a d 2 b c LogB 1 n a+bx 3 n2 - LogBe - a+bx c+dx F LogB a+bx c+dx d Ha + b xL b Hc + d xL F + bd a+bx + 3 n x LogB c+dx a+b x n M E c+d x a a d LogB F LogBe - n LogA d b n F - n LogB + xF LogB a+bx bc-ad F + 3 Hb c - a dL LogB bc+bdx 2 + a 2 c + xF Log@a + b xD + 2 a d LogB b a+bx c+dx 2 a c + xF Log@c + d xD - 2 b c LogB b d Ha + b xL -b c + a d F + 6 Hb c - a dL LogB + xF Log@a + b xD - 2 a d d F + 2 b c LogB F - 2 b c PolyLogB2, bc-ad c+dx F - + xF - 2 a d LogB F + b d x LogB c+dx b Hc + d xL a+bx Log@c + d xD d a b c+dx c 2 d Ha+b xL F b Hc+d xL a+bx 2 a+b x EM c+d x + xF + b c LogB F + 2 a d Log@a + b xD LogB d Ha + b xL LogB b c-a d F b Hc+d xL 6 Hb c - a dL n3 PolyLogB3, 3 c n ILogAe I F Log@c + d xD - 2 b c LogB H- 6 b c + 6 a dL PolyLogB3, a+b x n 2 M E c+d x F - 2 a d PolyLogB2, a+bx c+dx F PolyLogB2, d b Hc + d xL bc-ad d Ha + b xL b Hc + d xL F+ + xF Log@c + d xD F + Problem ð486: Valid but suboptimal antiderivative: :LogBe c+dx Ha + b xL LogAe I b 1 bd a d n2 LogB F , x, 3, 0> n 2 a+bx a+b x n 2 M E c+d x a + 2 Hb c - a dL n LogAe I a+b x n M E c+d x bd 2 + xF + b c n2 LogB b c 2 + xF - 2 a d n2 LogB d a+bx 2 a d n Log@a + b xD LogBe c+dx a+bx 2 b c n LogBe c+dx LogB n a b c-a d F b Hc+d xL + 2 Hb c - a dL n2 PolyLogB2, bd + xF Log@a + b xD + 2 a d n2 LogB b n F + b d x LogBe F Log@c + d xD - 2 b c n2 LogB a+bx c+dx a Problem ð490: Valid but suboptimal antiderivative: + xF Log@a + b xD - 2 a d n2 LogB d F + 2 b c n2 LogB n 2 + xF LogB b c d Ha+b xL F b Hc+d xL b Hc + d xL bc-ad a F - 2 b c n2 PolyLogB2, + xF LogB d + xF Log@c + d xD - 2 b c n2 LogB b c d Ha + b xL -b c + a d c d Ha + b xL -b c + a d + xF Log@c + d xD d F - 2 a d n2 PolyLogB2, b Hc + d xL bc-ad F+ F 2.2 Logarithm Functions.nb : LogAe I a+b x n 3 M E c+d x , x, 11, 0> x a+bx - LogBe c+dx F LogB n 3 c+dx n a+bx 6 n2 LogBe c+dx n a+bx c+dx n a+bx 3 n LogBe c+dx c+dx bx LogBa F LogB F - n LogB F - n LogB n a+bx a b Hc + d xL c+dx c+dx F - n LogB c+dx dx 2 b c a c + xF - LogB LogB 2 1 a Hc + d xL c Ha + b xL bx LogB2 a LogB c Ha + b xL PolyLogB3, 1 + a+bx c+dx F LogB 3 a+bx c+dx a+bx 6 LogB c+dx a F LogBdx bx a bx a Hc + d xL F PolyLogB3, c c -b c + a d d Ha + b xL dx c F - LogB F - 2 LogB a+bx c+dx F - 6 LogB b Hc + d xL d Ha + b xL F - PolyLogB3, F LogB 3 c bc-ad bc+bdx a+bx c+dx F + 6 PolyLogB4, c+dx F + 3 LogB F PolyLogB3, a Hc + d xL a F + LogB d Ha + b xL c Ha + b xL bx F + LogB b Hc + d xL b Hc + d xL a+bx c bx a bx a d a + xF + LogB b c+dx c Ha + b xL a Hc + d xL F PolyLogB2, 2 F+ F - 6 PolyLogB4, d Ha + b xL b Hc + d xL F bx a a dx c dx c a a b F LogB F LogB a Hc + d xL c Ha + b xL a Hc + d xL a+bx c+dx c a Hc + d xL c Ha + b xL + xF + d F LogB1 + F PolyLogB2, 1 + F PolyLogB2, 1 + F- F + + xF + LogB + xF LogB F - 2 PolyLogB3, 1 + c Ha + b xL c d F + bx dx c + xF + LogB b c Ha + b xL c F- F - PolyLogB2, - + 2 - LogB a Hc + d xL dx a Hc + d xL 2 F + LogB- + xF - LogB c Ha + b xL F F - 2 LogB- F - PolyLogB3, 1 + a+bx F F + PolyLogB2, - F + - LogB- dx + xF + LogB1 + F + PolyLogB2, c Ha + b xL ac+bcx a b c F - 2 PolyLogB3, 1 + bcx-adx d Ha + b xL + xF + LogB F - PolyLogB2, dx F+ F + PolyLogB2, - dx d c a Hc + d xL b Hc + d xL dx d c Ha + b xL b Hc + d xL F + LogB1 + a Hc + d xL c + xF + LogB + xF PolyLogB2, 1 + a Hc + d xL d Ha + b xL c+dx a bx c+dx c Ha + b xL F - LogB1 + a+bx b F + LogB F - LogB F - LogB F PolyLogB2, n 2 a+bx F - 6 n3 PolyLogB4, a Hc + d xL d F + 2 LogB d Ha + b xL c Ha + b xL 2 a F + PolyLogB3, F + 3 n LogBe F PolyLogB3, + xF + Log@xD - LogB F LogB1 + -b c x + a d x c+dx Log@xD LogB1 + c n a+bx a F - PolyLogB2, F PolyLogB2, 2 c a Hc + d xL F + 6 n3 PolyLogB4, 2 F LogB a Hb c - a dL x - F bx 2 F - LogB- a Hc + d xL 3 F - 6 n2 LogBe d bx 1 F c+dx + xF - LogB1 + + xF PolyLogB2, 1 + b F a+bx a 3 LogB d Ha + b xL a+bx b n3 LogB b Hc + d xL a+bx F LogB- n 3 a+bx d Ha + b xL + xF + LogB- Log@xD LogB 2 LogB F + LogBe F PolyLogB3, Log@xD LogBe 3 n2 LogBe b Hc + d xL F PolyLogB2, n 2 a+bx 3 n LogBe bc-ad dx c bx a dx c F + dx c F+ F- F+ F 93 94 2.2 Logarithm Functions.nb Problem ð491: Valid but suboptimal antiderivative: : LogAe I a+b x n 2 M E c+d x , x, 7, 0> x F LogB n 2 a+bx - LogBe c+dx a+bx 2 n LogBe n c+dx n a+bx c+dx n a+bx 2 n - LogBe c+dx bx a F LogB b Hc + d xL F PolyLogB2, Log@xD LogBe n2 LogB- bc-ad a F - n LogB F + n LogB b Hc + d xL a+bx c+dx a+bx c+dx dx 2 b c a c + xF - LogB b a Hc + d xL c Ha + b xL 2 1 bx LogB2 LogB a a F c Ha + b xL PolyLogB3, 1 + bx Log@xD LogB1 + a F LogB c LogB- a dx c F - PolyLogB2, bx a a Hc + d xL F - LogB c F + LogB1 + dx c a+bx c+dx F - LogB1 + dx d c -b c + a d d Ha + b xL dx F - LogB F - 2 LogB c Ha + b xL a a Hc + d xL c b Hc + d xL c Ha + b xL d Ha + b xL F - PolyLogB3, b Hc + d xL a+bx c+dx bx a F + LogB F + LogB b Hc + d xL d Ha + b xL F c a bx a + xF - LogB d a + xF + LogB b F - PolyLogB3, 1 + dx c dx c a b F LogB F LogB a Hc + d xL c Ha + b xL c F + a+bx + xF + LogB d a bx dx c + xF + LogB F + c Ha + b xL c a b a Hc + d xL dx a F- F - PolyLogB2, - + 2 - LogB F - 2 LogB- F + LogB- bx 2 F + PolyLogB2, bx a Hc + d xL F F + PolyLogB2, - + xF + LogB F + - LogB- dx + xF + LogB1 + F + PolyLogB2, c F - 2 PolyLogB3, 1 + ac+bcx b dx c Ha + b xL F PolyLogB2, d Ha + b xL d bcx-adx n c+dx c + xF + LogB F - PolyLogB2, - + xF PolyLogB2, 1 + a+bx F + 2 n2 PolyLogB3, b bx c F + 2 n LogBe a 2 a Hc + d xL F + PolyLogB3, c Ha + b xL d F + LogB F LogB1 + a Hc + d xL + xF + Log@xD - LogB F + 2 LogB bx 2 F - LogB- a Hc + d xL + a bx LogB F - 2 n2 PolyLogB3, + xF - LogB1 + + xF PolyLogB2, 1 + 1 2 c+dx d a b F F Hb c - a dL x F LogB- n 2 a+bx d Ha + b xL + xF + LogB- Log@xD LogB 2 LogB F + LogBe c+dx F c + xF LogB a Hc + d xL c Ha + b xL + xF + d F LogB1 + F PolyLogB2, 1 + F PolyLogB2, 1 + F - 2 PolyLogB3, 1 + dx c bx a dx c F dx c F+ F+ F- Problem ð502: Valid but suboptimal antiderivative: :Ha + b xL3 LogB - e Ha + b xL 5 Hb c - a dL3 x 12 d3 c+dx + Hb c - a dL4 LogB F , x, 13, 0> 2 Hb c - a dL2 Ha + b xL2 - 12 b d2 b c-a d F b Hc+d xL 2 b d4 LogA e Ha+b xL E c+d x Hb c - a dL3 Ha + b xL LogA 2 b d3 + Ha + b xL4 LogA 4b e Ha+b xL 2 E c+d x e Ha+b xL E c+d x + + Hb c - a dL2 Ha + b xL2 LogA 4 b d2 11 Hb c - a dL4 Log@c + d xD 12 b d4 - e Ha+b xL E c+d x - Hb c - a dL Ha + b xL3 LogA Hb c - a dL4 PolyLogB2, 2 b d4 6bd d Ha+b xL F b Hc+d xL e Ha+b xL E c+d x - 2.2 Logarithm Functions.nb 1 - 5 b4 c3 d x + 17 a b3 c2 d2 x - 19 a2 b2 c d3 x + 7 a3 b d4 x + b4 c2 d2 x2 - 2 a b3 c d3 x2 + a2 b2 d4 x2 + 3 a4 d4 LogB 12 b d4 a 2 + xF b 3 b4 c4 LogB c 2 + xF + 12 a b3 c3 d LogB d c 2 + xF - 18 a2 b2 c2 d2 LogB d c 2 + xF + 12 a3 b c d3 LogB d c 2 + xF - 6 a b3 c3 d Log@a + b xD + d 21 a2 b2 c2 d2 Log@a + b xD - 26 a3 b c d3 Log@a + b xD + 11 a4 d4 Log@a + b xD - 6 a4 d4 LogB a + xF Log@a + b xD + 6 a4 d4 LogB c + xF Log@a + b xD d c d Ha + b xL e Ha + b xL e Ha + b xL 6 a4 d4 LogB + xF LogB F - 6 b4 c3 d x LogB F + 24 a b3 c2 d2 x LogB F - 36 a2 b2 c d3 x LogB F+ d -b c + a d c+dx c+dx c+dx e Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL 18 a3 b d4 x LogB F + 3 b4 c2 d2 x2 LogB F - 12 a b3 c d3 x2 LogB F + 9 a2 b2 d4 x2 LogB Fc+dx c+dx c+dx c+dx e Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL 2 2 b4 c d3 x3 LogB F + 2 a b3 d4 x3 LogB F + 6 a4 d4 Log@a + b xD LogB F + 12 a3 b d4 x LogB F + c+dx c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 18 a2 b2 d4 x2 LogB F + 12 a b3 d4 x3 LogB F + 3 b4 d4 x4 LogB F + 11 b4 c4 Log@c + d xD - 38 a b3 c3 d Log@c + d xD + c+dx c+dx c+dx a a 45 a2 b2 c2 d2 Log@c + d xD - 18 a3 b c d3 Log@c + d xD - 6 b4 c4 LogB + xF Log@c + d xD + 24 a b3 c3 d LogB + xF Log@c + d xD b b a a c c 36 a2 b2 c2 d2 LogB + xF Log@c + d xD + 24 a3 b c d3 LogB + xF Log@c + d xD + 6 b4 c4 LogB + xF Log@c + d xD - 24 a b3 c3 d LogB + xF Log@c + d xD + b b d d c c e Ha + b xL 36 a2 b2 c2 d2 LogB + xF Log@c + d xD - 24 a3 b c d3 LogB + xF Log@c + d xD + 6 b4 c4 LogB F Log@c + d xD d d c+dx e Ha + b xL e Ha + b xL e Ha + b xL 24 a b3 c3 d LogB F Log@c + d xD + 36 a2 b2 c2 d2 LogB F Log@c + d xD - 24 a3 b c d3 LogB F Log@c + d xD + c+dx c+dx c+dx a b Hc + d xL a b Hc + d xL a b Hc + d xL 6 b4 c4 LogB + xF LogB F - 24 a b3 c3 d LogB + xF LogB F + 36 a2 b2 c2 d2 LogB + xF LogB Fb bc-ad b bc-ad b bc-ad a b Hc + d xL d Ha + b xL b Hc + d xL 24 a3 b c d3 LogB + xF LogB F + 6 b c Ib3 c3 - 4 a b2 c2 d + 6 a2 b c d2 - 4 a3 d3 M PolyLogB2, F - 6 a4 d4 PolyLogB2, F b bc-ad -b c + a d bc-ad b e Ha + b xL Problem ð503: Valid but suboptimal antiderivative: :Ha + b xL2 LogB Hb c - a dL2 x 3 d2 + e Ha + b xL c+dx F , x, 10, 0> 2 2 Hb c - a dL2 Ha + b xL LogA 2 Hb c - a dL3 LogB b c-a d F b Hc+d xL 3 b d3 3 b d2 LogA e Ha+b xL E c+d x e Ha+b xL E c+d x + - Hb c - a dL Ha + b xL2 LogA 3bd Ha + b xL3 LogA 3b e Ha+b xL 2 E c+d x - e Ha+b xL E c+d x + Hb c - a dL3 Log@c + d xD b d3 + 2 Hb c - a dL3 PolyLogB2, 3 b d3 d Ha+b xL F b Hc+d xL 95 96 2.2 Logarithm Functions.nb 1 b3 c2 d x - 2 a b2 c d2 x + a2 b d3 x + a3 d3 LogB 3 b d3 a 2 + xF + b3 c3 LogB b c 2 + xF - 3 a b2 c2 d LogB d c 2 + xF + 3 a2 b c d2 LogB d 2 a b2 c2 d Log@a + b xD - 5 a2 b c d2 Log@a + b xD + 3 a3 d3 Log@a + b xD - 2 a3 d3 LogB a + xF Log@a + b xD + 2 a3 d3 LogB c 2 + xF + d c + xF Log@a + b xD d c d Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL 2 a3 d3 LogB + xF LogB F + 2 b3 c2 d x LogB F - 6 a b2 c d2 x LogB F + 4 a2 b d3 x LogB Fd -b c + a d c+dx c+dx c+dx e Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL 2 b3 c d2 x2 LogB F + a b2 d3 x2 LogB F + 2 a3 d3 Log@a + b xD LogB F + 3 a2 b d3 x LogB F + c+dx c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 2 3 a b2 d3 x2 LogB F + b3 d3 x3 LogB F - 3 b3 c3 Log@c + d xD + 7 a b2 c2 d Log@c + d xD - 4 a2 b c d2 Log@c + d xD + c+dx c+dx a a a c 2 b3 c3 LogB + xF Log@c + d xD - 6 a b2 c2 d LogB + xF Log@c + d xD + 6 a2 b c d2 LogB + xF Log@c + d xD - 2 b3 c3 LogB + xF Log@c + d xD + b b b d c c e Ha + b xL e Ha + b xL 6 a b2 c2 d LogB + xF Log@c + d xD - 6 a2 b c d2 LogB + xF Log@c + d xD - 2 b3 c3 LogB F Log@c + d xD + 6 a b2 c2 d LogB F Log@c + d xD d d c+dx c+dx e Ha + b xL a b Hc + d xL a b Hc + d xL 6 a2 b c d2 LogB F Log@c + d xD - 2 b3 c3 LogB + xF LogB F + 6 a b2 c2 d LogB + xF LogB Fc+dx b bc-ad b bc-ad a b Hc + d xL d Ha + b xL b Hc + d xL 6 a2 b c d2 LogB + xF LogB F - 2 b c Ib2 c2 - 3 a b c d + 3 a2 d2 M PolyLogB2, F - 2 a3 d3 PolyLogB2, F b bc-ad -b c + a d bc-ad b Problem ð504: Valid but suboptimal antiderivative: :Ha + b xL LogB - e Ha + b xL c+dx F , x, 7, 0> Hb c - a dL Ha + b xL LogA bd Ha + b xL2 LogA 2b e Ha+b xL 2 E c+d x 2 e Ha+b xL E c+d x + - Hb c - a dL2 LogB Hb c - a dL2 Log@c + d xD b d2 - b c-a d F b Hc+d xL b d2 LogA e Ha+b xL E c+d x + Hb c - a dL2 PolyLogB2, b d2 d Ha+b xL F b Hc+d xL 2.2 Logarithm Functions.nb ae+bex a x LogB c+dx F + 2 a2 LogA b Hb c - a dL 1 b x2 LogB 2 c+dx 2 a b + bd 2 b2 Hb c - a dL b2 Hb c - a dL c2 LogA a b c2 LogA + xE a2 Log@a + b xD x 2 + xE + -b J c2 Log@c + d xD d I +xM a b ad b c 2 a + xF LogB b bc-ad b2 b a+b x E b a b ad b F -d J ae c Log@c+d xD + xE LogB1 - b I +xM d -a+ bc d d2 N + x LogA bd c+dx c c d - bex c+dx a2 LogA x d + F + F + PolyLogB2, b2 Hb c - a dL - a b I +xM c d -a+ bc d c+d x E d F c + xF Log@a + b xD - 2 a d LogB b F - 2 b c LogB - + xF + LogB d d I +xM -c+ N + x LogA c + xF + LogB 2 c+dx b Hc + d xL a Log@a+b xD bd + xF + 2 a d LogB e Ha + b xL - a d 2 a d Log@a + b xD LogB x b - LogB F + PolyLogB2, + xF - b c LogB b 2 b c LogB 2 d2 H- b c + a dL a bd F - 2 d2 H- b c + a dL - -c+ a - a d LogB c d d2 Hb c - a dL - + xE LogB1 - 1 ae+bex c b F + 2 b c PolyLogB2, d Ha + b xL -b c + a d + xF Log@c + d xD + 2 b c LogB d F + 2 a d PolyLogB2, + xF LogB d c + xF Log@c + d xD + 2 b c LogB - + xF Log@a + b xD + 2 a d LogB d a + b Hc + d xL bc-ad e Ha + b xL c+dx F d Ha + b xL -b c + a d F Log@c + d xD + Problem ð505: Valid but suboptimal antiderivative: :LogB e Ha + b xL c+dx F , x, 3, 0> 2 Hb c - a dL LogB 1 a a d LogB bd 2 b c-a d F b Hc+d xL bd e Ha+b xL E c+d x c 2 + xF + b c LogB b + Ha + b xL LogA 2 e Ha + b xL c+dx b e Ha + b xL c+dx e Ha+b xL 2 E c+d x : e Ha+b xL 2 E c+d x a+bx F + b d x LogB d Ha+b xL F b Hc+d xL c b F Log@c + d xD - 2 b c LogB , x, 3, 0> bd + xF Log@a + b xD + 2 a d LogB e Ha + b xL c+dx a b c + xF Log@a + b xD - 2 a d LogB d F + 2 b c LogB + xF LogB Problem ð506: Valid but suboptimal antiderivative: LogA + 2 Hb c - a dL PolyLogB2, a + xF - 2 a d LogB d 2 a d Log@a + b xD LogB 2 b c LogB LogA 2 b Hc + d xL bc-ad d a c + xF Log@c + d xD - 2 b c LogB b F - 2 b c PolyLogB2, + xF LogB d Ha + b xL -b c + a d d Ha + b xL -b c + a d + xF Log@c + d xD d F - 2 a d PolyLogB2, F- F+ b Hc + d xL bc-ad F 97 98 2.2 Logarithm Functions.nb LogB- b c-a d F d Ha+b xL LogA b 1 a c 3 LogB 3b e Ha+b xL 2 E c+d x + xF + 3 LogB b 2 LogA + 2 + xF LogB d a 3 LogB b d Ha + b xL -b c + a d c + xF - LogB e Ha+b xL E c+d x + xF - LogB d e Ha + b xL c+dx b Hc + d xL PolyLogB2, b b Hc+d xL F d Ha+b xL 2 PolyLogB3, + b F + 3 Log@a + b xD - LogB F a LogB a c + xF + LogB b + xF - 2 LogB b + xF + LogB d c 2 b Hc+d xL F d Ha+b xL + xF LogB d d Ha + b xL -b c + a d e Ha + b xL c+dx F F - 2 PolyLogB2, 2 b Hc + d xL bc-ad F + F+ d bc-ad 2 a c b Hc + d xL a d Ha + b xL d Ha + b xL b Hc + d xL 3 LogB + xF - LogB + xF + LogB F + 2 LogB + xF PolyLogB2, F - 2 PolyLogB3, F - 6 PolyLogB3, F b d bc-ad b -b c + a d -b c + a d bc-ad c 6 LogB + xF PolyLogB2, Problem ð509: Valid but suboptimal antiderivative: :Ha + b xL2 LogB e Ha + b xL c+dx Hb c - a dL2 Ha + b xL LogA b d2 F , x, 16, 0> e Ha+b xL E c+d x Hb c - a dL Ha + b xL2 LogA 2bd 3 Hb c - a dL3 PolyLogB2, b d3 3 + e Ha+b xL 2 E c+d x d Ha+b xL F b Hc+d xL 3 Hb c - a dL3 LogB + + b c-a d F b Hc+d xL LogA b d3 Hb c - a dL3 LogB b c-a d F b Hc+d xL 2 Hb c - a dL3 LogA b d3 LogA e Ha+b xL E c+d x b d3 e Ha+b xL E c+d x + b d2 e Ha+b xL 2 E c+d x PolyLogB2, Hb c - a dL2 Ha + b xL LogA + Ha + b xL3 LogA d Ha+b xL F b Hc+d xL 3b - e Ha+b xL 3 E c+d x e Ha+b xL 2 E c+d x - - Hb c - a dL3 Log@c + d xD 2 Hb c - a dL3 PolyLogB3, b d3 b d3 d Ha+b xL F b Hc+d xL + 2.2 Logarithm Functions.nb 1 6 b3 c3 LogB 6 b d3 Hb c - a dL e c+dx e Ha + b xL F - 18 a b2 c2 d LogB c+dx e Ha + b xL F + 18 a2 b c d2 LogB c+dx e Ha + b xL F + 6 b3 c2 d x LogB Hb c - a dL e e Ha + b xL c+dx F - 6 a3 d3 LogB F - 12 a b2 c d2 x LogB Hb c - a dL e c+dx e Ha + b xL F + 6 a b2 c2 d LogB Problem ð523: Valid but suboptimal antiderivative: :Hc + d xL3 LogB 5 Hb c - a dL3 x e Ha + b xL c+dx + 12 b3 F , x, 13, 0> 2 Hb c - a dL2 Hc + d xL2 12 b2 d Hb c - a dL2 Hc + d xL2 LogA 4 b2 d Hc + d xL4 LogA 4d e Ha+b xL 2 E c+d x + e Ha+b xL E c+d x - + 5 Hb c - a dL4 Log@a + b xD 12 b4 d Hb c - a dL Hc + d xL3 LogA 6bd Hb c - a dL4 Log@c + d xD 2 b4 d - - Hb c - a dL3 Ha + b xL LogA 2 b4 e Ha+b xL E c+d x + Hb c - a dL4 LogB- Hb c - a dL4 PolyLogB2, 2 b4 d b Hc+d xL F d Ha+b xL e Ha+b xL E c+d x b c-a d F d Ha+b xL 2 b4 d - LogA e Ha+b xL E c+d x + F + 6 a2 b d3 x LogB e Ha + b xL c+dx e Ha + b xL F- F+ c+dx c+dx c+dx c+dx 2 e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 6 a b2 c2 d LogB F - 15 a2 b c d2 LogB F + 9 a3 d3 LogB F + 6 b3 c2 d x LogB F - 18 a b2 c d2 x LogB F + c+dx c+dx c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 3 12 a2 b d3 x LogB F - 3 b3 c d2 x2 LogB F + 3 a b2 d3 x2 LogB F + 2 a3 d3 LogB F + c+dx c+dx c+dx c+dx e Ha + b xL 3 e Ha + b xL 3 e Ha + b xL 3 e Ha + b xL bc-ad 6 a2 b d3 x LogB F + 6 a b2 d3 x2 LogB F + 2 b3 d3 x3 LogB F + 18 b3 c3 LogB F LogB Fc+dx c+dx c+dx c+dx bc+bdx e Ha + b xL bc-ad e Ha + b xL bc-ad e Ha + b xL bc-ad 54 a b2 c2 d LogB F LogB F + 54 a2 b c d2 LogB F LogB F - 18 a3 d3 LogB F LogB F+ c+dx bc+bdx c+dx bc+bdx c+dx bc+bdx e Ha + b xL 2 bc-ad e Ha + b xL 2 bc-ad e Ha + b xL 2 bc-ad 6 b3 c3 LogB F LogB F - 18 a b2 c2 d LogB F LogB F + 18 a2 b c d2 LogB F LogB Fc+dx bc+bdx c+dx bc+bdx c+dx bc+bdx e Ha + b xL 2 bc-ad e Ha + b xL d Ha + b xL d Ha + b xL 6 a3 d3 LogB F LogB F + 6 Hb c - a dL3 3 + 2 LogB F PolyLogB2, F - 12 Hb c - a dL3 PolyLogB3, F c+dx bc+bdx c+dx b Hc + d xL b Hc + d xL 12 a2 b c d2 LogB F + 6 a3 d3 LogB Hb c - a dL e 99 100 2.2 Logarithm Functions.nb 1 7 b4 c3 d x - 19 a b3 c2 d2 x + 17 a2 b2 c d3 x - 5 a3 b d4 x + b4 c2 d2 x2 - 2 a b3 c d3 x2 + a2 b2 d4 x2 + 12 a b3 c3 d LogB 12 b4 d a 2 + xF - 18 a2 b2 c2 d2 LogB b 12 a3 b c d3 LogB a 2 + xF - 3 a4 d4 LogB b a 2 + xF + 3 b4 c4 LogB b c b c b 2 + xF - 18 a b3 c3 d Log@a + b xD + 45 a2 b2 c2 d2 Log@a + b xD a + xF Log@a + b xD + 36 a2 b2 c2 d2 LogB b a 2 + xF + d 38 a3 b c d3 Log@a + b xD + 11 a4 d4 Log@a + b xD - 24 a b3 c3 d LogB 24 a3 b c d3 LogB a + xF Log@a + b xD + 6 a4 d4 LogB a b c a + xF Log@a + b xD b + xF Log@a + b xD + 24 a b3 c3 d LogB c + xF Log@a + b xD - 36 a2 b2 c2 d2 LogB d c c + xF Log@a + b xD + d d Ha + b xL F+ -b c + a d c d Ha + b xL c d Ha + b xL c d Ha + b xL 36 a2 b2 c2 d2 LogB + xF LogB F - 24 a3 b c d3 LogB + xF LogB F + 6 a4 d4 LogB + xF LogB Fd -b c + a d d -b c + a d d -b c + a d e Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL 18 b4 c3 d x LogB F + 36 a b3 c2 d2 x LogB F - 24 a2 b2 c d3 x LogB F + 6 a3 b d4 x LogB Fc+dx c+dx c+dx c+dx e Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL 9 b4 c2 d2 x2 LogB F + 12 a b3 c d3 x2 LogB F - 3 a2 b2 d4 x2 LogB F - 2 b4 c d3 x3 LogB F+ c+dx c+dx c+dx c+dx e Ha + b xL e Ha + b xL e Ha + b xL 2 a b3 d4 x3 LogB F + 24 a b3 c3 d Log@a + b xD LogB F - 36 a2 b2 c2 d2 Log@a + b xD LogB F+ c+dx c+dx c+dx e Ha + b xL e Ha + b xL e Ha + b xL 2 e Ha + b xL 2 24 a3 b c d3 Log@a + b xD LogB F - 6 a4 d4 Log@a + b xD LogB F + 12 b4 c3 d x LogB F + 18 b4 c2 d2 x2 LogB F + c+dx c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 2 12 b4 c d3 x3 LogB F + 3 b4 d4 x4 LogB F + 11 b4 c4 Log@c + d xD - 26 a b3 c3 d Log@c + d xD + 21 a2 b2 c2 d2 Log@c + d xD c+dx c+dx a c e Ha + b xL 6 a3 b c d3 Log@c + d xD + 6 b4 c4 LogB + xF Log@c + d xD - 6 b4 c4 LogB + xF Log@c + d xD - 6 b4 c4 LogB F Log@c + d xD b d c+dx a b Hc + d xL d Ha + b xL b Hc + d xL 6 b4 c4 LogB + xF LogB F - 6 b4 c4 PolyLogB2, F + 6 a d I- 4 b3 c3 + 6 a b2 c2 d - 4 a2 b c d2 + a3 d3 M PolyLogB2, F b bc-ad -b c + a d bc-ad 24 a3 b c d3 LogB + xF Log@a + b xD - 6 a4 d4 LogB d + xF Log@a + b xD - 24 a b3 c3 d LogB d + xF LogB d Problem ð524: Valid but suboptimal antiderivative: :Hc + d xL2 LogB Hb c - a dL x 2 3 b2 + e Ha + b xL c+dx F , x, 10, 0> 2 Hb c - a dL3 Log@a + b xD 2 Hb c - a dL3 LogB- 3 b3 d b c-a d F d Ha+b xL 3 b3 d LogA - 2 Hb c - a dL2 Ha + b xL LogA e Ha+b xL E c+d x 3 b3 + Hc + d xL3 LogA 3d e Ha+b xL E c+d x e Ha+b xL 2 E c+d x + - Hb c - a dL Hc + d xL2 LogA 3bd 2 Hb c - a dL3 Log@c + d xD 3 b3 d - e Ha+b xL E c+d x + 2 Hb c - a dL3 PolyLogB2, 3 b3 d b Hc+d xL F d Ha+b xL 2.2 Logarithm Functions.nb 1 b3 c2 d x - 2 a b2 c d2 x + a2 b d3 x + 3 a b2 c2 d LogB 3 b3 d a 2 + xF - 3 a2 b c d2 LogB b a 2 + xF + a3 d3 LogB b a 2 + xF + b3 c3 LogB b 4 a b2 c2 d Log@a + b xD + 7 a2 b c d2 Log@a + b xD - 3 a3 d3 Log@a + b xD - 6 a b2 c2 d LogB a c d + xF Log@a + b xD + 6 a2 b c d2 LogB b 2 a3 d3 LogB a + xF Log@a + b xD + 6 a b2 c2 d LogB b d Ha + b xL c + xF Log@a + b xD - 6 a2 b c d2 LogB d d Ha + b xL + xF Log@a + b xD b + xF Log@a + b xD + 2 a3 d3 LogB d c + xF LogB c + xF LogB F - 2 a3 d3 LogB a d Ha + b xL c + xF Log@a + b xD d Fd -b c + a d d -b c + a d d -b c + a d e Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL e Ha + b xL 4 b3 c2 d x LogB F + 6 a b2 c d2 x LogB F - 2 a2 b d3 x LogB F - b3 c d2 x2 LogB F + a b2 d3 x2 LogB F+ c+dx c+dx c+dx c+dx c+dx e Ha + b xL e Ha + b xL e Ha + b xL 6 a b2 c2 d Log@a + b xD LogB F - 6 a2 b c d2 Log@a + b xD LogB F + 2 a3 d3 Log@a + b xD LogB F+ c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 3 b3 c2 d x LogB F + 3 b3 c d2 x2 LogB F + b3 d3 x3 LogB F + 3 b3 c3 Log@c + d xD - 5 a b2 c2 d Log@c + d xD + c+dx c+dx c+dx a c e Ha + b xL 2 a2 b c d2 Log@c + d xD + 2 b3 c3 LogB + xF Log@c + d xD - 2 b3 c3 LogB + xF Log@c + d xD - 2 b3 c3 LogB F Log@c + d xD b d c+dx a b Hc + d xL d Ha + b xL b Hc + d xL 2 b3 c3 LogB + xF LogB F - 2 b3 c3 PolyLogB2, F - 2 a d I3 b2 c2 - 3 a b c d + a2 d2 M PolyLogB2, F b bc-ad -b c + a d bc-ad 6 a b2 c2 d LogB F + 6 a2 b c d2 LogB c 2 + xF - Problem ð525: Valid but suboptimal antiderivative: :Hc + d xL LogB - e Ha + b xL c+dx F , x, 7, 0> Hb c - a dL Ha + b xL LogA b2 Hc + d xL2 LogA 2d e Ha+b xL 2 E c+d x 2 e Ha+b xL E c+d x + + Hb c - a dL2 LogB- Hb c - a dL2 Log@c + d xD b2 d - b c-a d F d Ha+b xL b2 d LogA e Ha+b xL E c+d x + Hb c - a dL2 PolyLogB2, b2 d b Hc+d xL F d Ha+b xL c + xF LogB 101 102 2.2 Logarithm Functions.nb ae+bex c x LogB c+dx F + 2 a2 LogA d Hb c - a dL 1 d x2 LogB 2 c+dx 2 a b + bd 2 b2 Hb c - a dL b2 Hb c - a dL c2 LogA a b c2 LogA + xE a2 Log@a + b xD x 2 + xE + -b J c2 Log@c + d xD d I +xM a b ad b c 2 a + xF LogB b bc-ad b2 b a+b x E b a b ad b F -d J ae c Log@c+d xD + xE LogB1 - b I +xM d -a+ bc d d2 N + x LogA bd c+dx c c d - bex c+dx a2 LogA x d + F + F + PolyLogB2, b2 Hb c - a dL - a b I +xM c d -a+ bc d c+d x E d F c + xF Log@a + b xD - 2 a d LogB b F - 2 b c LogB - + xF + LogB d d I +xM -c+ N + x LogA c + xF + LogB 2 c+dx b Hc + d xL a Log@a+b xD bd + xF + 2 a d LogB e Ha + b xL - a d 2 a d Log@a + b xD LogB x b - LogB F + PolyLogB2, + xF - b c LogB b 2 b c LogB 2 d2 H- b c + a dL a bd F - 2 d2 H- b c + a dL - -c+ c - a d LogB c d d2 Hb c - a dL - + xE LogB1 - 1 ae+bex c b F + 2 b c PolyLogB2, d Ha + b xL -b c + a d + xF Log@c + d xD + 2 b c LogB d F + 2 a d PolyLogB2, + xF LogB d c + xF Log@c + d xD + 2 b c LogB - + xF Log@a + b xD + 2 a d LogB d a + b Hc + d xL bc-ad e Ha + b xL c+dx F d Ha + b xL -b c + a d F Log@c + d xD + Problem ð526: Valid but suboptimal antiderivative: :LogB e Ha + b xL c+dx F , x, 3, 0> 2 Hb c - a dL LogB 1 a a d LogB bd 2 b c-a d F b Hc+d xL bd e Ha+b xL E c+d x c 2 + xF + b c LogB b + Ha + b xL LogA 2 e Ha + b xL c+dx b e Ha + b xL c+dx e Ha+b xL 2 E c+d x : e Ha+b xL 2 E c+d x c+dx F + b d x LogB d Ha+b xL F b Hc+d xL c b F Log@c + d xD - 2 b c LogB , x, 3, 0> bd + xF Log@a + b xD + 2 a d LogB e Ha + b xL c+dx a b c + xF Log@a + b xD - 2 a d LogB d F + 2 b c LogB + xF LogB Problem ð527: Valid but suboptimal antiderivative: LogA + 2 Hb c - a dL PolyLogB2, a + xF - 2 a d LogB d 2 a d Log@a + b xD LogB 2 b c LogB LogA 2 b Hc + d xL bc-ad d a c + xF Log@c + d xD - 2 b c LogB b F - 2 b c PolyLogB2, + xF LogB d Ha + b xL -b c + a d d Ha + b xL -b c + a d + xF Log@c + d xD d F - 2 a d PolyLogB2, F- F+ b Hc + d xL bc-ad F 2.2 Logarithm Functions.nb LogB - b c-a d F b Hc+d xL 1 d c LogB 3d LogA e Ha+b xL 2 E c+d x 2 LogA - a 3 + xF + 3 - LogB b a b + xF - LogB d d Ha + b xL 6 PolyLogB3, -b c + a d d + xF + LogB d c + xF - LogB PolyLogB2, c + xF + LogB d 3 LogB e Ha+b xL E c+d x e Ha + b xL c+dx F + 3 LogB c 2 + xF d Ha+b xL F b Hc+d xL e Ha + b xL c+dx F c - LogB + d 2 a Log@c + d xD + 3 LogB a 2 + xF LogB b a + xF + LogB b 2 + xF LogB b + xF + 2 LogB d - LogB d F d Ha+b xL F b Hc+d xL 2 PolyLogB3, d Ha + b xL -b c + a d b Hc + d xL bc-ad F + 2 LogB c b Hc + d xL bc-ad F + 6 LogB F + 2 PolyLogB2, + xF PolyLogB2, d a + xF PolyLogB2, b d Ha + b xL -b c + a d b Hc + d xL bc-ad F - 2 PolyLogB3, Problem ð530: Valid but suboptimal antiderivative: :Hc + d xL2 LogB e Ha + b xL c+dx Hb c - a dL2 Ha + b xL LogA b3 2 Hb c - a dL3 LogB Hb c - a dL3 LogB- F , x, 16, 0> 3 e Ha+b xL E c+d x b c-a d F b Hc+d xL LogA b3 d b c-a d F d Ha+b xL b3 d Hb c - a dL3 PolyLogB2, b3 d LogA - Hb c - a dL3 LogB- e Ha+b xL E c+d x e Ha+b xL 2 E c+d x b Hc+d xL F d Ha+b xL - - b c-a d F d Ha+b xL b3 d LogA e Ha+b xL E c+d x Hb c - a dL2 Ha + b xL LogA b3 + Hc + d xL3 LogA 2 Hb c - a dL3 LogA 3d e Ha+b xL 3 E c+d x e Ha+b xL E c+d x b3 d - e Ha+b xL 2 E c+d x - - Hb c - a dL Hc + d xL2 LogA 2bd Hb c - a dL3 Log@c + d xD PolyLogB2, b3 d b Hc+d xL F d Ha+b xL - - e Ha+b xL 2 E c+d x + 2 Hb c - a dL3 PolyLogB2, 2 Hb c - a dL3 PolyLogB3, b3 d b3 d b Hc+d xL F d Ha+b xL d Ha + b xL -b c + a d F - d Ha+b xL F b Hc+d xL + 103 b Hc + d xL bc-ad F F- 104 2.2 Logarithm Functions.nb 1 6 b3 c3 LogB 6 b3 d Hb c - a dL e c+dx e Ha + b xL F - 18 a b2 c2 d LogB c+dx e Ha + b xL F + 18 a2 b c d2 LogB c+dx e Ha + b xL F + 6 b3 c2 d x LogB Hb c - a dL e e Ha + b xL c+dx F - 6 a3 d3 LogB F - 12 a b2 c d2 x LogB Hb c - a dL e c+dx e Ha + b xL F + 6 a b2 c2 d LogB e Ha + b xL c+dx F+ c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 6 a2 b d3 x LogB F - 12 a b2 c2 d LogB F + 21 a2 b c d2 LogB F - 9 a3 d3 LogB F c+dx c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 e Ha + b xL 2 12 b3 c2 d x LogB F + 18 a b2 c d2 x LogB F - 6 a2 b d3 x LogB F - 3 b3 c d2 x2 LogB F + c+dx c+dx c+dx c+dx e Ha + b xL 2 e Ha + b xL 3 e Ha + b xL 3 e Ha + b xL 3 3 a b2 d3 x2 LogB F + 6 a b2 c2 d LogB F - 6 a2 b c d2 LogB F + 2 a3 d3 LogB F + c+dx c+dx c+dx c+dx e Ha + b xL 3 e Ha + b xL 3 e Ha + b xL 3 e Ha + b xL bc-ad 6 b3 c2 d x LogB F + 6 b3 c d2 x2 LogB F + 2 b3 d3 x3 LogB F - 18 b3 c3 LogB F LogB F+ c+dx c+dx c+dx c+dx bc+bdx e Ha + b xL bc-ad e Ha + b xL bc-ad e Ha + b xL bc-ad 54 a b2 c2 d LogB F LogB F - 54 a2 b c d2 LogB F LogB F + 18 a3 d3 LogB F LogB F+ c+dx bc+bdx c+dx bc+bdx c+dx bc+bdx e Ha + b xL 2 bc-ad e Ha + b xL 2 bc-ad e Ha + b xL 2 bc-ad 6 b3 c3 LogB F LogB F - 18 a b2 c2 d LogB F LogB F + 18 a2 b c d2 LogB F LogB Fc+dx bc+bdx c+dx bc+bdx c+dx bc+bdx e Ha + b xL 2 bc-ad e Ha + b xL d Ha + b xL d Ha + b xL 6 a3 d3 LogB F LogB F + 6 Hb c - a dL3 - 3 + 2 LogB F PolyLogB2, F - 12 Hb c - a dL3 PolyLogB3, F c+dx bc+bdx c+dx b Hc + d xL b Hc + d xL 12 a2 b c d2 LogB F + 6 a3 d3 LogB Hb c - a dL e Problem ð538: Valid but suboptimal antiderivative: : LogB d Ha+b xL F b Hc+d xL , x, 1, 0> cf+dfx PolyLogB2, df 1 b c-a d F b Hc+d xL c - LogB 2df 2 LogB a 2 + xF - 2 LogB d d Ha + b xL b Hc + d xL c + xF Log@c + d xD + 2 LogB b F Log@c + d xD + 2 LogB + xF Log@c + d xD + d a + xF LogB b b Hc + d xL bc-ad Problem ð549: Valid but suboptimal antiderivative: :Hf + g xL3 LogBe a+bx c+dx F , x, 15, 0> n 2 F + 2 PolyLogB2, d Ha + b xL -b c + a d F F- 2.2 Logarithm Functions.nb - Hb c - a dL2 Hb c + a dL g3 n2 x 6 b3 d3 + Hb c - a dL2 g2 H4 b d f - b c g - a d gL n2 x 4 b3 d3 a2 Hb c - a dL g2 H4 b d f - b c g - a d gL n2 Log@a + b xD 4 b4 d2 Hb c - a dL g3 n x3 LogAe I 6bd Hb f - a gL4 n LogB- a+b x n M E c+d x b c-a d F d Ha+b xL 2 b4 g - LogAe I a+b x n M E c+d x 4 b2 d4 2 d4 g 1 d Ha+b xL F b Hc+d xL Hb c - a dL2 g3 n2 x2 a3 Hb c - a dL g3 n2 Log@a + b xD - 12 b2 d2 Hb c - a dL g2 H4 b d f - b c g - a d gL n x2 LogAe I 4 b2 d2 2 b4 d3 + Hf + g xL4 LogAe I 4g - + a+b x n 2 M E c+d x - Hd f - c gL4 n LogAe I a+b x n M E c+d x a+b x n M E c+d x + 6 b4 d - Hb c - a dL g Ia2 d2 g2 - a b d g H4 d f - c gL + b2 I6 d2 f2 - 4 c d f g + c2 g2 MM n Ha + b xL LogAe I c2 Hb c - a dL g2 H4 b d f - b c g - a d gL n2 Log@c + d xD Hd f - c gL4 n2 PolyLogB2, - + LogB 2 d4 g b c-a d F b Hc+d xL + a+b x n M E c+d x + c3 Hb c - a dL g3 n2 Log@c + d xD 6 b d4 Hb c - a dL2 g Ia2 d2 g2 - a b d g H4 d f - c gL + b2 I6 d2 f2 - 4 c d f g + c2 g2 MM n2 Log@c + d xD Hb f - a gL4 n2 PolyLogB2, 2 b4 g 105 - 2 b4 d4 b Hc+d xL F d Ha+b xL 12 b4 c2 d2 f g2 n2 x - 24 a b3 c d3 f g2 n2 x + 12 a2 b2 d4 f g2 n2 x - 5 b4 c3 d g3 n2 x + 5 a b3 c2 d2 g3 n2 x + 5 a2 b2 c d3 g3 n2 x - 5 a3 b d4 g3 n2 x + 12 b4 d4 b4 c2 d2 g3 n2 x2 - 2 a b3 c d3 g3 n2 x2 + a2 b2 d4 g3 n2 x2 + 12 a b3 d4 f3 n2 LogB a 2 + xF - 18 a2 b2 d4 f2 g n2 LogB b 4 4 3 2 3 a d g n LogB a 2 4 3 3 2 c + xF + 12 b c d f n LogB 2 4 2 2 2 2 a 2 + xF + 12 a3 b d4 f g2 n2 LogB b c + xF - 18 b c d f g n LogB 2 4 3 2 2 + xF b c 2 a + xF + 12 b c d f g n LogB 2 4 4 3 2 c + xF - 3 b c g n LogB 2 + xF - b d d d d 36 a b3 c d3 f2 g n2 Log@a + b xD + 36 a2 b2 d4 f2 g n2 Log@a + b xD + 24 a b3 c2 d2 f g2 n2 Log@a + b xD + 12 a2 b2 c d3 f g2 n2 Log@a + b xD 36 a3 b d4 f g2 n2 Log@a + b xD - 6 a b3 c3 d g3 n2 Log@a + b xD - 3 a2 b2 c2 d2 g3 n2 Log@a + b xD - 2 a3 b c d3 g3 n2 Log@a + b xD + 11 a4 d4 g3 n2 Log@a + b xD a a a 24 a b3 d4 f3 n2 LogB + xF Log@a + b xD + 36 a2 b2 d4 f2 g n2 LogB + xF Log@a + b xD - 24 a3 b d4 f g2 n2 LogB + xF Log@a + b xD + b b b a c c 6 a4 d4 g3 n2 LogB + xF Log@a + b xD + 24 a b3 d4 f3 n2 LogB + xF Log@a + b xD - 36 a2 b2 d4 f2 g n2 LogB + xF Log@a + b xD + b d d c c c d Ha + b xL 24 a3 b d4 f g2 n2 LogB + xF Log@a + b xD - 6 a4 d4 g3 n2 LogB + xF Log@a + b xD - 24 a b3 d4 f3 n2 LogB + xF LogB F+ d d d -b c + a d c d Ha + b xL c d Ha + b xL c d Ha + b xL 36 a2 b2 d4 f2 g n2 LogB + xF LogB F - 24 a3 b d4 f g2 n2 LogB + xF LogB F + 6 a4 d4 g3 n2 LogB + xF LogB Fd -b c + a d d -b c + a d d -b c + a d a+bx n a+bx n a+bx n 36 b4 c d3 f2 g n x LogBe F + 36 a b3 d4 f2 g n x LogBe F + 24 b4 c2 d2 f g2 n x LogBe Fc+dx c+dx c+dx 24 a2 b2 d4 f g2 n x LogBe a+bx n c+dx 12 a b3 d4 f g2 n x2 LogBe a+bx c+dx 2 a b3 d4 g3 n x3 LogBe a+bx c+dx n n F - 6 b4 c3 d g3 n x LogBe F + 3 b4 c2 d2 g3 n x2 LogBe a+bx c+dx a+bx c+dx F + 24 a b3 d4 f3 n Log@a + b xD LogBe - n F + 6 a3 b d4 g3 n x LogBe n a+bx c+dx F - 3 a2 b2 d4 g3 n x2 LogBe a+bx c+dx n n F - 12 b4 c d3 f g2 n x2 LogBe a+bx c+dx n F - 2 b4 c d3 g3 n x3 LogBe F - 36 a2 b2 d4 f2 g n Log@a + b xD LogBe + a+bx c+dx + n F+ a+bx c+dx a+bx c+dx n F+ n F+ - 106 2.2 Logarithm Functions.nb 24 a3 b d4 f g2 n Log@a + b xD LogBe a+bx c+dx a+bx 18 b4 d4 f2 g x2 LogBe n F - 6 a4 d4 g3 n Log@a + b xD LogBe F + 12 b4 d4 f g2 x3 LogBe n 2 c+dx a+bx c+dx a+bx c+dx n F + 12 b4 d4 f3 x LogBe F + 3 b4 d4 g3 x4 LogBe n 2 a+bx c+dx a+bx F + n 2 c+dx F + 36 b4 c2 d2 f2 g n2 Log@c + d xD - n 2 36 a b3 c d3 f2 g n2 Log@c + d xD - 36 b4 c3 d f g2 n2 Log@c + d xD + 12 a b3 c2 d2 f g2 n2 Log@c + d xD + 24 a2 b2 c d3 f g2 n2 Log@c + d xD + 11 b4 c4 g3 n2 Log@c + d xD - 2 a b3 c3 d g3 n2 Log@c + d xD - 3 a2 b2 c2 d2 g3 n2 Log@c + d xD - 6 a3 b c d3 g3 n2 Log@c + d xD + a a a 24 b4 c d3 f3 n2 LogB + xF Log@c + d xD - 36 b4 c2 d2 f2 g n2 LogB + xF Log@c + d xD + 24 b4 c3 d f g2 n2 LogB + xF Log@c + d xD b b b a c c 4 4 3 2 4 3 3 2 4 2 2 2 2 6 b c g n LogB + xF Log@c + d xD - 24 b c d f n LogB + xF Log@c + d xD + 36 b c d f g n LogB + xF Log@c + d xD b d d c c a+bx n 24 b4 c3 d f g2 n2 LogB + xF Log@c + d xD + 6 b4 c4 g3 n2 LogB + xF Log@c + d xD - 24 b4 c d3 f3 n LogBe F Log@c + d xD + d d c+dx c+dx 24 b4 c d3 f3 n2 LogB n a+bx 36 b4 c2 d2 f2 g n LogBe a + xF LogB b F Log@c + d xD - 24 b4 c3 d f g2 n LogBe b Hc + d xL F + 36 b4 c2 d2 f2 g n2 LogB n a+bx c+dx a + xF LogB F Log@c + d xD + 6 b4 c4 g3 n LogBe b Hc + d xL F - 24 b4 c3 d f g2 n2 LogB a+bx c+dx a n F Log@c + d xD - + xF LogB bc-ad b bc-ad b a b Hc + d xL d Ha + b xL 6 b4 c4 g3 n2 LogB + xF LogB F + 6 b4 c I- 4 d3 f3 + 6 c d2 f2 g - 4 c2 d f g2 + c3 g3 M n2 PolyLogB2, F+ b bc-ad -b c + a d b Hc + d xL 6 a d4 I- 4 b3 f3 + 6 a b2 f2 g - 4 a2 b f g2 + a3 g3 M n2 PolyLogB2, F bc-ad b Hc + d xL bc-ad Problem ð550: Valid but suboptimal antiderivative: :Hf + g xL2 LogBe a+bx c+dx Hb c - a dL2 g2 n2 x 3 b2 d2 + F , x, 12, 0> n 2 a2 Hb c - a dL g2 n2 Log@a + b xD - 3 b3 d Hb c - a dL g2 n x2 LogAe I 3bd 2 Hb c - a dL g H3 b d f - b c g - a d gL n Ha + b xL LogAe I 3 b3 d2 Hf + g xL3 LogAe I 3g a+b x n 2 M E c+d x - 2 Hd f - c gL3 n LogAe I a+b x n M E c+d x 3 d3 g 2 Hb c - a dL2 g H3 b d f - b c g - a d gL n2 Log@c + d xD 3 b3 d3 a+b x n M E c+d x - + LogB a+b x n M E c+d x 2 Hb f - a gL3 n LogB- b c-a d F d Ha+b xL 3 b3 g b c-a d F b Hc+d xL - LogAe I a+b x n M E c+d x c2 Hb c - a dL g2 n2 Log@c + d xD 2 Hd f - c gL3 n2 PolyLogB2, 3 d3 g + + 3 b d3 d Ha+b xL F b Hc+d xL - 2 Hb f - a gL3 n2 PolyLogB2, 3 b3 g b Hc+d xL F d Ha+b xL F+ 2.2 Logarithm Functions.nb 1 b3 c2 d g2 n2 x - 2 a b2 c d2 g2 n2 x + a2 b d3 g2 n2 x + 3 a b2 d3 f2 n2 LogB 3 b3 d3 a 2 + xF - 3 a2 b d3 f g n2 LogB b 3 b3 c d2 f2 n2 LogB c 2 + xF - 3 b3 c2 d f g n2 LogB d c 2 + xF + b3 c3 g2 n2 LogB d c a 2 + xF + a3 d3 g2 n2 LogB b a 2 + xF + b 2 + xF - 6 a b2 c d2 f g n2 Log@a + b xD + 6 a2 b d3 f g n2 Log@a + b xD + d 2 a b2 c2 d g2 n2 Log@a + b xD + a2 b c d2 g2 n2 Log@a + b xD - 3 a3 d3 g2 n2 Log@a + b xD - 6 a b2 d3 f2 n2 LogB a + xF Log@a + b xD + b 6 a2 b d3 f g n2 LogB a + xF Log@a + b xD - 2 a3 d3 g2 n2 LogB b c a b c + xF Log@a + b xD + 6 a b2 d3 f2 n2 LogB c + xF Log@a + b xD d c d Ha + b xL F+ -b c + a d d Ha + b xL c d Ha + b xL a+bx n 6 a2 b d3 f g n2 LogB + xF LogB F - 2 a3 d3 g2 n2 LogB + xF LogB F - 6 b3 c d2 f g n x LogBe F+ d -b c + a d d -b c + a d c+dx 6 a2 b d3 f g n2 LogB + xF Log@a + b xD + 2 a3 d3 g2 n2 LogB d c d 6 a b2 d3 f g n x LogBe a+bx c+dx a b2 d3 g2 n x2 LogBe + xF Log@a + b xD - 6 a b2 d3 f2 n2 LogB a+bx n c+dx 2 a3 d3 g2 n Log@a + b xD LogBe n F + 2 b3 c2 d g2 n x LogBe d n a+bx c+dx F + 6 a b2 d3 f2 n Log@a + b xD LogBe a+bx n c+dx F + 3 b3 d3 f2 x LogBe + xF LogB F - 2 a2 b d3 g2 n x LogBe a+bx n c+dx a+bx c+dx a+bx c+dx F - b3 c d2 g2 n x2 LogBe F - 6 a2 b d3 f g n Log@a + b xD LogBe F + 3 b3 d3 f g x2 LogBe n 2 n a+bx c+dx n a+bx c+dx a+bx c+dx F+ F + b3 d3 g2 x3 LogBe n 2 n a+bx c+dx F+ F + n 2 6 b3 c2 d f g n2 Log@c + d xD - 6 a b2 c d2 f g n2 Log@c + d xD - 3 b3 c3 g2 n2 Log@c + d xD + a b2 c2 d g2 n2 Log@c + d xD + 2 a2 b c d2 g2 n2 Log@c + d xD + a a a 6 b3 c d2 f2 n2 LogB + xF Log@c + d xD - 6 b3 c2 d f g n2 LogB + xF Log@c + d xD + 2 b3 c3 g2 n2 LogB + xF Log@c + d xD b b b c c c 6 b3 c d2 f2 n2 LogB + xF Log@c + d xD + 6 b3 c2 d f g n2 LogB + xF Log@c + d xD - 2 b3 c3 g2 n2 LogB + xF Log@c + d xD d d d a+bx n a+bx n a+bx n 6 b3 c d2 f2 n LogBe F Log@c + d xD + 6 b3 c2 d f g n LogBe F Log@c + d xD - 2 b3 c3 g2 n LogBe F Log@c + d xD c+dx c+dx c+dx 6 b3 c d2 f2 n2 LogB a + xF LogB b b Hc + d xL bc-ad F + 6 b3 c2 d f g n2 LogB 2 b3 c I3 d2 f2 - 3 c d f g + c2 g2 M n2 PolyLogB2, d Ha + b xL -b c + a d Problem ð551: Valid but suboptimal antiderivative: :Hf + g xL LogBe a+bx c+dx F , x, 9, 0> n 2 a + xF LogB b b Hc + d xL bc-ad F - 2 b3 c3 g2 n2 LogB a + xF LogB b F - 2 a d3 I3 b2 f2 - 3 a b f g + a2 g2 M n2 PolyLogB2, b Hc + d xL b Hc + d xL bc-ad bc-ad F F- 107 108 2.2 Logarithm Functions.nb - Hb c - a dL g n Ha + b xL LogAe I b2 d Hd f - c gL2 n LogAe I a+b x n M E c+d x a+b x n M E c+d x LogB d2 g n a+bx f x LogBe c+dx F - n LogB a+bx 2 f n LogBe n c+dx a+bx n 2 g n LogBe c+dx 2 1 gn a+bx 2 x LogB 2 c+dx -d J x d - b c-a d F b Hc+d xL a+bx c+dx F - n LogB F - n LogB a+bx c+dx d I +xM f n2 x LogB b -c+ a+bx c+dx F - 1 2 bd d a+bx 2 b c LogB c+dx a+bx c+dx 1 x2 LogB 2 F+ a+bx c+dx a b 2 - + b2 bd b -c+ ad b c 2 + xF - b c LogB -b c + a d F- Hb c - a dL 1 2 c d Hb c - a dL a 2 + xE F + 2g d Ha+b xL F b Hc+d xL Hd f - c gL2 n2 PolyLogB2, a+bx c+dx F a+bx Ha + b xL LogAe I b F , x, 3, 0> a+b x n 2 M E c+d x d2 a2 Log@a + b xD x + bd -b J x b - b2 Hb c - a dL a Log@a+b xD b2 c d a 2 a+b x E b a b I +xM c d bc d a + xF LogB b c+dx b Hc + d xL bc-ad a + a+b x n M E c+d x bd LogB b c-a d F b Hc+d xL + bx + xF + LogB + c+dx F + PolyLogB2, b2 Hb c - a dL b I +xM c d -a+ bc d F c+dx F + + c + xF Log@a + b xD - 2 a d LogB F - 2 b c LogB - d b a+bx + c + xF + LogB b -a+ + xF + 2 a d LogB d2 Hb c - a dL N + x LogA - LogB + xE LogB1 - - c2 Log@c + d xD - bd Hb c - a dL a2 LogA F - 2 a d Log@a + b xD LogB 2 Hb c - a dL n LogAe I b2 g + + xF Log@a + b xD + d a c + xF Log@c + d xD + 2 b c LogB b F + 2 b c PolyLogB2, d Ha + b xL -b c + a d 2 Hb c - a dL n2 PolyLogB2, bd + xF Log@c + d xD + d F + 2 a d PolyLogB2, n 2 c+dx Hb f - a gL2 n2 PolyLogB2, + Problem ð552: Valid but suboptimal antiderivative: :LogBe - - 2 c2 Log@c + d xD - d F Log@c + d xD + 2 b c LogB + a+b x n 2 M E c+d x b2 c d - a b d2 2 d2 H- b c + a dL d I +xM Hf + g xL2 LogAe I d2 g F - n LogB a2 Log@a + b xD x + F + PolyLogB2, d Ha + b xL - a+b x n M E c+d x Hb c - a dL Ha d Log@a + b xD - b c Log@c + d xDL c2 LogA + xE b + xF LogB n c+dx 2 b2 Hb c - a dL a b2 g a+bx g x2 LogBe a2 LogA - a d LogB LogAe I b2 d2 x LogB F b c-a d F d Ha+b xL Hb c - a dL2 g n2 Log@c + d xD d2 H- b c + a dL c 2 a d LogB ad b F c+d x E d a + xE LogB1 - + 2 c+dx N + x LogA 1 + a+bx bd a b 2 F - Hb c - a dL d2 c2 LogA F 2 c Log@c+d xD + Hb f - a gL2 n LogB- d Ha+b xL F b Hc+d xL b Hc + d xL bc-ad F b Hc+d xL F d Ha+b xL 2.2 Logarithm Functions.nb 1 a d n2 LogB bd a 2 + xF + b c n2 LogB b c 2 + xF - 2 a d n2 LogB d a+bx 2 a d n Log@a + b xD LogBe c+dx a+bx 2 b c n LogBe c+dx n a + xF Log@a + b xD + 2 a d n2 LogB b n F + b d x LogBe a+bx c+dx F Log@c + d xD - 2 b c n2 LogB a + xF Log@a + b xD - 2 a d n2 LogB d F + 2 b c n2 LogB n 2 + xF LogB b c b Hc + d xL bc-ad a F - 2 b c n2 PolyLogB2, d Ha + b xL -b c + a d c : - a+b x n 2 M E c+d x F - 2 a d n2 PolyLogB2, , x, 7, 0> f+gx LogAe I a+b x n 2 M E c+d x 2 n LogAe I LogB g a+b x n M E c+d x b c-a d F b Hc+d xL + PolyLogB2, g LogAe I a+b x n 2 M E c+d x LogB g a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL F Hb c-a dL Hf+g xL Hb f-a gL Hc+d xL F - 2 n2 PolyLogB3, + g 2 n LogAe I d Ha+b xL F b Hc+d xL a+b x n M E c+d x PolyLogB2, g 2 n2 PolyLogB3, - d Ha+b xL F b Hc+d xL + a Hd f-c gL+b d f x-b c g x g d Ha + b xL -b c + a d + xF Log@c + d xD d Problem ð553: Valid but suboptimal antiderivative: LogAe I + xF LogB d + xF Log@c + d xD - 2 b c n2 LogB b c Hb f-a gL Hc+d xL F b Hc + d xL bc-ad F+ F 109 110 2.2 Logarithm Functions.nb 1 - n2 LogB g -b c + a d d Ha + b xL a 2 n LogB F LogB a n2 LogB 2 g Hc + d xL -d f + c g n2 LogB b Hf + g xL bf-ag F LogB 2 Hd f - c gL Ha + b xL c d n2 LogB g Hc + d xL -d f + c g n2 LogB F LogB 2 n c+dx F LogB 2 Hd f - c gL Ha + b xL a+bx n 2 n LogBe c+dx F LogB F + n LogB Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL b b df-cg F - 2 n2 LogB a+bx d c+dx n a+bx -d f + c g a b -d f + c g Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL c + xF LogB b Hf + g xL bf-ag c + xF LogB + xF LogB b F PolyLogB2, Hd f - c gL Ha + b xL g Hc + d xL F LogB Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL F + 2 n LogBe Hb f - a gL Hc + d xL d Hf + g xL -d f + c g F LogB F LogB df-cg a a+bx c+dx g Hc + d xL -d f + c g d F + 2 n2 LogB Hd f - c gL Ha + b xL d + xF Log@f + g xD + n2 LogB F Log@f + g xD + LogBe Hb f - a gL Hc + d xL + xF LogB g Hc + d xL Hd f - c gL Ha + b xL F LogB F LogB F - 2 n2 LogB H- b c + a dL Hf + g xL n + xF LogBe g Hc + d xL a b c F + 2 n2 LogB d Hf + g xL F PolyLogB2, 2 + xF Log@f + g xD - 2 n2 LogB c+dx F + 2 n2 LogB F LogB a + xF LogBe bf-ag df-cg 2 2 a b Hf + g xL d Hf + g xL Hb f - a gL Hc + d xL 2 n2 LogB F + 2 n LogB bf-ag a+bx + xF LogBe F + n2 LogB F Log@f + g xD + 2 n LogB b Hf + g xL Hb f - a gL Hc + d xL 2 n LogB n c+dx + xF LogB b Hd f - c gL Ha + b xL a+bx + xF LogBe b n2 LogB Hb f - a gL Hc + d xL n F + n LogB F - 2 n2 LogB F + 2 n2 PolyLogB3, F Log@f + g xD - c+dx + xF LogB b b Hf + g xL bf-ag df-cg F LogB d a d Hf + g xL g Hc + d xL -d f + c g F- c 2 + xF LogB d df-cg F+ Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL d Ha + b xL F LogB b Hf + g xL d Hf + g xL df-cg F+ d Hf + g xL b Hc + d xL 2 + xF Log@f + g xD - n 2 a+bx F - n2 LogB c F PolyLogB2, F PolyLogB2, F - 2 n2 PolyLogB3, bf-ag F- g Ha + b xL -b f + a g b Hc + d xL d Ha + b xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL Problem ð554: Valid but suboptimal antiderivative: : LogAe I a+b x n 2 M E c+d x Hf + g xL2 Ha + b xL LogAe I , x, 3, 0> a+b x n 2 M E c+d x Hb f - a gL Hf + g xL + 2 Hb c - a dL n LogAe I - b d f2 n2 LogB a 2 + xF + b c f g n2 LogB LogB Hb f - a gL Hd f - c gL 1 g H- b f + a gL H- d f + c gL Hf + g xL a+b x n M E c+d x a 2 Hb c-a dL Hf+g xL Hb f-a gL Hc+d xL + xF - b d f g n2 x LogB a F + 2 Hb c - a dL n2 PolyLogB2, Hd f-c gL Ha+b xL Hb f-a gL Hc+d xL Hb f - a gL Hd f - c gL 2 + xF + b c g2 n2 x LogB a 2 + xF + 2 b d f2 n2 LogB F a c + xF d 2 a c a c a c c 2 a d f g n2 LogB + xF LogB + xF + 2 b d f g n2 x LogB + xF LogB + xF - 2 a d g2 n2 x LogB + xF LogB + xF - b d f2 n2 LogB + xF + b d b d b d d 2 2 2 c c c a a+bx n a d f g n2 LogB + xF - b d f g n2 x LogB + xF + a d g2 n2 x LogB + xF + 2 b d f2 n LogB + xF LogBe Fd d d b c+dx b b b + b - F- + xF LogB b - F- F+ F 2.2 Logarithm Functions.nb a + xF LogBe b 2 b d f2 n LogB c+dx c a+bx n + xF LogBe d 2 a d g2 n x LogB c+dx c a+bx + xF LogBe d a c g2 LogBe n a+bx 2 b c f g n LogB c+dx a + xF LogB -d f + c g -b c + a d 2 a d f g n2 LogB 2 F LogB d Ha + b xL g Hc + d xL 2 a d g2 n2 x LogB g Hc + d xL a + xF LogB b Hd f - c gL Ha + b xL a+bx n c+dx a+bx 2 b c g2 n x LogBe F LogB n c+dx 2 b d f g n2 x LogB g Hc + d xL -d f + c g 2 a d f g n2 LogB a + xF LogB b 2 b d f2 n LogBe a+bx n c+dx 2 a d g2 n x LogBe a+bx c+dx F LogB bf-ag n df-cg F LogB df-cg b g Hc + d xL -d f + c g 2 F + b c f g n2 LogB + xF LogB b a g Hc + d xL -d f + c g g Hc + d xL -d f + c g F + 2 b d f2 n2 LogB n -d f + c g -d f + c g F LogB bf-ag n F LogB g Hc + d xL -d f + c g c+dx g Hc + d xL g Hc + d xL -d f + c g Hd f - c gL Ha + b xL F - a d f g n2 LogB 2 a + xF LogB b 2 F+ F+ F+ -d f + c g Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL b Hf + g xL + xF LogB b F- b Hf + g xL bf-ag F + 2 b c f g n2 LogB a+bx -d f + c g a d Hf + g xL df-cg F + 2 b d f g n x LogBe F - 2 b c f g n2 LogB F LogB b + xF LogB a+bx c+dx g Hc + d xL -d f + c g F- F LogB + xF LogB a b n c+dx g Hc + d xL F - 2 b d f2 n2 LogB + g Hc + d xL F + 2 F+ F - 2 b d f g n x LogBe d Hf + g xL F+ F - a d f g n2 LogB bf-ag a F- F - n 2 c+dx Hd f - c gL Ha + b xL F + 2 a d g2 n2 x LogB df-cg a+bx Hb f - a gL Hc + d xL Hb f - a gL Hc + d xL bf-ag df-cg F- Hd f - c gL Ha + b xL b Hf + g xL d Hf + g xL F LogB d Hb f - a gL Hc + d xL bf-ag n a+bx + xF LogBe Hd f - c gL Ha + b xL b Hf + g xL d Hf + g xL c F - 2 a d g2 n2 x LogB F LogB df-cg c+dx F LogB b Hf + g xL F LogB c+dx Hb f - a gL Hc + d xL F + 2 b d f2 n2 LogB g Hc + d xL a+bx F LogB 2 F LogB g Hc + d xL b F LogB Hd f - c gL Ha + b xL b Hf + g xL b -d f + c g Hb f - a gL Hc + d xL c+dx + xF LogB b d Ha + b xL bf-ag a+bx + xF LogB F + b c f g n2 LogB d Ha + b xL Hd f - c gL Ha + b xL a a n a+bx + xF LogBe F + a d f g LogBe c+dx -b c + a d a n 2 a+bx -b c + a d Hb f - a gL Hc + d xL F + 2 b c g2 n2 x LogB - F - 2 b d f g n x LogB F + 2 a d f g n2 LogB + xF LogB F - 2 a d f g n LogBe d Hf + g xL n F - 2 b c g2 n x LogB F + b c f g LogBe n 2 F - 2 b c f g n2 LogB F - 2 b d f g n2 x LogB d Hf + g xL a F - 2 b d f2 n2 LogB b Hf + g xL df-cg a+bx F + 2 b c f g n LogBe bf-ag d Hf + g xL c+dx F + 2 b d f g n2 x LogB b Hf + g xL F LogB d F - 2 b c f g n2 LogB F - a d g2 n2 x LogB 2 bf-ag a+bx + xF LogBe F - 2 b c g2 n2 x LogB Hd f - c gL Ha + b xL b Hf + g xL F LogB c F - 2 b c g2 n2 x LogB Hb f - a gL Hc + d xL bf-ag c+dx g Hc + d xL -d f + c g Hd f - c gL Ha + b xL b Hf + g xL b c+dx g Hc + d xL n a+bx + xF LogBe F + 2 a d g2 n2 x LogB Hb f - a gL Hc + d xL F LogB a -d f + c g Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL 2 b d f2 n LogBe b Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL 2 a d f g n2 LogB + xF LogB Hb f - a gL Hc + d xL F LogB F LogB -d f + c g b c g2 n2 x LogB a F - a d g2 n2 x LogB -b c + a d -d f + c g g Hc + d xL -d f + c g d Ha + b xL 2 a d g2 n2 x LogB F - b d f2 LogBe F - 2 b c f g n2 LogB g Hc + d xL 2 a d f g n2 LogB n n 2 b b c g2 n2 x LogB F + 2 a d f g n LogB c+dx a+bx 2 b c g2 n2 x LogB F + 2 b d f g n x LogB n F + 2 b Hf + g xL bf-ag b Hf + g xL bf-ag d Hf + g xL df-cg F+ F LogB F LogB F+ F- d Hf + g xL df-cg d Hf + g xL df-cg F+ F- F+ 111 112 2.2 Logarithm Functions.nb 2 b d f g n2 x LogB g Hc + d xL -d f + c g F LogB d Hf + g xL df-cg 2 Hb c - a dL g n2 Hf + g xL PolyLogB2, 2 a d f g n2 PolyLogB2, b Hc + d xL d Ha + b xL F - 2 b c g2 n2 x LogB g Ha + b xL -b f + a g g Hc + d xL -d f + c g F LogB d Hf + g xL df-cg F - 2 Hb c - a dL g n2 Hf + g xL PolyLogB2, b Hc + d xL F - 2 b c g2 n2 x PolyLogB2, d Ha + b xL F+ g Hc + d xL -d f + c g F + 2 a d g2 n2 x PolyLogB2, F - 2 b c f g n2 PolyLogB2, b Hc + d xL d Ha + b xL F Problem ð555: Valid but suboptimal antiderivative: : LogAe I a+b x n 2 M E c+d x Hf + g xL3 , x, 17, 0> Hb c - a dL g n Ha + b xL LogAe I a+b x n M E c+d x Hb f - a gL2 Hd f - c gL Hf + g xL d2 n LogAe I a+b x n M E c+d x LogB g Hd f - c gL2 b c-a d F b Hc+d xL - b2 n LogB- LogAe I g Hb f - a gL2 - Hb c - a dL2 g n2 Log@c + d xD Hb f - a gL2 Hd f - c gL2 Hb c - a dL H2 b d f - b c g - a d gL n2 LogB- Hb f - a gL Hd f - c gL 2 Hb c - a dL H2 b d f - b c g - a d gL n2 LogB- g Ha+b xL b f-a g 2 g Hc+d xL Hb f - a gL2 Hd f - c gL2 d f-c g Hb c - a dL H2 b d f - b c g - a d gL n2 PolyLogB2, * * * b c-a d F d Ha+b xL Hb f - a gL2 Hd f - c gL2 F Log@f + g xD F Log@f + g xD b Hf+g xL b f-a g F + + + a+b x n M E c+d x - LogAe I 2 g Hf + g xL2 :Hf + g xL2 LogBe a+bx c+dx F , x, 21, 0> n 3 + Hb c - a dL2 g n2 Log@f + g xD Hb f - a gL2 Hd f - c gL2 - Hb c - a dL H2 b d f - b c g - a d gL n LogAe I Hb f - a gL Hd f - c gL 2 d2 n2 PolyLogB2, + d Ha+b xL F b Hc+d xL g Hd f - c gL2 2 + Hb f - a gL2 Hd f - c gL2 * * * a+b x n M E c+d x b2 n2 PolyLogB2, Log@f + g xD b Hc+d xL F d Ha+b xL g Hb f - a gL2 Hb c - a dL H2 b d f - b c g - a d gL n2 PolyLogB2, Result of integration not displayed since its leaf count is 18299 Problem ð556: Valid but suboptimal antiderivative: a+b x n 2 M E c+d x d Hf+g xL d f-c g F + - b Hc + d xL d Ha + b xL F+ 2.2 Logarithm Functions.nb Hb c - a dL2 g2 n2 Ha + b xL LogAe I a+b x n M E c+d x b3 d2 Hb c - a dL g2 n x2 LogAe I 2bd Hb f - a gL3 n LogB- a+b x n 2 M E c+d x b c-a d F d Ha+b xL b3 g - - a2 Hb c - a dL g2 n2 LogB- b3 d a+b x n 2 M E c+d x Hf + g xL3 LogAe I + 3g b3 d3 + 2 Hd f - c gL3 n2 LogAe I 2 Hb f - a gL3 n2 LogAe I a+b x n M E c+d x PolyLogB2, d3 g a+b x n M E c+d x PolyLogB2, b3 g d Ha+b xL F b Hc+d xL a+bx c+dx g x2 ILogAe I n F - n LogB a+b x n M E c+d x a+bx c+dx a+bx c+dx - n LogA F LogBe F 2 a+b x EM c+d x bd a+bx c+dx c LogB + xF LogB d a+bx c+dx b2 d2 d Ha + b xL a+bx c+dx a+bx c+dx n F - n LogB a+bx 2bd a+b x n M E c+d x a+bx c+dx - n LogA F 2 a+b x EM c+d x n a+bx c+dx F + n LogB + xF LogB b a+bx c+dx - - F b Hc+d xL F d Ha+b xL d Ha+b xL F b Hc+d xL - d Ha+b xL F b Hc+d xL - 2 Hb f - a gL3 n3 PolyLogB3, b3 g b Hc+d xL F d Ha+b xL + 2 1 + g2 x3 LogBe 3 a+bx c+dx n F - n LogB a+bx c+dx F F + n Hb c - a dL g H- 3 b d f + b c g + a d gL - b2 d2 f2 LogB a+b x n M E c+d x - n LogA a+b x EMM c+d x 3 1 + b2 d2 a+bx c+dx F + - c 2 + xF + b c LogB a 2 + xF - 2 a d LogB d a+bx F + b d x LogB c+dx b Hc + d xL bc-ad F b c-a d F b Hc+d xL b c-a d F b Hc+d xL b3 d3 2 a+b x EM c+d x b a LogB LogB Log@c + d xD a a d LogB F + 2 a d Log@a + b xD LogB - - I- b c g n + a d g n + 2 b d f ILogAe I F Log@c + d xD - 2 b c LogB 3 f g n2 - LogBe n a+b x n 2 M E c+d x a+b x n M E c+d x 2 Hb c - a dL2 g H3 b d f - b c g - a d gL n3 PolyLogB2, 2 Hd f - c gL3 n3 PolyLogB3, - n LogA + d3 g d Ha+b xL F b Hc+d xL c+dx F + n LogB -b c + a d 2 b c LogB 1 n Hd f - c gL3 n LogAe I a2 Hb c - a dL g2 n3 PolyLogB2, a+b x n M E c+d x a+b x n 2 M E c+d x b d3 d3 g b2 d2 f2 LogBe d3 3 f2 n2 - LogBe + 2 c I3 d2 f2 - 3 c d f g + c2 g2 M n ILogAe I 1 - + - c2 Hb c - a dL g2 n2 LogAe I b3 d b Hc+d xL F d Ha+b xL b3 n x I3 f2 + 3 f g x + g2 x2 M LogB + a+b x n 3 M E c+d x b c-a d F b Hc+d xL c2 Hb c - a dL g2 n3 PolyLogB2, a I3 b2 f2 - 3 a b f g + a2 g2 M n Log@a + b xD ILogAe I x LogBe LogB b d3 a+b x n M E c+d x a+b x n M E c+d x Hb c - a dL g H3 b d f - b c g - a d gL n Ha + b xL LogAe I 2 Hb c - a dL2 g H3 b d f - b c g - a d gL n2 LogAe I b3 d3 LogAe I b3 d2 LogAe I Hb c - a dL3 g2 n3 Log@c + d xD b c-a d F d Ha+b xL 113 - a2 d2 LogB a b a+bx c+dx F + 2 b c LogB 2 F - 2 b c PolyLogB2, 2 + xF - b2 c2 LogB b c + xF Log@a + b xD + 2 a d LogB c a b d Ha + b xL -b c + a d 2 + xF Log@a + b xD - 2 a d d c + xF Log@c + d xD - 2 b c LogB F - 2 a d PolyLogB2, d b Hc + d xL bc-ad F - + xF - 2 a b c d Log@a + b xD + 2 a2 d2 Log@a + b xD + d + + xF Log@c + d xD - - + 114 2.2 Logarithm Functions.nb d Ha + b xL F+ d -b c + a d c+dx a+bx a+bx a+bx 2 2 a b d2 x LogB F - 2 a2 d2 Log@a + b xD LogB F + b2 d2 x2 LogB F + 2 b2 c2 Log@c + d xD - 2 a b c d Log@c + d xD c+dx c+dx c+dx a c a+bx a b Hc + d xL 2 b2 c2 LogB + xF Log@c + d xD + 2 b2 c2 LogB + xF Log@c + d xD + 2 b2 c2 LogB F Log@c + d xD + 2 b2 c2 LogB + xF LogB F+ b d c+dx b bc-ad d Ha + b xL b Hc + d xL 1 a+bx n a+bx 2 b2 c2 PolyLogB2, F + 2 a2 d2 PolyLogB2, F g2 n2 - LogBe F + n LogB F -b c + a d bc-ad c+dx c+dx b3 d3 2 2 a c b3 c2 d x - 2 a b2 c d2 x + a2 b d3 x + a3 d3 LogB + xF + b3 c3 LogB + xF + 2 a b2 c2 d Log@a + b xD + a2 b c d2 Log@a + b xD - 3 a3 d3 Log@a + b xD b d 2 a2 d2 LogB a + xF Log@a + b xD - 2 a2 d2 LogB b + xF Log@a + b xD + 2 a2 d2 LogB c + xF LogB d d Ha + b xL F - 2 b2 c d x LogB a+bx F-b c + a d c+dx a+bx a+bx a+bx a+bx 2 2 a2 b d3 x LogB F - b3 c d2 x2 LogB F + a b2 d3 x2 LogB F + 2 a3 d3 Log@a + b xD LogB F + b3 d3 x3 LogB F c+dx c+dx c+dx c+dx c+dx a c 3 b3 c3 Log@c + d xD + a b2 c2 d Log@c + d xD + 2 a2 b c d2 Log@c + d xD + 2 b3 c3 LogB + xF Log@c + d xD - 2 b3 c3 LogB + xF Log@c + d xD b d a+bx a b Hc + d xL d Ha + b xL b Hc + d xL 2 b3 c3 LogB F Log@c + d xD - 2 b3 c3 LogB + xF LogB F - 2 b3 c3 PolyLogB2, F - 2 a3 d3 PolyLogB2, F + c+dx b bc-ad -b c + a d bc-ad 2 a3 d3 LogB a + xF Log@a + b xD + 2 a3 d3 LogB b 1 f2 n3 LogB bd a+bx c+dx f g n3 LogB a+bx c+dx F 2 d Ha + b xL LogB a+bx c+dx d Ha + b xL b Hc + d xL F d2 Ia2 - b2 x2 M LogB F - b3 + xF Log@a + b xD - 2 a3 d3 LogB g2 n3 6 b3 c3 LogB bc-ad d3 c+dx bc+bdx 2 c+dx + xF LogB F + 6 Hb c - a dL LogB F + 2 b3 c2 d x LogB a+bx c+dx F PolyLogB2, d2 F + 6 Hb c - a dL2 LogB a+bx F PolyLogB2, F - 18 a b2 c2 d LogB bc-ad c+dx bc-ad bc+bdx d Ha + b xL b Hc + d xL F + 3 Hb c - a dL LogB c+dx bc-ad c+dx F - 6 a3 d3 LogB a+bx c+dx a+bx a+bx a+bx F + 6 b3 c2 d x LogB d Ha + b xL b Hc + d xL F+ F d Ha + b xL + Hb c + a dL LogB F + I- 6 b2 c2 + 6 a2 d2 M PolyLogB3, F + 18 a2 b c d2 LogB F + 6 a3 d3 LogB a+bx a+bx d Ha + b xL b Hc + d xL bc-ad c+dx F+ - a+bx F - 12 a b2 c d2 x LogB F + F+ c+dx c+dx c+dx c+dx a+bx 2 a+bx 2 a+bx 2 a+bx 2 6 a2 b d3 x LogB F + 6 a b2 c2 d LogB F + 3 a2 b c d2 LogB F - 9 a3 d3 LogB F + 6 b3 c2 d x LogB F c+dx c+dx c+dx c+dx c+dx a+bx 2 a+bx 2 a+bx 2 a+bx 3 a+bx 3 6 a2 b d3 x LogB F - 3 b3 c d2 x2 LogB F + 3 a b2 d3 x2 LogB F + 2 a3 d3 LogB F + 2 b3 d3 x3 LogB F + c+dx c+dx c+dx c+dx c+dx a+bx bc-ad a+bx bc-ad a+bx bc-ad 18 b3 c3 LogB F LogB F - 18 a b2 c2 d LogB F LogB F - 18 a2 b c d2 LogB F LogB F+ c+dx bc+bdx c+dx bc+bdx c+dx bc+bdx a+bx bc-ad a+bx 2 bc-ad a+bx 2 bc-ad 18 a3 d3 LogB F LogB F + 6 b3 c3 LogB F LogB F - 6 a3 d3 LogB F LogB F+ c+dx bc+bdx c+dx bc+bdx c+dx bc+bdx 6 a b2 c2 d LogB F - 12 a2 b c d2 LogB bc-ad 1 c+dx c d F + 3 Hb c - a dL LogB b2 a+bx 6 Hb c - a dL b c - a d + Hb c + a dL LogB 1 c d a+bx H- 6 b c + 6 a dL PolyLogB3, 6 c a+bx bc-ad bc+bdx F + 2.2 Logarithm Functions.nb 6 3 Hb c - a dL2 Hb c + a dL + 2 Ib3 c3 - a3 d3 M LogB a+bx c+dx F PolyLogB2, d Ha + b xL b Hc + d xL F - 12 Ib3 c3 - a3 d3 M PolyLogB3, d Ha + b xL b Hc + d xL F Problem ð557: Valid but suboptimal antiderivative: :Hf + g xL LogBe - a+bx c+dx F , x, 12, 0> n 3 3 Hb c - a dL g n Ha + b xL LogAe I 2 b2 d Hf + g xL2 LogAe I 2g a+b x n 3 M E c+d x - 3 Hb f - a gL2 n2 LogAe I d Ha+b xL F b Hc+d xL - b c-a d F d Ha+b xL 2 b2 g a+b x n M E c+d x LogB b2 d2 a+b x n M E c+d x b2 g + 3 Hb f - a gL2 n LogB- 3 Hb c - a dL2 g n2 LogAe I 3 Hb c - a dL2 g n3 PolyLogB2, b2 d2 a+b x n 2 M E c+d x 3 Hd f - c gL2 n2 LogAe I PolyLogB2, b Hc+d xL F d Ha+b xL + LogAe I b c-a d F b Hc+d xL a+b x n M E c+d x - a+b x n 2 M E c+d x 3 Hd f - c gL2 n LogAe I d2 g 3 Hd f - c gL2 n3 PolyLogB3, a+b x n 2 M E c+d x 2 d2 g PolyLogB2, d2 g + d Ha+b xL F b Hc+d xL d Ha+b xL F b Hc+d xL - LogB b c-a d F b Hc+d xL - 3 Hb f - a gL2 n3 PolyLogB3, b2 g b Hc+d xL F d Ha+b xL 115 116 2.2 Logarithm Functions.nb 1 2 3 Hb c - a dL g n x ILogAe I bd a+b x n M E c+d x 3 a2 g n Log@a + b xD ILogAe I b2 3 g n x2 LogB a+bx c+dx 6 c f n ILogAe I 1 F LogBe a+b x n M E c+d x 6 f n2 - LogBe bd a+bx n c + xF LogB d a+bx c+dx 1 3 g n2 - LogBe b2 d2 c+dx 2 a2 d2 LogB a n a+bx + 6 f n x LogB a+bx c+dx + a+bx c+dx F c+dx F 2 F LogBe c+dx 3 c2 g n ILogAe I a a d LogB a+b x n M E c+d x c 2 + xF + b c LogB a+bx c+dx a + xF LogB F b a+bx n c+dx F - n LogB - n LogA d2 a a+bx c+dx 2 a+b x EM c+d x F - a+bx c+dx 3 F 2 + + g x2 LogBe a+bx Log@c + d xD c+dx 2 + xF - b2 c2 LogB b c a+bx c+dx F 2 d Ha + b xL LogB H- 6 b c + 6 a dL PolyLogB3, g n3 LogB a+bx c+dx a+bx c+dx d Ha + b xL b Hc + d xL F d2 Ia2 - b2 x2 M LogB F + 3 Hb c - a dL LogB F - a+bx c+dx 6 Hb c - a dL b c - a d + Hb c + a dL LogB bc-ad bc+bdx 1 a 2 b d Ha + b xL -b c + a d d b Hc + d xL F - 2 a d PolyLogB2, bc-ad a+bx c+dx F PolyLogB2, Problem ð558: Valid but suboptimal antiderivative: bc+bdx d Ha + b xL b Hc + d xL + xF Log@c + d xD F - + xF - 2 a b c d Log@a + b xD + 2 a2 d2 Log@a + b xD + c + xF LogB d Ha + b xL F + 6 Hb c - a dL LogB bc-ad - c + xF Log@c + d xD - 2 b c LogB F - 2 b2 c d x LogB a+bx c+dx F PolyLogB2, b2 d2 F + 6 Hb c - a dL2 LogB 3 + xF Log@a + b xD - 2 a d a+bx d -b c + a d c+dx a+bx a+bx a+bx 2 2 a b d2 x LogB F - 2 a2 d2 Log@a + b xD LogB F + b2 d2 x2 LogB F + 2 b2 c2 Log@c + d xD c+dx c+dx c+dx a c a+bx 2 a b c d Log@c + d xD - 2 b2 c2 LogB + xF Log@c + d xD + 2 b2 c2 LogB + xF Log@c + d xD + 2 b2 c2 LogB F Log@c + d xD + b d c+dx a b Hc + d xL d Ha + b xL b Hc + d xL 1 2 b2 c2 LogB + xF LogB F + 2 b2 c2 PolyLogB2, F + 2 a2 d2 PolyLogB2, F + b bc-ad -b c + a d bc-ad bd 2 f n3 LogB c+dx F d 2 d + xF Log@a + b xD + 2 a2 d2 LogB a+bx c + xF Log@a + b xD + 2 a d LogB F + 2 b c LogB 2 F - n LogB - b a+bx n c+dx a F - 2 b c PolyLogB2, d 2 a+b x EM c+d x F - n LogB 2 F + b d x LogB c+dx b Hc + d xL - a2 d2 LogB + xF Log@a + b xD - 2 a2 d2 LogB a+bx d bc-ad c - n LogA + xF - 2 a d LogB b b n a+bx + 2 f x LogBe F + 2 a d Log@a + b xD LogB F + n LogB a+b x n M E c+d x b Log@c + d xD F Log@c + d xD - 2 b c LogB a+bx 6 a f n Log@a + b xD ILogAe I + 2 a+b x EM c+d x F - n LogB F + n LogB -b c + a d 2 b c LogB n 2 a+b x EM c+d x d Ha + b xL 2 a+b x EM c+d x - n LogA c+dx c+dx LogB a+b x n M E c+d x - n LogA d a+bx - n LogA F + 3 Hb c - a dL LogB a+bx c+dx d Ha + b xL b Hc + d xL F+ F+ F d Ha + b xL + Hb c + a dL LogB F + I- 6 b2 c2 + 6 a2 d2 M PolyLogB3, d Ha + b xL b Hc + d xL F bc-ad bc+bdx F + 2.2 Logarithm Functions.nb :LogBe F , x, 4, 0> n 3 a+bx c+dx Ha + b xL LogAe I b a+b x n 3 M E c+d x 6 Hb c - a dL n2 LogAe I + 3 Hb c - a dL n LogAe I a+b x n M E c+d x bd PolyLogB2, bd 3 a n Log@a + b xD ILogAe a+b x n I M E c+d x -n b a+bx n x LogBe c+dx 1 F - n LogB a+bx 3 n2 - LogBe bd c+dx c LogB + xF LogB d a+bx c+dx n3 LogB bd a+bx c+dx d Ha + b xL a+bx c+dx F 2 d Ha+b xL F b Hc+d xL 2 a+b x LogA EM c+d x F F + n LogB -b c + a d 2 b c LogB 1 n 3 - - a+bx c+dx F LogB a+bx c+dx d Ha + b xL b Hc + d xL F + bd F LogBe a+bx + 3 n x LogB c+dx a+b x n M E c+d x a a d LogB - n LogA d c 2 b a + xF LogB b d Ha+b xL F b Hc+d xL a+bx c+dx 2 a+b x EM c+d x n F - n LogB bc-ad F + 3 Hb c - a dL LogB bc+bdx + c + xF Log@a + b xD + 2 a d LogB b a+bx c+dx 2 a -b c + a d c + xF Log@c + d xD - 2 b c LogB b d Ha + b xL F + 6 Hb c - a dL LogB F - 2 a d PolyLogB2, a+bx c+dx F PolyLogB2, Problem ð559: Valid but suboptimal antiderivative: : - LogAe I a+b x n 3 M E c+d x , x, 11, 0> f+gx LogAe I a+b x n 3 M E c+d x 3 n LogAe I LogB g a+b x n 2 M E c+d x 6 n2 LogAe I a+b x n M E c+d x b c-a d F b Hc+d xL + PolyLogB2, g PolyLogB3, g LogAe I a+b x n 3 M E c+d x LogB g a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL F F Hb c-a dL Hf+g xL Hb f-a gL Hc+d xL + F 6 n2 LogAe I - 3 n LogAe I a+b x n M E c+d x 6 n3 PolyLogB4, g a+b x n 2 M E c+d x PolyLogB3, g d Ha+b xL F b Hc+d xL PolyLogB2, g d Ha+b xL F b Hc+d xL + xF Log@a + b xD - 2 a d d F + 2 b c LogB F - 2 b c PolyLogB2, bc-ad 2 a + xF - 2 a d LogB F + b d x LogB c+dx b Hc + d xL c+dx F - d a+bx a+bx Log@c + d xD 2 + xF + b c LogB F + 2 a d Log@a + b xD LogB d Ha + b xL LogB b c-a d F b Hc+d xL 6 Hb c - a dL n3 PolyLogB3, 3 c n ILogAe I F Log@c + d xD - 2 b c LogB H- 6 b c + 6 a dL PolyLogB3, a+b x n 2 M E c+d x d Ha+b xL F b Hc+d xL + - 6 n3 PolyLogB4, + g Hd f-c gL Ha+b xL Hb f-a gL Hc+d xL F d b Hc + d xL bc-ad d Ha + b xL b Hc + d xL F+ + xF Log@c + d xD F + 117 118 2.2 Logarithm Functions.nb 1 n a+bx LogBe g c+dx F - n LogB n a+bx 3 n LogBe c+dx a LogB + xF LogB b a+bx c+dx F - n LogB b Hf + g xL bf-ag 3 n2 - LogBe n a+bx c+dx F 3 Log@f + g xD + a+bx c+dx F F - LogB F + n LogB 2 a - LogB b c + xF LogB d a+bx c+dx c d Hf + g xL df-cg F a+bx + xF Log@f + g xD + LogB -b c + a d LogB d Ha + b xL + xF Log@f + g xD + LogB d g Ha + b xL F + PolyLogB2, F LogB -b f + a g Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL c+dx F - PolyLogB2, F - LogB 2 a F Log@f + g xD + g Hc + d xL -d f + c g 2 F + + xF Log@f + g xD + b F Log@f + g xD c+dx 2 a+bx 2 a b Hf + g xL a a+bx b Hf + g xL 2 LogB + xF LogB F Log@f + g xD - LogB F Log@f + g xD + LogB + xF LogB F - 2 LogB + xF LogB F LogB Fd c+dx c+dx b bf-ag b c+dx bf-ag a 2 LogB c + xF LogB b c d a+bx a 2 LogB + xF LogB b LogB b Hf + g xL bf-ag c LogB 2 g Hc + d xL -d f + c g LogB F + LogB F LogB 2 c+dx 2 LogB df-cg F LogB 2 c+dx 3 a+bx c+dx a+bx 6 LogB c+dx c d F + 2 LogB F PolyLogB2, 2 F PolyLogB3, 2 F - LogB -d f + c g b Hf + g xL bf-ag c+dx g Hc + d xL -d f + c g F LogB F LogB c+dx Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL b Hf + g xL bf-ag a df-cg b F + 2 LogB Hd f - c gL Ha + b xL c+dx -d f + c g F + LogB F + 2 LogB F - 2 PolyLogB3, Hb f - a gL Hc + d xL a+bx c+dx F + 6 PolyLogB4, c a + xF LogB b d Hf + g xL g Hc + d xL -d f + c g d Hf + g xL df-cg F- Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL d Ha + b xL a+bx c+dx F LogB df-cg b Hc + d xL F PolyLogB3, b Hc + d xL -d f + c g + xF LogB F LogB F + 3 LogB d Ha + b xL g Hc + d xL d Hb f - a gL Hc + d xL a+bx F - 2 LogB + xF LogB d Hf + g xL Hb c - a dL Hf + g xL F - 6 LogB Problem ð560: Valid but suboptimal antiderivative: 2 g Hc + d xL Hd f - c gL Ha + b xL 3 F LogB F - 2 LogB F - 2 LogB Hb f - a gL Hc + d xL F LogB a+bx + xF LogB b F PolyLogB2, a+bx a g Hc + d xL a+bx Hd f - c gL Ha + b xL F PolyLogB2, bc+bdx F LogB H- b c + a dL Hf + g xL Hd f - c gL Ha + b xL bc-ad F + LogB + xF LogB Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL F LogB bf-ag F + 2 LogB d Hf + g xL Hb f - a gL Hc + d xL a+bx 3 LogB F + LogB b Hf + g xL Hd f - c gL Ha + b xL df-cg Hd f - c gL Ha + b xL a+bx F LogB Hb f - a gL Hc + d xL d Hf + g xL Hb f - a gL Hc + d xL 2 LogB n3 LogB g Hc + d xL 2 + xF Log@f + g xD + 2 LogB d -d f + c g + xF LogB d LogB c + xF Log@f + g xD - LogB b Hc + d xL F+ F - 6 PolyLogB4, F LogB d Hf + g xL F PolyLogB2, F PolyLogB2, d Ha + b xL Hd f - c gL Ha + b xL df-cg F- g Ha + b xL F PolyLogB2, F + 2 PolyLogB3, 2 F+ Hb f - a gL Hc + d xL -b f + a g b Hc + d xL d Ha + b xL F+ F- Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL d Ha + b xL b Hc + d xL Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL F- F F - F 2.2 Logarithm Functions.nb : LogAe I a+b x n 3 M E c+d x Hf + g xL2 Ha + b xL LogAe I , x, 4, 0> a+b x n 3 M E c+d x Hb f - a gL Hf + g xL 6 Hb c - a dL n2 LogAe I * * * + 3 Hb c - a dL n LogAe I a+b x n M E c+d x a+b x n 2 M E c+d x LogB Hb f - a gL Hd f - c gL PolyLogB2, Hb f - a gL Hd f - c gL a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL Hb c-a dL Hf+g xL Hb f-a gL Hc+d xL F - F + 6 Hb c - a dL n3 PolyLogB3, a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL Hb f - a gL Hd f - c gL Result of integration not displayed since its leaf count is 5537 F * * * Problem ð561: Valid but suboptimal antiderivative: : LogAe I a+b x n 3 M E c+d x Hf + g xL3 , x, 19, 0> 3 Hb c - a dL g n Ha + b xL LogAe I a+b x n 2 M E c+d x 2 Hb f - a gL2 Hd f - c gL Hf + g xL 3 b2 n LogB- 3 Hb c - a dL H2 b d f - b c g - a d gL n LogAe I a+b x n 2 M E c+d x 2 Hb f - a gL2 Hd f - c gL2 3 Hb c - a dL H2 b d f - b c g - a d gL n LogAe I a+b x n 2 M E c+d x 2 Hb f - a gL2 Hd f - c gL2 3 Hb c - a dL H2 b d f - b c g - a d gL n2 LogAe I 3 b2 n2 LogAe I a+b x n M E c+d x Hb f - a gL2 Hd f - c gL2 PolyLogB2, g Hb f - a gL2 3 d2 n3 PolyLogB3, d Ha+b xL F b Hc+d xL g Hd f - c gL2 3 b2 n3 PolyLogB3, b Hc+d xL F d Ha+b xL g Hb f - a gL2 3 H- b c + a dL n ILogAe I a+b x n M E c+d x + - a+b x n M E c+d x b Hc+d xL F d Ha+b xL + b c-a d F d Ha+b xL LogAe I 2 g Hb f - a gL2 LogB LogB b c-a d F b Hc+d xL + Hb f-a gL Hc+d xL F 2 g Hf + g xL2 + d Ha+b xL F b Hc+d xL 3 d2 n2 LogAe I + + - LogB g Hd f - c gL2 Hb f - a gL2 Hd f - c gL2 3 b2 n Log@a + b xD ILogAe I Hb c-a dL Hf+g xL Hb f-a gL Hc+d xL 2 g H- b f + a gL2 Hb f-a gL Hc+d xL - F b c-a d F b Hc+d xL + F + a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL F - + a Hd f-c gL+b d f x-b c g x a+b x n M E c+d x LogB - Hb f-a gL Hc+d xL PolyLogB2, Hb f - a gL2 Hd f - c gL2 d Ha+b xL F b Hc+d xL d Ha+b xL F b Hc+d xL Hd f-c gL Ha+b xL Hb f - a gL2 Hd f - c gL2 a+b x n M E c+d x a+b x n 2 M E c+d x 2 g Hd f - c gL2 PolyLogB2, 3 Hb c - a dL2 g n3 PolyLogB2, 3 Hb c - a dL H2 b d f - b c g - a d gL n3 PolyLogB3, 2 a+b x EM c+d x + 3 d2 n LogAe I a+b x n M E c+d x a+b x n M E c+d x 3 Hb c - a dL H2 b d f - b c g - a d gL n2 LogAe I Hb f - a gL2 Hd f - c gL2 2 H- b f + a gL H- d f + c gL Hf + g xL a+b x n 3 M E c+d x Hb f - a gL2 Hd f - c gL2 3 Hb c - a dL H2 b d f - b c g - a d gL n3 PolyLogB3, - n LogA - LogAe I 3 Hb c - a dL2 g n2 LogAe I Hb c-a dL Hf+g xL PolyLogB2, a+b x n 2 M E c+d x - n LogA F 2 a+b x EM c+d x - + - 119 120 2.2 Logarithm Functions.nb 3 n LogA a+b x E c+d x ILogAe I a+b x n M E c+d x - n LogA 2 g Hf + g xL2 2 a+b x EM c+d x ILogAe I - 3 H- b c + a dL H- 2 b d f + b c g + a d gL n ILogAe I a+b x n M E c+d x a+b x n M E c+d x - n LogA 2 g Hf + g xL2 - n LogA 2 H- b f + a gL2 H- d f + c gL2 2 a+b x EM c+d x g I +xM g2 b n a+bx 3 n2 LogBe c+dx F - n LogB a+bx c+dx F 2 ag b M 3 1- 3 d2 n ILogAe I - g I +xM d I-f+ I- LogA cg d a b M 3 1- + g K +xO a I-f+ b ag -f+ b ag b M I +xM a b 4 1- g K +xO cg d g I +xM a b -f+ I- f + ag 2 b ag b M 2 g2 1 - I-f+ d -f+ I- f + cg d M d M 4 1- F g K +xO 2 c cg d g K +xO a 2 I-f+ b ag -f+ b a c+d x + g + xF b I- f + 2 + xM 1- ag b d -f+ cg - d + 2 I- f + cg d a I b M 2g c d M 3 g K +xO 1- 3 1- 3 c I d M 3 LogA + xE - g K +xO d cg I-f+ d cg -f+ d 1 g 2 I- f + ag b M LogB1 - + xM g K +xO a + xE - g I +xM b -f+ ag b - I- f + + xM c g I +xM c d -f+ cg d LogB -f+ b cg d M d F b ag ag b M + b 2 - 2 F a c - LogB + xF + LogB g I +xM a 4 1a b ag b ag b M 2 2 b -f+ g I +xM + F ag b I- f + ag b M I- f + g I +xM c LogB1 - d -f+ 2 I- f + cg d cg d M 2 F 1- g I +xM ag b a g I +xM b ag b M 2 F F - 2 + xF + b b 1 + g g + xF d c+dx ag c + LogB LogB b -f+ bx + c+dx + xM a 3 -f+ - a b a PolyLogB2, a + xF + LogB d 2gI + xM + xF + d I-f+ b ag 2 a b -f+ a 1- g2 I 1 + 1- g K +xO -f+ a b b 2 bx EM c+d x a g I +xM cg c b Log@c + d xD 2g -f+ c LogA d M ag 2g + xE + LogA + LogB I-f+ d -f+ 2 g Hf + g xL2 c I d 4 c d cg 2 g I +xM 2 g I +xM b + LogB1- d + 2 a+b x EM c+d x LogB1- a 2 c 2 d - c + xE + LogA LogB1 - g2 I +xM c - n LogA 2 g H- d f + c gL2 Log@f + g xD c c a+b x n M E c+d x a a I-f+ 3 a+b x EM c+d x I- f + cg d c I d M 3 g I +xM c LogB1 - + xM g I +xM c 1- d -f+ d -f+ cg d - I- f + cg d cg d M 2 F - 2.2 Logarithm Functions.nb g I +xM c PolyLogB2, d -f+ I- f + cg d M 2 LogB cg d F 1 f2 bf-ag LogB2 c LogB g b H- d f + c gL + xF - LogBd dgI c d c d bgI a b 2 g g Hb f - a gL J + xM + xM 2abx Hb f-a gL2 F + LogB- 1 c d + xM b H- d f + c gL I Hb f - a gL x J c d a I b + + xM dgI Hb f-a gL3 N - b b Hf + g xL2 + + xM aJ + + 2 b H-d f+c gL2 I +xM + a b Hf+g xL Hb f-a gL2 b Hf + g xL N - +x a b +x +x c d + xM a b + xM + a b c d a b N LogB- F - 2 LogB + xM - c d + xM a I b + xM F - LogB b H- d f + c gL I + xF + LogB- b F + PolyLogB3, bx b f-a g + xF + LogB b b Hf + g xL bf-ag + c d Hb f-a gL2 + xM d Hb f - a gL I c d b H- d f + c gL I 2cdx H-d f+c gL2 a b - d Hf + g xL + a b b H- d f + c gL N + xM + xM + xM c d c d b H- d f + c gL I c d + xM + xM a b + xM + xM a b + xM -d f + c g F - + xF - bx b f-a g + b Hf + g xL Hb f - a gL J a b Hf+g xL Hb f-a gL2 2abx Hb f-a gL2 + b Hf + g xL + F + d Hb f - a gL J N d Hf + g xL c + xF LogB b H-d f+c gL3 F F + a LogB F+ - + xM F + xM a I b d Hb f - a gL I 2 c2 d Hf+g xL c d F + LogB- d Hb f - a gL I b H- d f + c gL I a b Hf+g xL b H- d f + c gL I F + + xM dgI -d f + c g d Hb f - a gL I b Hf + g xL adI H- d f + c gL J a b a F - PolyLogB2, - aJ a H- b c + a dL Hf + g xL F + LogB +x Hb f-a gL2 c d -d f + c g c d a b Hf+g xL c H-b c+a dL Hf+g xL bx b f-a g + xM F - PolyLogB3, bx b f-a g dgI F - LogB- F PolyLogB2, b H- d f + c gL2 I +x a N + xM 2 bf-ag c d Hb f - a gL I a H-b c+a dL x Hb f-a gL2 a b a b LogB b H- d f + c gL bf-ag bgI 1 d Hb f - a gL I b Hf + g xL b Hf + g xL b H-d f+c gL I +xM a b Hf+g xL c d + xM Hb f - a gL x J b + c d -d f + c g H- b c + a dL Hf + g xL bx b f-a g + xM b H- d f + c gL I 2 a2 b Hf+g xL + xM - F + LogB F+ F LogB- + xM -d f + c g d Hb f - a gL I 2 b H- d f + c gL a b a I b c d F PolyLogB2, - + xM F - PolyLogB3, b Hf + g xL d Hb f - a gL I bd b H- d f + c gL + xM dgI -b c + a d LogB b Hf + g xL bf-ag F + LogB 2 d Hb f - a gL I bf-ag 1 -d f + c g F + xF LogB d d Hf + g xL + xM a I b -d f + c g PolyLogB3, - c + xF LogB b F - LogB- d Hb f - a gL I PolyLogB2, a 2 LogB b Hf + g xL 1 1 - 121 H- d f + c gL x J- N + 2 a2 b Hf+g xL Hb f-a gL3 dx -d f+c g d Hf + g xL2 + N + c d Hf+g xL H-d f+c gL2 N - 122 2.2 Logarithm Functions.nb c J- dx -d f+c g c d Hf+g xL H-d f+c gL2 + d Hf + g xL Hb f - a gL x J bx b f-a g + b Hf + g xL 2 b H- d f + c gL x I LogB- - 1 c d + xM a b Hf+g xL Hb f-a gL2 c d a b + xM dg LogB 2 - H-b c+a dL x a F +2 - -d f + c g Hb f - a gL J + b H-d f+c gL2 I +xM d Hb f - a gL I + xM H- d f + c gL2 dgI c d + xM 2 H- d f + c gL - 2abx Hb f-a gL2 + b Hf + g xL 2 a2 b Hf+g xL Hb f-a gL3 + c H-b c+a dL Hf+g xL b H-d f+c gL2 I +xM bx b f-a g + a d I +xM d b H-d f+c gL I +xM a + a b Hf+g xL Hb f-a gL2 N - a b c d -d f+c g c d g I +xM c d gL2 d I +xM c + d -d f+c g Hb f - a gL x J H-d f+c d Hb f-a gL K dgI bx b f-a g + b Hf + g xL2 + gL2 c d bx b f-a g a b H- b c + a dL Hf + g xL H-b c+a dL x b H-d f+c gL I +xM a + + a b Hf+g xL Hb f-a gL2 d I +xM d -d f+c g dgI bx b f-a g + dx -d f+c g c d JLogB a b Hf+g xL Hb f-a gL2 b Hf+g xL N + b f-a g + xM O + Hb f-a gL2 N - aJ a b + b c d g I +xM N c H-d f+c d gL2 d I +xM c + LogB- -d f+c g dgI c d + Hb f-a gL2 b Hf + g xL N d Hf+g xL -d f+c g c d Hf+g xL H-d f+c gL2 d a b H-d f+c gL I +xM b + xM dx -d f+c g d Hb f-a gL I +xM c d + d Hf + g xL a b Hf+g xL - a c d Hf+g xL dx - b H-d f+c gL3 I +xM H-d f+c gL2 -d f+c g N b H-d f+c gL2 I +xM d Hf+g xL bx b f-a g Hb f-a gL3 c H-b c+a dL Hf+g xL + F - LogB- H-d f+c gL K- 2 a2 b Hf+g xL 2 c2 H-b c+a dL Hf+g xL H- d f + c gL J- b Hf+g xL + xM a b Hf+g xL + H- d f + c gL - c + - b H-d f+c gL2 I +xM d Hf + g xL b Hf + g xL c 2 c H-b c+a dL x + xM H- d f + c gL JF + b Hf + g xL 2 H- b c + a dL Hf + g xL d c d g I +xM a b 2abx Hb f-a gL2 b d g I +xM Hb f - a gL J + xM F+ Hb f - a gL J 1 + xM - + xM H- d f + c gL - H-d f+c + xM c LogB b c d a b bcI a -d f + c g - + + xM b H- d f + c gL I b dI F + N N c b c d Hb f-a gL2 b Hf + g xL + c d b H- d f + c gL I a b Hf+g xL Hb f - a gL J d a cdgI + d Hb f - a gL I b Hf + g xL b H-d f+c gL I +xM -d f + c g F bx b f-a g c + xF + LogB + xM aJ c d Hb f-a gL I +xM + xM b 1 N 2 + xM a c I d F LogB- + xM H- b c + a dL Hf + g xL2 H- d f + c gL - - 2 LogB c d -d f + c g + xM - b H- d f + c gL I a b LogB dgI b d Hb f - a gL I b H- d f + c gL I dgI a b N FN c d Hf+g xL H-d f+c gL2 F + N + O + + H- d f + c gL J 2cdx H-d f+c gL2 - d Hf + g xL + 2 c2 d Hf+g xL H-d f+c gL3 N - 2.2 Logarithm Functions.nb H- d f + c gL x JH- d f + c gL LogB d Hf + g xL2 2 c2 d g I +xM c H-d f+c gL3 d dgI b Hf + g xL bf-ag H- d f + c gL d - g2 c d F - LogB- c d g I +xM c d c I d a b b H- d f + c gL I c d Hf+g xL N H-d f+c gL2 + 2 c d I +xM + c J- c d c - d Hf + g xL -d f + c g d I +xM F 2 d c d g I +xM H-d f+c gL2 d + xM g Hb f - a gL b H-d f+c gL3 d gL3 c d + xM I +xM a b H-d f+c gL2 I +xM + xM b H-d f+c gL2 c - gL2 + + I +xM b b H-d f+c gL2 dgI c d d I +xM - bcI LogB c d g I +xM H-d f+c gL2 d c - dgI + bf-ag c d F - LogB- d I +xM c + d -d f+c g - + xM d Hf + g xL -d f + c g F + + c d Hb f-a gL I +xM -d f+c g - a d I +xM c a b + xM dgI b H-d f+c gL2 d a I +xM b d Hb f - a gL I c d + xM -d f + c g d + xM + xM b Hf + g xL b c + d -d f+c g a a d c + d d b H-d f+c gL I +xM c d g I +xM LogB F + c H-d f+c gL2 b H-d f+c gL I +xM a a d I +xM c d F + xM c d d + xM c + b c d 2 a c d I +xM d d c a d I +xM c c d a H-d f+c + xM + xM c d Hb f-a gL I +xM 2 c d I +xM 2 b c d + xM d I +xM c 2 c d I +xM - c d c + b - c - b H-d f+c gL2 I +xM c I d 2 c2 d Hb f-a gL I +xM c d c d g I +xM H-d f+c gL2 d g2 I H-d f+c gL3 dgI -d f + c g H- d f + c gL - d dgI + xF + LogB b -d f + c g d a -d f+c g - 2 LogB c c d a 2 c2 d g I +xM dgI c d Hb f-a gL I +xM c d Hb f - a gL2 I d + xF + LogB d I +xM + - + xM b d Hb f - a gL I 2 c2 d g I +xM c d c d d I +xM -d f+c g a - 2 LogB N c + H- d f + c gL + c dgI d d + xM - H-d f+c c d g I +xM H-d f+c gL2 1 -d f+c g d2 c c d Hf+g xL H-d f+c gL2 d Hf + g xL dgI + c a b + c + + xM - a b dx -d f+c g c H-d f+c gL2 + xM a b H- d f + c gL I H- d f + c gL - + xM H-d f+c gL2 2 b H- d f + c gL2 I - dx -d f+c g c d c + d b H-d f+c gL I +xM a b - + xM F+ H- d f + c gL - c d g I +xM c H-d f+c d g2 I d c d gL2 + xM d I +xM c + d -d f+c g LogB- d Hb f - a gL I c d b H- d f + c gL I a b + xM + xM F 123 124 2.2 Logarithm Functions.nb - LogB b Hf + g xL bf-ag F + LogB- 2 b H- d f + c gL I 2bcI + xM - bdI a b d Hb f - a gL - b H-d f+c gL3 I +xM Hb f-a gL2 a b g I +xM Hb f-a gL2 b b gL2 H- d f + c gL2 - I +xM + b d Hb f - a gL I b H-d f+c b I +xM + xM d a I +xM b bgI a b c d g I +xM a d I +xM dgI b H-d f+c gL I +xM a b f-a g b b g I +xM b b f-a g + xM d I +xM c d -d f+c g + a b + xM a d I +xM c c + d Hb f-a gL I +xM d a b H-d f+c gL I +xM F + d b H-d f+c gL I +xM a b b f-a g + Hb f - a gL - F 2 - Hb f - a gL - d g I +xM c LogB1 - d -d f+c g 2 + xM adI c d + xM b H- d f + c gL I a b F a b b f-a g 2 a2 b g I +xM a Hb f-a gL3 b a Hb f-a b gL2 b I +xM a - b g2 I b a b dgI c d + xM -d f + c g c d F - - a b g I +xM a Hb f-a gL2 b + LogB 2 F + LogB Hb f-a gL2 2 + xF - LogBb g I +xM a b b f-a g + xM b g I +xM a b b f-a g a + xF + LogB- F b b f-a g + xM LogB1 + 2 b H- d f + c gL2 b a b b g I +xM a LogB1 + d b a b 2 b b f-a g F d Hb f - a gL I c + xM + xM b I +xM a - b2 g2 I 2 a b I +xM LogB1 + b f-a g d2 g Hb f - a gL I F a - bgI 1 + LogB1 + xM a b g I +xM b F b g I +xM LogB1 + b b I +xM b H-d f+c gL I +xM a b Hb f - a gL2 - b f-a g a c d b H- d f + c gL I a + b g I +xM LogB1 + + xM + xM Hb f-a gL2 b a d2 g2 I H- b c + a dL Hf + g xL d - b + xM + xM a b g I +xM b + b b c d LogB- a c d c c d b H-d f+c gL2 I +xM b f-a g a c d F - LogB- F d a b H-d f+c gL I +xM I +xM 2 c + + xM d a d Hb f-a gL I +xM d a 2 b c + d a b I +xM - b f-a g a b c d c gL2 I d b H-d f+c gL I +xM b c d Hb f-a gL I +xM b + xM 1 + H-d f+c gL2 b H- d f + c gL2 I gL2 a - b c d b c d Hb f-a gL I +xM a c d Hb f - a gL I b H-d f+c gL I +xM gL2 d Hb f-a gL I +xM c LogB- a b H-d f+c d2 Hb f - a + xM d bf-ag b I +xM - c d a d I +xM c + xM b a d I +xM b Hf + g xL a a d Hb f - a gL I d Hb f - a gL2 I b bg b H-d f+c gL2 I +xM c d Hb f-a gL I +xM 2 a b LogB- a c d a + xM F + LogB a I b d c a b a a b g I +xM Hb f-a gL2 a b g I +xM 2 H- d f + c gL - 2 a c d I +xM c - b b H-d f+c + xM 2 a c d Hb f-a gL I +xM + xM 1 2 c2 d Hb f-a gL I +xM c a b -b c + a d a - -d f + c g F + c a b 2 a b H- d f + c gL I LogB d Hf + g xL 2 b2 H- d f + c gL2 I + c d b H- d f + c gL I F + xM a b + xM c d + xM - + a +x b - cdgI c d + xM H- d f + c gL2 d Hb f - a gL I c d b H- d f + c gL I a b + xM + xM F + dI -d f + c g F 2.2 Logarithm Functions.nb - c d g I +xM H- d f + c gL c H-d f+c d dgI c d g I +xM c c - H-d f+c b2 H- d f + c gL2 I dgI a b c d d d -d f+c g F - c gL2 c d Hb f - a gL I b H- d f + c gL2 I d Hb f - a gL d a I +xM c d Hb f-a gL I +xM c d a b H-d f+c gL2 I +xM b a b H- d f + c gL I + xM + b H-d f+c gL I +xM a b + xM a b c d - c LogB1 + b H-d f+c gL I +xM a Hb f - a gL - a b g I +xM a Hb f-a gL3 b a Hb f-a gL2 d a b c d b H-d f+c gL2 I +xM b 2 a2 b g I +xM + xM 2 a b I +xM Hb f-a gL2 b bgI a b b I +xM b b f-a g b g2 I a b + xM - b a b b f-a g + xM b g I +xM a PolyLogB2, - b b f-a g + xM + F + F c d + xM a b + xM a F F F c d b H- d f + c gL I adI c d + xM a b + xM F - b H- d f + c gL b H- d f + c gL I 2 a c d I +xM d b H-d f+c gL2 I +xM a d Hb f - a gL I a b + xM LogB1 + b c d + xM c - - + d Hb f - a gL I + b H-d f+c gL3 I +xM d Hb f - a gL d Hb f-a gL I +xM c d a b H-d f+c gL I +xM b + xM F - - d Hb f-a gL I +xM d a b H-d f+c gL I +xM a b g I +xM a a - d -d f+c g + LogB- d b + xM b g I +xM c 2 c2 d Hb f-a gL I +xM c LogB1 + a d g I +xM b d a d PolyLogB2, - F d -d f+c g + xM c a b b H-d f+c gL I +xM a d I +xM c d c b H- d f + c gL2 I + xM d Hb f-a gL I +xM b H-d f+c gL I +xM a - a b LogB1 - gL2 + xM c d Hb f - a gL I c + c d c d LogB1 - -d f+c g d a b d Hb f - a gL2 I a - dgI b H-d f+c gL I +xM c d d H-d f+c d g I +xM c d - gL3 c 2 c d I +xM c d d Hb f-a gL I +xM + xM b H- d f + c gL I a d I +xM d I +xM + b 2 c d gL2 d g2 I + xM adI + xM d Hb f - a gL I H-d f+c d b b H- d f + c gL I + xM c d Hb f-a gL I +xM + xM H-d f+c c LogB1 + a - c d g I +xM 2 d c a b Hb f - a gL - b c d c I d a d I +xM c + b b H- d f + c gL I 1+ d - H- d f + c gL - + d2 Hb f - a gL2 I c + xM d g I +xM c d Hb f-a gL I +xM b H-d f+c -d f+c g c -d f+c g + xM 2 b a b H-d f+c 2 c2 d g I +xM H- d f + c gL d + gL2 d g I +xM LogB1 - + xM d I +xM c d c d -d f+c g a bcI - -d f+c g + xM 1 - c d g I +xM c d c c d d I +xM + gL2 +x d I +xM c - c d +x gL2 125 Hb f-a gL2 b b H- d f + c gL I a b b I +xM F + a - PolyLogB2, - b f-a g bgI + xM - b g I +xM a b a b b b f-a g + xM 2 c2 d Hb f-a gL I +xM b H-d f+c c d gL3 d Hb f - a gL I c d I +xM a b + xM F - 2 a c d I +xM c - d b H-d f+c gL2 I +xM a b + - 126 2.2 Logarithm Functions.nb bcI c d Hb f-a gL I +xM d Hb f - a gL I 2 c2 d g I +xM H-d f+c gL3 d c d g I +xM c H-d f+c gL2 d d b c d c d c c I d PolyLogB2, c d Hb f-a gL I +xM d a + xM b H-d f+c gL2 I +xM d Hb f - a gL I + xM - b H- d f + c gL I a b + xM a b d a b H-d f+c gL2 I +xM b b H-d f+c gL3 I +xM b c d Hb f-a gL I +xM b + xM b H-d f+c gL2 I +xM a a d I +xM I +xM b a d I +xM d b H-d f+c b H-d f+c gL I +xM a c d d gL2 d a b H-d f+c gL2 I +xM b + bcI c d d -d f+c g a b d a b H-d f+c gL2 I +xM d Hb f - a gL2 I c d + xM c I d d Hb f-a gL I +xM d a b H-d f+c gL I +xM c d d a b H-d f+c gL2 I +xM +x a b +x d a b H-d f+c gL I +xM F - - d Hb f-a gL I +xM d a b H-d f+c gL I +xM b F - F- d b H-d f+c gL2 I +xM a b + xM a d I +xM - c d b H-d f+c gL I +xM a a d I +xM PolyLogB2, dgI a b c d + xM -d f + c g d b H-d f+c gL I +xM - + xM F - PolyLogB2, - c PolyLogB2, - c I d + xM bf-ag c + a b c - b c bgI + b + xM d Hb f-a gL I +xM F + b c d Hb f-a gL I +xM PolyLogB2, c b b H-d f+c gL3 I +xM d Hb f - a gL b b H-d f+c gL I +xM d PolyLogB2, - 2 a c d I +xM c a b c a 2 c2 d Hb f-a gL I +xM + xM - c d F d -d f+c g + xM b PolyLogB2, - d c d d Hb f - a gL + xM + xM a d I +xM dgI d g I +xM c PolyLogB2, d Hb f - a gL2 I b a + xM c I +xM + xM c d c d Hb f-a gL I +xM + xM a b d b H-d f+c gL I +xM b c a b PolyLogB2, - a b d Hb f - a gL I + a d Hb f - a gL I c d Hb f-a gL I +xM H-d f+c gL2 c - d I +xM c c d d b H-d f+c gL I +xM c + - a d I +xM + b d Hb f - a gL2 I c d g I +xM a b H- d f + c gL I 2 a c d I +xM c d + - b + xM a b H-d f+c gL2 I +xM c + b H-d f+c gL2 I +xM gL3 d a + xM b H- d f + c gL I d c d a c d - b + xM c a b - c d Hb f-a gL I +xM c a b c c - c d c I d F F 2 a c d I +xM c a b H-d f+c - + xM 2 c2 d Hb f-a gL I +xM c d Hb f-a gL I +xM a b H- d f + c gL I b H-d f+c gL I +xM 2 c2 d Hb f-a gL I +xM c + xM c d d d Hb f - a gL2 I + xM - -d f+c g a d I +xM d Hb f - a gL a b d c + b a b d g I +xM + xM c a b -d f+c g c d -d f+c g g2 d PolyLogB2, + xM d I +xM + c H-d f+c gL2 dgI a b H- d f + c gL I d g I +xM c d - + + xM 2 c d I +xM c a b H- d f + c gL I a b b H-d f+c gL I +xM a b b H- d f + c gL I bcI d + b H-d f+c gL2 I +xM H- d f + c gL - - c d a + xM H- d f + c gL bcI a d I +xM c a b d Hb f - a gL I c d b H- d f + c gL I a b + xM + xM F - F+ 2.2 Logarithm Functions.nb 1 2 g2 Hb f - a gL J bx b f-a g H- d f + c gL J- - Hb f - a gL J a b Hf+g xL Hb f-a gL2 + b Hf + g xL dx -d f+c g bx b f-a g Hb f - a gL J + bx b f-a g a b Hf+g xL Hb f-a gL2 + b Hf + g xL 2 H- d f + c gL cdgI c d c d g I +xM H-d f+c gL2 d + xM 2 b H- d f + c gL H- d f + c gL a +x b LogB- + dI N a b Hf+g xL d I +xM + xE LogA d -d f+c g N LogB c d g I +xM c H-d f+c d gL2 + d g I +xM d -d f+c g c I d - 2 LogB c d -d f+c g b H- d f + c LogB- + xM + + xM d a b H-d f+c gL I +xM b H- d f + c gL I c d g I +xM c d gL2 + a b + xM d I +xM c d -d f+c g b + xM c d c d F - LogB- + xM c d + xM F a I b F J- LogB a I b Hb f - a gL - a b g I +xM a Hb f-a b gL2 b f-a g b b f-a g a b dg + xE + LogB- d Hb f-a gL I +xM c I d d a b H-d f+c gL I +xM b + xM + xM + xM F LogBb -d f+c g FN 2dgI b H- d f + c gL dgI c d + xM -d f+c g c d + xM a I b + xM FN F LogB1 - + xM d g I +xM c d -d f+c g F -d f + c g F - b Hf + g xL bf-ag + xM F+ c d + xM a I b + xM F + 2 1 d Hb f - a gL I F + LogB- 1 - d Hb f - a gL I -b c + a d bd a I b + xM d Hb f-a gL I +xM c d a b H-d f+c gL I +xM c d + xM F LogB1 + d Hb f - a gL I c d + xM c d + xM d Hf + g xL -d f + c g b H- d f + c gL F + LogB 1 - + xM H- d f + c gL b a b a I b b H- d f + c gL d Hf + g xL F LogB + xE - LogB- bgI + xM F - LogB d Hf+g xL F + LogB- c d c d d Hb f - a gL I LogB- F - LogB- -d f + c g c d d Hb f - a gL I 1 + b Hf + g xL LogA F + a b H- d f + c gL b I +xM + b H-d f+c gL2 I +xM d Hb f - a gL I a - c d bf-ag b Hf+g xL + xM c d N c H-b c+a dL Hf+g xL d Hf+g xL LogB c LogA + + xM LogB- dgI dgI LogB + xM b H- d f + c gL F + b f-a g Hb f-a gL2 -d f + c g a b H- d f + c gL c adI LogB H- b c + a dL Hf + g xL adI d Hb f-a gL I +xM c I d N H-b c+a dL x b Hf+g xL a b Hf+g xL -d f + c g b H-d f+c gL I +xM -d f + c g + b b + xF + LogB + xM a I b + xM H- b c + a dL Hf + g xL H-d f+c c d + xF + LogB c d Hf+g xL + xM - dgI + a H-d f+c gL2 + + xM dg a gL2 I b a b b c d Hb f - a gL I c d dx -d f+c g a d I +xM F - 2 LogB + xM F JLogB c LogB + xM + c d b H- d f + c gL I 2 b dgI bx b f-a g b Hf + g xL d Hf + g xL b H- d f + c gL +x Hb f - a gL J 1 -d f + c g -d f + c g a + xE 2 2dg c d c d + c + c d Hb f - a gL I H- d f + c gL - a b H- d f + c gL J- - N Hb f-a gL2 c H- d f + c gL - c d Hf+g xL H-d f+c gL2 + d Hf + g xL b Hf + g xL 1 N LogA b Hf + g xL bf-ag b g I +xM a b b f-a g F F- - F + 127 128 2.2 Logarithm Functions.nb b H- d f + c gL 1+ c d Hb f - a gL I a b d Hb f - a gL I b H- d f + c gL I a b d Hb f - a gL H-d f+c c I d + xM LogB- c d -d f+c g dgI c d + xF - LogB- + xM -d f + c g c d a I b + xM + xM dgI c d + xM F d -d f+c g a b 1 LogB 2 a I b + xM + xM d Hb f - a gL I a c d b H- d f + c gL I - dgI c d c d a b -d f + c g + xM + xM d Hb f - a gL I a I b c d bf-ag bgI a b bf-ag c d +x a b +x a d I +xM + + xF + LogB I +xM b a d I +xM d a dgI + xM c d + xM b Hf + g xL bf-ag F F + LogB- b H- d f + c gL I a + xF + LogB- b F - PolyLogB2, - a b + xM d -d f+c g + xM c d +x a b +x F- PolyLogB2, - b c d PolyLogB2, b c H- b c + a dL Hf + g xL F + LogB + b H-d f+c gL I +xM a c d d g I +xM c d + xM -d f + c g F - LogB + b H-d f+c gL I +xM a b c d a PolyLogB2, d Hb f - a gL I F - LogB- + xM a I b gL2 d Hb f - a gL I + xM b H-d f+c gL2 I +xM + xM F LogB + b H-d f+c d a + xM + xM c d Hb f-a gL I +xM + xM c d Hb f-a gL I +xM b Hf + g xL F PolyLogB2, - c d a b c a b b H- d f + c gL F - 2 LogB F + xM - c + xM b H- d f + c gL + xM + xM + F b + xM F + LogB b b f-a g adI + xM a I b F PolyLogB2, - a b b b f-a g b H- d f + c gL I b H- d f + c gL I F LogB- bd c d F b H- d f + c gL LogB b H- d f + c gL I - c -b c + a d 2 b g I +xM + xM d g I +xM b g I +xM + xM c d Hb f - a gL I F+ -d f + c g d Hb f - a gL I F + LogB- c d F - + xM a b c d b H- d f + c gL I a PolyLogB2, - PolyLogB2, - 2 + xM + xM b b a b bgI b H-d f+c gL I +xM +x b Hf + g xL F + LogB b H- d f + c gL c d c I d b f-a g a a bf-ag d Hb f - a gL I dgI d b - gL2 + xM b H- d f + c gL I + xF LogB Hb f-a a b LogB- a b d Hb f - a gL I + xM b I +xM a PolyLogB2, b H- d f + c gL d Hf + g xL d 2, d I +xM + a b g I +xM a d I +xM d Hb f - a gL I d Hb f - a gL -d f + c g LogB gL2 c d b H- d f + c gL I c + b H-d f+c gL2 I +xM d 1 + xM c d a d b c F+ c LogB- a b adI + Hb f - a gL - c d Hb f-a gL I +xM c + xF LogB 2 + xM c d g I +xM a g3 a b + xM b PolyLogB2, - 2 LogB + xM + xM 1 1 c d b H- d f + c gL I H- d f + c gL - b H- d f + c gL2 I +x c d d Hb f-a gL I +xM c d a b H-d f+c gL I +xM b LogB b Hf + g xL bf-ag d Hf + g xL -d f + c g F + d Hb f - a gL I c d b H- d f + c gL I d Hb f - a gL I c d b H- d f + c gL I a b + xM + xM a b F + + xM + xM F - F- F PolyLogB + F + F - 2.2 Logarithm Functions.nb bgI PolyLogB3, - a b + xM bf-ag 1 4g g Hb f - a gL J LogB dgI c d bx b f-a g + xM -d f + c g - 1 2 Hb f - a gL J a b Hf+g xL b Hf + g xL F - 2 LogB + b Hf + g xL bx b f-a g H- d f + c gL 1 d Hb f - a gL I - LogB H-d f+c gL2 d cdgI c d + xM bf-ag LogB- -d f+c g c d + + xM dI b H- d f + c gL a I b + xM + xM + xM F + dx -d f+c g + b H- d f + c gL I d g I +xM c LogB c d d -d f+c g c I d + + xM - b a +x b bd a I b b Hf+g xL b f-a g N LogB + a F - LogB- b H- d f + c gL2 I + xM a b + xM + c d g I +xM c H-d f+c d gL2 c d + xM a b + xM + b Hf + g xL bf-ag Hb f-a gL2 dgI + c d + xM c d H- d f + c gL J- F LogB- F d I +xM 2dgI d c d c d F + c d a b + c d Hf+g xL H-d f+c gL2 + xM + xM d Hb f - a gL I b H- d f + c gL N F+ c d a I b + xM + xM F + 2 + xM F - LogB- + xM a b LogB dgI d Hb f-a gL I +xM d a b H-d f+c gL I +xM dgI b c d c d a b + xM + xM H- b c + a dL Hf + g xL b H- d f + c gL I - -d f + c g + xM a b + xM F F J- LogB + xM b H- d f + c gL I d Hf + g xL -d f + c g + xM c LogB- adI F - LogB- LogB- b Hf + g xL c -d f+c g dx -d f+c g b H- d f + c gL I bf-ag c d + xM 1 + LogB adI + xM a I b d Hb f - a gL I b FN c d d Hf + g xL + a b H- d f + c gL I + b H- d f + c gL b H-d f+c gL2 I +xM -d f+c g + xM N d Hb f - a gL I c H-b c+a dL Hf+g xL d Hf+g xL -d f + c g c d Hb f - a gL I F + LogB dgI c d a b Hf+g xL -d f + c g H-b c+a dL x + xF + LogB H- d f + c gL - + xM bx b f-a g H- b c + a dL Hf + g xL b H- d f + c gL2 I -b c + a d +x F + PolyLogB3, - b c d Hb f - a gL I F + a b b H-d f+c gL I +xM b +x F LogB c d Hf+g xL H-d f+c gL2 a - 2 LogB +x b Hf + g xL + xM a -d f + c g a b F JLogB + xM d Hf + g xL b H- d f + c gL c d c d -d f + c g b H- d f + c gL d Hb f - a gL I dgI c d Hb f - a gL J 1 d Hf + g xL 2dg F + LogB- 1 N d + xM + xE H- d f + c gL J- - d I +xM H- d f + c gL2 b Hf + g xL d Hb f - a gL I c d N F - PolyLogB3, 2 c + c d -d f + c g Hb f-a gL2 c + xM + b a b Hf+g xL c d g I +xM c d -d f + c g + xE LogA + xF + LogB Hb f-a gL2 + a b a a b Hf+g xL b Hf + g xL H- d f + c gL - N LogA Hb f-a gL2 + bx b f-a g Hb f - a gL J F - PolyLogB3, dgI 129 F + b Hf+g xL b f-a g F - F + LogB- d Hf+g xL -d f+c g FN - 130 2.2 Logarithm Functions.nb Hb f - a gL - a b g I +xM Hb f-a b H- d f + c gL b H- d f + c gL Hb f - a gL - gL2 c d g I +xM Hb f-a gL2 b LogB c H-d f+c d + xM 1 LogB- LogB b bgI gL2 + a b c d b H- d f + c gL F - LogB- d Hb f - a gL I PolyLogB2, dgI c d + xM a b + xM + c d + xM a b + xM d Hf + g xL -d f + c g + xM F b f-a g d g I +xM d -d f+c g F F+ 1 LogB 2 bd c d F + LogB- a b + xM + xM F + - a I b dgI c d dgI c d + xM -d f + c g + xM c d + xM a b + xM a b + xM c d b H- d f + c gL I a b + xM + xM + F d g I +xM c d -d f+c g F b H-d f+c a b c d b H-d f+c a b + b H-d f+c gL I +xM a b d a I +xM b H-d f+c gL I +xM dgI c d F - LogB- +x a b +x + xM F - LogB b H- d f + c gL I a + xF + LogB- b F - PolyLogB2, - a b PolyLogB2, d g I +xM d -d f+c g c d +x a b +x F- d Hb f-a gL I +xM c d a b H-d f+c gL I +xM b b Hf + g xL bf-ag H- b c + a dL Hf + g xL F + LogB c d + xM F + xM b b f-a g + xM + xM + xM b g I +xM a b c d F+ c d b c d + xM PolyLogB2, - c PolyLogB2, - a -d f + c g a b a d I +xM + xM d a b + xM c + xF + LogB b H- d f + c gL I c d b H-d f+c gL I +xM b + xM b a a d I +xM a c d + c d b H- d f + c gL I a d I +xM PolyLogB2, d Hb f - a gL I b d Hb f - a gL I d + I +xM gL2 + xM c d a d Hb f - a gL I + xM d Hb f - a gL I F PolyLogB2, gL2 c d Hb f-a gL I +xM + xM b H-d f+c gL2 I +xM bf-ag + xM c a b d a + xM a b c d F LogB1 + d Hb f - a gL I b bgI + xM c d Hb f-a gL I +xM + xM c bf-ag c d c a b c d Hb f-a gL I +xM F LogB- d Hb f - a gL I b H- d f + c gL I b H- d f + c gL I F - 2 LogB 1 - d Hb f - a gL I adI b Hf + g xL F PolyLogB2, - d Hb f - a gL I + xM b H- d f + c gL I + xM F + LogB a b LogB- b H- d f + c gL I -d f + c g -b c + a d LogB F b H- d f + c gL I F + LogB 2 b H- d f + c gL I -d f + c g b b H- d f + c gL2 I b d Hb f - a gL I + xM b g I +xM b b f-a g + xM c d Hb f - a gL I +x c I d c d b H- d f + c gL I a a b Hf + g xL + xM a I b adI + xM bf-ag d c d PolyLogB2, -d f+c g b H-d f+c gL I +xM b c d d a + xM a F LogB1 - c c I d b g I +xM F LogB1 + d Hb f-a gL I +xM + xE + LogB- c b H- d f + c gL I + xF - LogB- + xM + xM d I +xM dgI b H- d f + c gL a b a b PolyLogB2, - b f-a g + xF LogB b Hf + g xL LogA b H-d f+c gL I +xM b a d bf-ag c b I +xM - c b 2 d d Hb f - a gL + xF LogB LogB d I +xM -d f+c g dg c d g I +xM a g2 + d a + xE - LogB- bgI b H- d f + c gL2 I a 1 1 b f-a g c d c d Hb f - a gL I b PolyLogB2, - LogA c d +x c d c b c H-d f+c gL2 a b g I +xM d Hb f-a gL I +xM a - a H- d f + c gL d Hb f - a gL I b I +xM a F + F + LogB- d Hb f - a gL I c d b H- d f + c gL I d Hb f - a gL I + xM a b c d b H- d f + c gL I a b + xM + xM + xM F F d Hf + g xL -d f + c g F - - F + F F - - 2.2 Logarithm Functions.nb PolyLogB3, - bgI a b + xM bf-ag a n3 - b f + d f F - PolyLogB3, bx a c+dx c d + xM -d f + c g F - PolyLogB3, c d +x a b +x F + PolyLogB3, - d Hb f - a gL I c d b H- d f + c gL I a b + xM + xM F - 3 bx +g a-c + dgI + c+dx c+dx c+dx LogB a bx + c+dx c+dx Ib f - a g - d f F 1 a I c+d x bx M c+d x + +cg a b2 f - 2 d f LogB bx + c+dx a a I c+d x c+dx bx adg df b Hd f - c gL a a c+dx a - c g - 3 + LogB bx + a bx + c+dx H- d f + c gL I c+dx a c+d x + bf-ag a 6 a d g 1 + LogB bx + c+dx c+dx PolyLogB 2, F+ c+dx bx M c+d x F + bx c+dx a c+dx bx + c+dx a c+dx bx + c+dx bx + c+dx c+dx F a bx a 3 + LogB + c+dx c+dx F + 2 d f LogB bx + a c+dx F F +g 3a-c F + d f 3 + 2 LogB F + b c g - 1 + LogB c+dx c+dx + F + b c g - 2 + LogB a LogB bx c+dx a c+dx c+dx c+dx c+dx c+dx bx + + bx bx + a a + c+dx a g c g 3 - 2 LogB LogB1 + F + c g - 3 + 2 LogB c+dx 3 a d g 2 + LogB Hd f - c gL 3 + LogB + c+dx + 2 bx MM c+d x c+dx bx + c+dx F F - 2 d f LogB F - 2 d f LogB c+dx a c+dx + a bx + c+dx a c+dx bx + c+dx c+dx F bx + F + F c+dx F + 131 132 2.2 Logarithm Functions.nb 2, Hd f - c gL I a c+d x + bf-ag bx M c+d x F+ 6 H2 b d f - b c g - a d gL PolyLogB3, Hb f - a gL2 Hd f - c gL2 Hf + g xL3 ad Hc + d xL2 Hd f - c gL I a c+d x bf-ag + bx M c+d x F 2 bdx Hc + d xL2 + b -b + c+dx d a bx + c+dx c+dx Problem ð562: Valid but suboptimal antiderivative: :LogBe a+bx c+dx Ha + b xL LogAe I b F , x, 6, 0> n 4 a+b x n 4 M E c+d x 24 Hb c - a dL n3 LogAe I + 4 Hb c - a dL n LogAe I a+b x n M E c+d x bd a+b x n 3 M E c+d x bd PolyLogB3, d Ha+b xL F b Hc+d xL + LogB b c-a d F b Hc+d xL + 12 Hb c - a dL n2 LogAe I 24 Hb c - a dL n4 PolyLogB4, bd d Ha+b xL F b Hc+d xL a+b x n 2 M E c+d x bd PolyLogB2, d Ha+b xL F b Hc+d xL - 2.2 Logarithm Functions.nb n a+bx x LogBe c+dx 4 n I- LogAe I a+b x n M E c+d x a a d LogB F - n LogB a+bx c+dx + n LogA + xF + b c LogB b Ia d Log@a + b xD + b d x LogA bd a+b x E c+d x - b c Log@c + d xDM d F + b d x LogB 6 n2 LogBe bd c + xF Log@a + b xD - 2 a d LogB d a+bx F + 2 b c LogB 2 a+bx c+dx c + xF Log@a + b xD + 2 a d LogB b a+bx 1 + a 2 + xF - 2 a d LogB 2 a d Log@a + b xD LogB n F - n LogB + xF LogB d a a+bx c+dx d Ha + b xL -b c + a d c a+bx 4 n3 - LogBe bd n c+dx 6 Hb c - a dL LogB n4 a d LogB a+bx c+dx PolyLogB2, F + n LogB a+bx c+dx c+dx F PolyLogB2, F + b d x LogB 4 a+bx d Ha + b xL b Hc + d xL a+bx c+dx F a+bx LogB d Ha + b xL b Hc + d xL c+dx F - 24 Hb c - a dL LogB d Ha + b xL LogB 2 a+bx c+dx F + H- 6 b c + 6 a dL PolyLogB3, F + 4 b c LogB 4 F + xF Log@c + d xD - 2 b c LogB F+ F 2 + xF Log@c + d xD c+dx b d a+bx a b Hc + d xL d Ha + b xL b Hc + d xL 2 b c LogB F Log@c + d xD - 2 b c LogB + xF LogB F - 2 b c PolyLogB2, F - 2 a d PolyLogB2, F c+dx b bc-ad -b c + a d bc-ad c+dx 1 4 3 a+b x EM c+d x c 2 F 133 a+bx c+dx a+bx c+dx F LogB 3 bc-ad bc+bdx F PolyLogB3, F + 3 Hb c - a dL LogB d Ha + b xL b Hc + d xL F - 4 a d LogB d Ha + b xL b Hc + d xL F + a+bx c+dx bc-ad bc+bdx 1 F + bd F LogB bc-ad 3 F + 24 b c PolyLogB4, bc+bdx d Ha + b xL b Hc + d xL F + 12 Hb c - a dL LogB F - 24 a d PolyLogB4, a+bx c+dx F 2 d Ha + b xL b Hc + d xL F Problem ð563: Valid but suboptimal antiderivative: :LogBe a+bx c+dx Ha + b xL LogAe I b F , x, 7, 0> n 5 a+b x n 5 M E c+d x 60 Hb c - a dL n3 LogAe I + 5 Hb c - a dL n LogAe I a+b x n 4 M E c+d x bd a+b x n 2 M E c+d x bd PolyLogB3, d Ha+b xL F b Hc+d xL + LogB b c-a d F b Hc+d xL + 20 Hb c - a dL n2 LogAe I 120 Hb c - a dL n4 LogAe I a+b x n M E c+d x bd a+b x n 3 M E c+d x bd PolyLogB4, PolyLogB2, d Ha+b xL F b Hc+d xL - d Ha+b xL F b Hc+d xL - 120 Hb c - a dL n5 PolyLogB5, bd d Ha+b xL F b Hc+d xL 134 2.2 Logarithm Functions.nb 5 a n Log@a + b xD ILogAe I b n a+bx x LogBe c+dx 1 F - n LogB c+dx c LogB + xF LogB d a+bx c+dx 10 n3 LogBe bd c+dx n c+dx F LogB 3 24 Hb c - a dL LogB n5 a d LogB bd a+bx c+dx c+dx c+dx a+bx c+dx bc+bdx c+dx 5 F F F a+bx c+dx d Ha + b xL b Hc + d xL a+b x n M E c+d x 3 a + xF LogB b a+bx a+bx c+dx c+dx d Ha + b xL b Hc + d xL F + b d x LogB F PolyLogB2, 3 F - 120 a d LogB d Ha + b xL b Hc + d xL a+bx c+dx a+bx Ha + b xLm Hc + d xL-2-m LogAe I a+b x n M E c+d x Ha + b xL1+m Ie I c+dx a+bx c+dx F LogB 1+m n c+dx c+dx bc+bdx H1+mL LogBe J b Hc + d xL n a+b x c+d x N F n F b Hc + d xL b Hc + d xL F - bc+bdx F LogB 3 F - 24 a d PolyLogB4, a+bx c+dx F PolyLogB3, 2 F - 120 b c PolyLogB5, F + + xF Log@c + d xD - d b Hc + d xL bc-ad F + bd c+dx F- bc-ad 1 a+bx F - 5 a d LogB c+dx F - 2 a d PolyLogB2, F + 3 Hb c - a dL LogB d Ha + b xL a+bx c + xF Log@c + d xD - 2 b c LogB -b c + a d F + 4 b c LogB 4 a b d Ha + b xL d Ha + b xL + xF Log@a + b xD - 2 a d d 2 b Hc + d xL d Ha + b xL c + xF Log@a + b xD + 2 a d LogB F + 2 b c LogB d Ha + b xL F - 60 Hb c - a dL LogB F PolyLogB4, Hc + d xL-1-m ExpIntegralEiB Hb c - a dL n a+bx bc-ad 4 , x, 1, 0> a+b x n M M c+d x c+dx F PolyLogB2, Problem ð564: Unable to integrate: : a+bx a+bx 2 + b d Ha + b xL LogB 4 4 a 2 + xF - 2 a d LogB F - 2 b c PolyLogB2, F + 24 b c PolyLogB4, F + 5 b c LogB 5 2 c+dx F - F + H- 6 b c + 6 a dL PolyLogB3, a+bx a d LogB F a+bx Log@c + d xD F + b d x LogB c+dx b Hc + d xL LogB F - n LogB d bc-ad 2 c+dx c 2 + xF + b c LogB F + 12 Hb c - a dL LogB c+dx d n a+bx 4 a+b x EM c+d x b b Hc + d xL a+bx F LogBe - n LogA a a d LogB d Ha + b xL F PolyLogB3, F + b d x LogB 20 Hb c - a dL LogB PolyLogB4, a+bx a+bx bc-ad a+bx c+dx 5 c n ILogAe I F PolyLogB2, F + n LogB a+bx + 5 n x LogB F + 2 a d Log@a + b xD LogB F - n LogB c+dx c+dx a+bx n a+bx a+bx 4 a d LogB - 4 a+b x EM c+d x F Log@c + d xD - 2 b c LogB c+dx 5 n4 - LogBe F 5 F + n LogB d Ha + b xL a+bx 6 Hb c - a dL LogB - n LogA a+bx -b c + a d 2 b c LogB 1 n a+bx 10 n2 - LogBe bd 1 a+b x n M E c+d x bc-ad bc+bdx d Ha + b xL b Hc + d xL F LogB 4 F- bc-ad bc+bdx d Ha + b xL b Hc + d xL F + F+ F + 120 b c LogB d Ha + b xL b Hc + d xL a+bx c+dx F F + 120 a d PolyLogB5, d Ha + b xL b Hc + d xL F 2.2 Logarithm Functions.nb á Ha + b xLm Hc + d xL-2-m LogAe I âx a+b x n M E c+d x Problem ð577: Unable to integrate: : LogA LogA á a E a+b x LogA 2 cx E a+b x x Ha + b xL 2 cx E a+b x a LogA E a+b x , x, 3, 0> PolyLogA2, a 2 cx LogA E a+b x x Ha + b xL bx E a+b x 2 LogA + cx E a+b x PolyLogA3, a bx E a+b x 2 PolyLogA4, a bx E a+b x âx Problem ð578: Valid but suboptimal antiderivative: : - LogAe I a+b x n M E c+d x Hc + d xL Hf + g xL LogAe I a+b x n M E c+d x , x, 2, 0> LogB Hb c-a dL Hf+g xL Hb f-a gL Hc+d xL df-cg F n PolyLogB2, - Hd f-c gL Ha+b xL Hb f-a gL Hc+d xL df-cg F 1 2df-2cg c - n LogB a 2 + xF - 2 n LogB d c + xF Log@c + d xD + 2 n LogB b d a 2 n LogB c+dx c + xF Log@f + g xD - 2 n LogB b c 2 n LogB a+bx + xF Log@c + d xD + 2 LogBe + xF LogB d d Hf + g xL df-cg a+bx + xF Log@f + g xD - 2 LogBe d F + 2 n PolyLogB2, d Ha + b xL -b c + a d c+dx n n F Log@c + d xD + 2 n LogB F Log@f + g xD - 2 n LogB F - 2 n PolyLogB2, g Ha + b xL -b f + a g a + xF LogB b a + xF LogB b b Hf + g xL bf-ag F + 2 n PolyLogB2, g Hc + d xL -d f + c g F b Hc + d xL bc-ad F+ Problem ð579: Valid but suboptimal antiderivative: : - LogAe I a+b x n 2 M E c+d x Hc + d xL Hf + g xL LogAe I a+b x n 2 M E c+d x , x, 3, 0> LogB df-cg Hb c-a dL Hf+g xL Hb f-a gL Hc+d xL F - 2 n LogAe I a+b x n M E c+d x PolyLogB2, df-cg a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL F 2 n2 PolyLogB3, + a Hd f-c gL+b d f x-b c g x Hb f-a gL Hc+d xL df-cg F F+ 135 136 2.2 Logarithm Functions.nb 1 - 2 n2 LogB 3df-3cg c 3 + xF + 3 n2 LogB d 6 n2 LogB a c + xF Log@c + d xD + 3 n2 LogB 6 n2 LogB a+bx n c+dx a c 2 F - 3 n LogB c d c+dx a + xF Log@c + d xD - 6 n LogB c b Hc + d xL F + 6 n LogB a+bx F Log@c + d xD - 3 n2 LogB a a+bx n n c+dx n 2 c+dx F + 3 n2 LogB a+bx + xF LogBe b F Log@c + d xD + 3 LogBe n a+bx 2 + xF LogBe d + xF LogBe d d Ha + b xL -b c + a d d c 6 n LogB 2 + xF LogB d + xF LogB b c F LogB a 2 + xF LogB b b Hc + d xL a 2 + xF Log@c + d xD b F Log@c + d xD + b Hc + d xL bc-ad F + 3 n2 LogB F+ -b c + a d F LogB Hb f - a gL Hc + d xL bc-ad b c+dx bc-ad d Ha + b xL Hd f - c gL Ha + b xL 2 a c c 3 n2 LogB + xF Log@f + g xD + 6 n2 LogB + xF LogB + xF Log@f + g xD - 3 n2 LogB + xF Log@f + g xD + b b d d a a+bx n c a+bx n a+bx n 2 6 n LogB + xF LogBe F Log@f + g xD - 6 n LogB + xF LogBe F Log@f + g xD - 3 LogBe F Log@f + g xD + b c+dx d c+dx c+dx + xF LogB b a 3 n2 LogB d 2 a 2 + xF LogB b 3 n2 LogB g Hc + d xL -d f + c g 3 n2 LogB b Hf + g xL bf-ag F LogB 2 Hd f - c gL Ha + b xL 6 n2 LogB g Hc + d xL -d f + c g c 6 n n LogB F LogB 2 + xF + LogBe n c+dx c + xF PolyLogB2, Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL 6 n2 PolyLogB3, b d Ha + b xL -b c + a d n bf-ag -d f + c g df-cg F LogB a+bx c+dx bc-ad Problem ð580: Valid but suboptimal antiderivative: : LogAe I a+b x n 3 M E c+d x Hc + d xL Hf + g xL , x, 5, 0> + xF LogB + xF LogB F - 3 n2 LogB F - 6 n LogBe n F PolyLogB2, F LogB d Hf + g xL g Hc + d xL -d f + c g F - 6 n2 LogB b Hc + d xL c a b bf-ag F - 6 n2 LogB Hd f - c gL Ha + b xL d df-cg -b c + a d d Ha + b xL F - 6 n2 PolyLogB3, b b Hf + g xL Hb f - a gL Hc + d xL + xF LogB d Hf + g xL b Hc + d xL F LogB F LogB d Ha + b xL F + 6 n LogBe F PolyLogB2, a F + 6 n2 LogB F PolyLogB2, bc-ad g Hc + d xL F - 6 n2 LogB d Hf + g xL b Hc + d xL n c+dx F - 6 n2 LogB Hd f - c gL Ha + b xL a+bx a+bx + xF LogBe b Hf + g xL F LogB + xF LogBe a Hb f - a gL Hc + d xL d 6 n2 LogB F LogB c+dx d 6 n2 LogB bf-ag a+bx + xF LogBe d F - 6 n LogB b Hf + g xL Hb f - a gL Hc + d xL c 6 n LogB + xF LogB df-cg F LogB c+dx -d f + c g Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL F - 6 n2 PolyLogB3, bf-ag d Hf + g xL df-cg 2 F+ c 2 + xF LogB d F LogB bf-ag d Hf + g xL g Hc + d xL -d f + c g Hd f - c gL Ha + b xL 2 F+ F PolyLogB2, Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL d Hf + g xL df-cg g Ha + b xL -b f + a g F PolyLogB2, Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL F- Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL F 2 F+ F+ F LogB H- b c + a dL Hf + g xL F + 6 n2 PolyLogB3, b Hf + g xL df-cg Hd f - c gL Ha + b xL F PolyLogB2, d Ha + b xL -d f + c g Hb f - a gL Hc + d xL F + 6 n2 LogB b Hc + d xL g Hc + d xL F - 3 n2 LogB F LogB F + n LogB g Hc + d xL b F + 3 n2 LogB Hd f - c gL Ha + b xL n + xF LogB b Hf + g xL Hb f - a gL Hc + d xL a+bx a F - F+ F+ g Hc + d xL -d f + c g F+ 2.2 Logarithm Functions.nb - LogAe I a+b x n 3 M E c+d x df-cg 6 n2 LogAe I ILogAe I Hb c-a dL Hf+g xL LogB a+b x n M E c+d x a+b x M c+d x n Hb f-a gL Hc+d xL a+b x EM c+d x Log@c + d xD + df-cg 1 n a+bx 3 n LogBe 2df-2cg c+dx a 2 LogB + xF LogB b + xF LogB d 1 d Hf + g xL df-cg n a+bx n2 - LogBe df-cg F - n LogB b Hc + d xL bc-ad c 2 LogB c+dx a+b x n 2 M E c+d x Hb f-a gL Hc+d xL df-cg E - n LogA - 3 n LogAe I a Hd f-c gL+b d f x-b c g x PolyLogB3, 3 F c+dx a F Hb f-a gL Hc+d xL df-cg Hd f-c gL Ha+b xL Hb f-a gL Hc+d xL - df-cg a+b x M c+d x n 2 E - n LogA 3 a+b x EM c+d x c c+dx F + Log@f + g xD - a 2 LogB + xF + 2 LogB d c b d Ha + b xL -b c + a d c + xF Log@c + d xD - 2 LogB d c+dx a+bx + xF Log@f + g xD + 2 LogB d F + 2 PolyLogB2, c 3 - 2 LogB a+bx + xF Log@c + d xD - 2 LogB c b a+bx F F -d f + c g + xF Log@f + g xD + 2 LogB F - 2 PolyLogB2, F + n LogB a Hd f-c gL+b d f x-b c g x 6 n3 PolyLogB4, ILogAe I a+bx F - 2 LogB F PolyLogB2, 137 + xF + 3 LogB d g Ha + b xL -b f + a g 2 + xF LogB d c+dx F Log@f + g xD + 2 LogB F - 2 PolyLogB2, d Ha + b xL -b c + a d g Hc + d xL -d f + c g F - 3 LogB c 2 F - a+bx + xF LogB d c+dx a F Log@c + d xD - + xF LogB b b Hf + g xL bf-ag F- F+ F Log@c + d xD + d d b c+dx 2 a+bx a+bx 2 a b Hc + d xL a c b Hc + d xL 6 LogB + xF LogB F Log@c + d xD + 3 LogB F Log@c + d xD - 3 LogB + xF LogB F + 6 LogB + xF LogB + xF LogB F+ d c+dx c+dx b bc-ad b d bc-ad 2 a a+bx b Hc + d xL -b c + a d Hb f - a gL Hc + d xL 2 a 6 LogB + xF LogB F LogB F + 3 LogB F LogB F - 3 LogB + xF Log@f + g xD + b c+dx bc-ad d Ha + b xL Hd f - c gL Ha + b xL b a 3 LogB a 2 + xF Log@c + d xD - 6 LogB b c c + xF LogB c + xF Log@c + d xD + 3 LogB 2 a + xF Log@c + d xD - 6 LogB a+bx + xF LogB b F Log@f + g xD c+dx d c+dx 2 2 a b Hf + g xL a a+bx b Hf + g xL a g Hc + d xL 3 LogB F Log@f + g xD + 3 LogB + xF LogB F - 6 LogB + xF LogB F LogB F - 6 LogB + xF LogB F c+dx b bf-ag b c+dx bf-ag b -d f + c g a 6 LogB c + xF LogB b a+bx LogB b Hf + g xL bf-ag 3 LogB d Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL a+bx + xF LogB d 6 LogB g Hc + d xL -d f + c g c 6 LogB c+dx F LogB + xF + LogB d a 2 + xF Log@f + g xD + 6 LogB d F + 3 LogB c 6 LogB c + xF Log@f + g xD - 3 LogB g Hc + d xL -d f + c g F LogB 2 F LogB F LogB 2 b Hf + g xL bf-ag d Hf + g xL df-cg Hd f - c gL Ha + b xL a+bx c+dx F PolyLogB2, + b b Hf + g xL bf-ag F - 6 LogB F + 6 LogB Hb f - a gL Hc + d xL F LogB a+bx + xF LogB F - 6 LogB a b c b d Hf + g xL df-cg g Hc + d xL -d f + c g F - 3 LogB F - 6 LogB F LogB + xF LogB d + xF LogB -b c + a d -d f + c g + xF LogB a d Ha + b xL g Hc + d xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL d Hf + g xL df-cg F LogB F + 3 LogB d Hf + g xL df-cg Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL a+bx c+dx F + LogB F Log@f + g xD - 6 LogB c d 2 2 bf-ag g Hc + d xL -d f + c g df-cg F LogB 2 Hd f - c gL Ha + b xL Hd f - c gL Ha + b xL F+ d Hf + g xL H- b c + a dL Hf + g xL Hb f - a gL Hc + d xL + a+bx + xF LogB b Hf + g xL + xF LogB F - 3 LogB F LogB F LogB c F PolyLogB2, F+ d Hf + g xL df-cg F+ g Ha + b xL -b f + a g F+ F+ + 138 2.2 Logarithm Functions.nb c 6 LogB + xF PolyLogB2, d 6 LogB Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL 6 PolyLogB3, d Ha + b xL -b c + a d 1 b Hc + d xL n3 LogB df-cg a+bx c+dx a+bx 6 LogB c+dx F PolyLogB2, F PolyLogB3, a+bx c+dx b Hc + d xL d Ha + b xL F - 6 PolyLogB3, F LogB 3 bc-ad F + 6 LogB F - 6 LogB b Hc + d xL bc-ad Hb c - a dL Hf + g xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL F PolyLogB2, g Hc + d xL -d f + c g Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL F - 6 PolyLogB3, F + 3 LogB F + 6 LogB a+bx c+dx F + 6 PolyLogB4, 2 Hd f - c gL Ha + b xL F PolyLogB2, b Hc + d xL d Ha + b xL F PolyLogB2, Hb f - a gL Hc + d xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL F + 6 PolyLogB3, Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL Hd f - c gL Ha + b xL Hb f - a gL Hc + d xL F : b c-a d F b Hc+d xL LogA e Ha+b xL 2 E c+d x Hc + d xL Ha g + b g xL LogA - á e Ha+b xL 2 E c+d x PolyLogB2, Hd f - c gL Ha + b xL F- Hb c - a dL g b c-a d LogB F b Hc+d xL , x, 4, 0> d Ha+b xL F b Hc+d xL e Ha+b xL 2 LogA E c+d x Hc + d xL Ha g + b g xL 2 LogA e Ha+b xL E c+d x PolyLogB3, Hb c - a dL g + âx d Ha+b xL F b Hc+d xL 2 PolyLogB4, d Ha+b xL F b Hc+d xL Hb c - a dL g - Problem ð582: Valid but suboptimal antiderivative: : - LogAe I a+b x n 2 M E c+d x LogB b c-a d F b Hc+d xL Hc + d xL Ha g + b g xL LogAe I a+b x n 2 M E c+d x PolyLogB2, Hb c - a dL g , x, 4, 0> d Ha+b xL F b Hc+d xL + 2 n LogAe I a+b x n M E c+d x PolyLogB3, Hb c - a dL g d Ha+b xL F b Hc+d xL 2 n2 PolyLogB4, - d Ha+b xL F b Hc+d xL Hb c - a dL g F- Hb f - a gL Hc + d xL Problem ð581: Unable to integrate: LogB F PolyLogB2, g Hc + d xL -d f + c g F - F+ 2.2 Logarithm Functions.nb 139 1 3 Hb c - a dL g a+bx LogB c+dx c - LogB F 3 LogBe c+dx a 2 + xF - 2 LogB d a+bx c+dx c + xF Log@c + d xD + 2 LogB b d Ha + b xL 2 PolyLogB2, -b c + a d a 2 3 LogB + xF LogB b c F + n LogBe b Hc + d xL a 2 + xF + 2 LogB d + xF LogB - LogB d + xF + LogB b a+bx c+dx F LogB 3 c+dx a a+bx c+dx F + n2 LogB d Ha + b xL -b c + a d bc-ad bc+bdx F + 3 LogB F - n LogB a+bx c+dx b b Hc + d xL a+bx c+dx a+bx c+dx a+bx c+dx + xF PolyLogB2, bc-ad a 2 + xF F LogB + xF Log@c + d xD + 2 LogB n a+bx F + 6 LogB b c 3 LogB n d bc-ad - LogB n2 LogB F - 3 n LogBe n 2 a+bx F d Ha + b xL -b c + a d F + 2 PolyLogB2, F + 2 LogB c c LogB F PolyLogB2, 2 bc-ad 2 LogB bc+bdx d Ha + b xL b Hc + d xL 2 c+dx a + xF LogB b c b Hc + d xL bc-ad a+bx d Ha + b xL -b c + a d F - 2 PolyLogB3, a+bx c+dx F PolyLogB3, c+dx F+ c Hb + a xL2 x2 6 b LogA à LogB b E b+a x F , x, 5, 0> bc-ad d Ha + b xL b Hc + d xL 3 LogB a c Hb + a xL2 x2 c Hb+a xL2 x2 F 2 + x LogB c Hb + a xL2 x2 F âx 3 F + 3 24 b LogB c Hb+a xL2 x2 F PolyLogA2, a ax E b+a x 48 b PolyLogA3, + a Problem ð591: Unable to integrate: :LogB c x2 F , x, 5, 0> 3 Hb + a xL 2 c x2 x LogB à LogB Hb + a xL2 c x2 Hb + a xL2 F + 3 6 b LogB F âx 3 c x2 Hb+a xL2 a F LogA 2 b E b+a x 24 b LogB + c x2 Hb+a xL2 F PolyLogA2, a ax E b+a x 48 b PolyLogA3, a ax E b+a x F b Hc + d xL Problem ð588: Unable to integrate: :LogB c+dx + xF - LogB d F - 6 PolyLogB3, a+bx + xF + LogB c b F - n LogB b Hc + d xL d + xF - LogB n bc-ad + xF + LogB a F - 6 LogB a+bx LogBe b F - 3 LogB -b c + a d 3 a 3 d d Ha + b xL F+ F Log@c + d xD + 2 LogB + xF + 3 - LogB + xF PolyLogB2, d F ax E b+a x F F+ F a+bx c+dx F 2 Log@c + d xD + - F + 6 PolyLogB4, d Ha + b xL b Hc + d xL F 2 140 2.2 Logarithm Functions.nb Problem ð596: Valid but suboptimal antiderivative: : - Ka + b LogB 1- 1-c x 1+c x c2 Ka + b LogB x2 1-c x 1+c x 4bc LogB 1-c x 1+c x - FO 3 , x, 2, 0> FO 4 F 4 a3 + 6 a2 b LogB 1-c x 1+c x F + 4 a b2 LogB 1-c x 1+c x 4c F + b3 LogB 2 1-c x 1+c x F 3 Problem ð597: Valid but suboptimal antiderivative: : - Ka + b LogB 1-c x 1+c x 1 - c2 x2 Ka + b LogB 1-c x 1+c x 3bc LogB - 1-c x 1+c x FO 2 , x, 2, 0> FO 3 F 3 a2 + 3 a b LogB 1-c x 1+c x 3c F + b2 LogB 1-c x 1+c x F 2 Problem ð608: Unable to integrate: 9x3 LogA1 + e Ifc Ha+b xL M E, x, 5, 0= n x3 PolyLogA2, - e Ifc Ha+b xL M E n - + 3 x2 PolyLogA3, - e Ifc Ha+b xL M E n n - b2 c2 n2 Log@fD2 b c n Log@fD 3 c Ha+b xL M E âx à x LogA1 + e If 6 x PolyLogA4, - e Ifc Ha+b xL M E b3 c3 n3 Log@fD3 n Problem ð609: Unable to integrate: 9x2 LogA1 + e Ifc Ha+b xL M E, x, 4, 0= n x2 PolyLogA2, - e Ifc Ha+b xL M E n - b c n Log@fD 2 c Ha+b xL M E âx à x LogA1 + e If n + 2 x PolyLogA3, - e Ifc Ha+b xL M E n b2 c2 n2 Log@fD2 2 PolyLogA4, - e Ifc Ha+b xL M E n - b3 c3 n3 Log@fD3 6 PolyLogA5, - e Ifc Ha+b xL M E n + b4 c4 n4 Log@fD4 2.2 Logarithm Functions.nb Problem ð610: Unable to integrate: 9x LogA1 + e Ifc Ha+b xL M E, x, 3, 0= n x PolyLogA2, - e Ifc Ha+b xL M E n - + n b2 c2 n2 Log@fD2 b c n Log@fD à x LogA1 + e If PolyLogA3, - e Ifc Ha+b xL M E M E âx c Ha+b xL n Problem ð611: Unable to integrate: 9LogA1 + e Ifc Ha+b xL M E, x, 3, 0= n PolyLogA2, - e Ifc Ha+b xL M E n - b c n Log@fD à LogA1 + e If M E âx c Ha+b xL n Problem ð613: Unable to integrate: 9x3 LogAd + e Ifc Ha+b xL M E, x, 6, 0= n 1 4 x4 LogAd + e Ifc Ha+b xL M E - 1 n 3 x2 PolyLogB3, - e Ifc Ha+b xL M n x4 LogB1 + 4 n d b2 c2 n2 Log@fD2 3 c Ha+b xL M E âx à x LogAd + e If F e Ifc Ha+b xL M F- d 6 x PolyLogB4, - x3 PolyLogB2, - b3 c3 n3 Log@fD3 n d b c n Log@fD e Ifc Ha+b xL M d e Ifc Ha+b xL M n F 6 PolyLogB5, + F + e Ifc Ha+b xL M d b4 c4 n4 Log@fD4 n F n Problem ð614: Unable to integrate: 9x2 LogAd + e Ifc Ha+b xL M E, x, 5, 0= n 1 3 x3 LogAd + e If M E- c Ha+b xL n 2 c Ha+b xL M E âx à x LogAd + e If n 1 3 x3 LogB1 + e Ifc Ha+b xL M n d F- x2 PolyLogB2, - e Ifc Ha+b xL M b c n Log@fD d n F 2 x PolyLogB3, + e Ifc Ha+b xL M d b2 c2 n2 Log@fD2 n F 2 PolyLogB4, - e Ifc Ha+b xL M d b3 c3 n3 Log@fD3 n F 141 142 2.2 Logarithm Functions.nb Problem ð615: Unable to integrate: 9x LogAd + e Ifc Ha+b xL M E, x, 4, 0= n 1 2 x2 LogAd + e Ifc Ha+b xL M E n à x LogAd + e If 1 e Ifc Ha+b xL M n x2 LogB1 + 2 d M E âx c Ha+b xL n F- x PolyLogB2, - e Ifc Ha+b xL M d b c n Log@fD Problem ð616: Unable to integrate: 9LogAd + e Ifc Ha+b xL M E, x, 4, 0= n x LogAd + e If à LogAd + e If M E - x LogB1 + c Ha+b xL n M E âx c Ha+b xL n e Ifc Ha+b xL M n d F- PolyLogB2, - e Ifc Ha+b xL M d b c n Log@fD n F Problem ð653: Valid but suboptimal antiderivative: 8Cos@xD Log@Cos@xDD, x, 3, 0< ArcTanh@Sin@xDD - Sin@xD + Log@Cos@xDD Sin@xD x x x x - LogBCosB F - SinB FF + LogBCosB F + SinB FF - Sin@xD + Log@Cos@xDD Sin@xD 2 2 2 2 Problem ð742: Valid but suboptimal antiderivative: : H1 + Log@xDL5 , x, 1, 0> x 1 6 H1 + Log@xDL6 5 Log@xD2 Log@xD + 10 Log@xD3 + 2 5 Log@xD4 + 3 + Log@xD5 + Log@xD6 2 Problem ð781: Valid but suboptimal antiderivative: : LogAc I1 + x2 M E n 1 + x2 , x, 5, 0> 6 n F PolyLogB3, + e Ifc Ha+b xL M d b2 c2 n2 Log@fD2 n F 2.2 Logarithm Functions.nb F + ArcTan@xD LogAc I1 + x2 M E + ä n PolyLogB2, - 2ä ä n ArcTan@xD2 + 2 n ArcTan@xD LogB n ä-x 1 - 4 n ArcTan@xD Log@- ä + xD - ä n Log@- ä + xD2 + 2 ä n Log@- ä + xD LogB- 4 1 2 1 2 ä n LogB 2 ä+x ä-x 143 F ä Hä + xLF - 4 n ArcTan@xD Log@ä + xD - H1 + ä xLF Log@ä + xD + ä n Log@ä + xD2 + 4 ArcTan@xD LogAc I1 + x2 M E + 2 ä n PolyLogB2, n 1 äx + 2 2 F - 2 ä n PolyLogB2, - 1 2 ä Hä + xLF Problem ð782: Valid but suboptimal antiderivative: : LogB x2 1+x2 1 + x2 F , x, 6, 0> 2x ä ArcTan@xD2 - 2 ArcTan@xD LogB ä+x 1 F + ArcTan@xD LogB x2 1+ x2 F + ä PolyLogB2, ä-x ä+x F ä Log@- ä + xD2 - ä Log@ä + xD2 + 4 ArcTan@xD - 2 Log@xD + Log@- ä + xD + Log@ä + xD + LogB x2 1 + x2 4 1 2 ä Log@- ä + xD LogB2 ä Hä + xLF + PolyLogB2, 1 äx + 2 2 4 ä HLog@1 - ä xD Log@xD + PolyLog@2, ä xDL + 2 ä LogB 1 2 F - F - 4 ä HLog@1 + ä xD Log@xD + PolyLog@2, - ä xDL + H1 + ä xLF Log@ä + xD + PolyLogB2, - 1 2 ä Hä + xLF Problem ð785: Valid but suboptimal antiderivative: : LogB c x2 a+b x2 a + b x2 ä ArcTanB F , x, 6, 0> b x a a b F 2 2 ArcTanB a - b x - 8 ArcTanB a a b ä F LogB a 1 4 b x a 2 ä LogB- b x 4 ArcTanB 1 a b ä b x 2 F LogB b x a + b x F Log@xD + 4 ArcTanB + xF LogB b 2 ä 2 c x2 a+b x2 a F F + 2 ä LogB ArcTanB b x a + a b x a ä a ä PolyLogB2, + a ä a + xF + ä LogB- a +ä b x a -ä b x b F b x 2 + xF + 4 ArcTanB b a b 1 ä b x + 2 b x F a ä + xF LogB b c x2 a+b x2 b a ä F - 4 ä PolyLogB2, - F LogB- F LogB 2 a F + 4 ä Log@xD LogB1 - F + 4 ä PolyLogB2, ä b x a ä b x a F + 2 ä PolyLogB2, 1 F LogB a ä ä b x 2 a a + xF - ä LogB 2 + xF - b F - 4 ä Log@xD LogB1 + - 2 ä b ä b x a F - 2 ä PolyLogB2, F+ 1 ä b x + 2 2 a F 144 2.2 Logarithm Functions.nb Problem ð786: Valid but suboptimal antiderivative: : LogB1 + 1-a x ä 1+a x 1 - a2 x2 PolyLogB2, - F , x, 1, 0> 1-a x ä 1+a x a F 1 ä 1-ax 4 ArcTanh@a xD LogB1 + 4a 1+ax F + PolyLogA2, - ã-2 ArcTanh@a xD E - 2 IArcTanh@a xD ILogA1 + ã-2 ArcTanh@a xD E - LogA1 - ä ã-ArcTanh@a xD E + LogA1 + ä ã-ArcTanh@a xD EM - PolyLogA2, - ä ã-ArcTanh@a xD E + PolyLogA2, ä ã-ArcTanh@a xD EM Problem ð787: Valid but suboptimal antiderivative: : LogB1 - 1-a x ä 1+a x 1 - a2 x2 PolyLogB2, ä F , x, 1, 0> 1-a x 1+a x a F 1 ä 1-ax 4 ArcTanh@a xD LogB1 4a 1+ax F + PolyLogA2, - ã-2 ArcTanh@a xD E - 2 IArcTanh@a xD ILogA1 + ã-2 ArcTanh@a xD E + LogA1 - ä ã-ArcTanh@a xD E - LogA1 + ä ã-ArcTanh@a xD EM + PolyLogA2, - ä ã-ArcTanh@a xD E - PolyLogA2, ä ã-ArcTanh@a xD EM Test complete! 2.2 Logarithm Functions.nb Exponential and log function test suite statistics * * * Indefinite Integration Test Suite Results Integration function: Time and date of test: Mathematica version: * * * Mathematica's built-in Integrate function 13:21 18 June 2010 10.0 for Microsoft Windows H64-bitL HDecember 4, 2014L Largest result size: 19 104 leaves Optimal size: 1119 leaves Longest compute time: 111.291113 seconds Integrand: Result size: 24 leaves File 2 Exponential Functions\Exponential Functions LogAe I Integrand: a+b x n 3 M E c+d x Hf + g xL3 a + b Log@c Hd He + f xLp Lq D Optimal Nonident Unintegrable Timeout Invalid Total Intsec Time 746 76 33 1 0 856 4.19 327.2 197 41 13 0 0 251 0.25 1013.8 2 Exponential Functions\Logarithm Functions 712 76 18 1 0 807 2.19 493.7 Totals 1655 193 64 2 0 1914 1.21 1834.7 Percentages 86.47% 10.08% 2 Exponential Functions\u Ha+b logHc Hd He+f xL^pL^qLL^n 3.34% 0.10% 0.00% 100.00% 145
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