RUBI 4.1 results for Charlwood’s
integrals
(*requires Rubi package loaded*)
1
Int@ArcSin@xD Log@xD, xD
-2
1 - x2 - x ArcSin@xD + ArcTanhB
1 - x2 F +
1 - x2 Log@xD + x ArcSin@xD Log@xD
2
x ArcSin@xD
IntB
, xF
1 - x2
x-
1 - x2 ArcSin@xD
3
IntBArcSinB
x+1 -
- x ArcSinB x -
x F, xF
1+x F+
SubstBIntB
1 - x2 + x
- 1 + x2 , xF, x,
2
Printed by Wolfram Mathematica Student Edition
1+x F
2
mma_rubi4.nb
4
IntBLogB1 + x
1
-2 x 10
5 N ArcTanB
J- 1 +
5 N ArcTanB
5
5
1
10
2
5
5 N ArcTanB
J2 +
1
2
2
J1 +
1 + x2 F, xF
J- 1 +
J1 +
xF +
1+
5
2
2
xF +
1+
5
-1 +
5
5 J- 1 +
5 N
1
5
2
-1 +
5 J1 +
5
-1 +
J- 2 +
5
5
5 N ArcTanhB
2
5 N
ArcTanhB
1 + x2 F + x LogB1 + x
2
1+
ArcTanB
2
xF +
1 + x2 F +
2
1 + x2 F + 2
2
5 N ArcTanhB
5 N ArcTanhB
2
1+
5
2
xF +
-1 +
5
1 + x2 F -
1 + x2 F
5
Cos@xD2
IntB
, xF
Cos@xD4 + Cos@xD2 + 1
1
EllipticPiB
2
2ä
3
3ä+
3
J3 - ä
3 N, ArcSinB
I1 + Tan@xD2 M
2ä
-
3ä+
3
3ä-
3
Tan@xDF,
3ä+
3
2 Tan@xD2
2 Tan@xD2
1+
1+
3-ä
3+ä
3
3
6
IntBTan@xD
1
2
1 + Tan@xD4 , xF
ArcSinhATan@xD2 E -
1-Tan@xD2
ArcTanhB
1+Tan@xD4
2
2
F
1
+
1 + Tan@xD4
2
Printed by Wolfram Mathematica Student Edition
F I3 + 3 Tan@xD2 + Tan@xD4 M “
3 + 3 Tan@xD2 + Tan@xD4
I1 + Tan@xD2 M
2
mma_rubi4.nb
7
Tan@xD
IntB
, xF
Sec@xD3
2
-
ArcTanhB
3
+1
1 + Sec@xD3 F
8
IntB
Tan@xD2 + 2 Tan@xD + 2 , xF
1
ArcSinh@1 + Tan@xDD -
ä
1 - 2 ä ArcTanhB
2
2
1
ä
1 + 2 ä ArcTanhB
2
2
H4 - 2 äL + H2 - 2 äL Tan@xD
2 + 2 Tan@xD + Tan@xD2
1-2ä
H4 + 2 äL + H2 + 2 äL Tan@xD
2 + 2 Tan@xD + Tan@xD2
1+2ä
F
F+
9
IntBSin@xD ArcTanB
1
2
Sec@xD - 1 F, xF
ArcTanB - 1 + Sec@xD F - ArcTanB - 1 + Sec@xD F Cos@xD +
1
Cos@xD
- 1 + Sec@xD
2
10
x3 Exp@ArcSin@xDD
IntB
, xF
1 - x2
3
ãArcSin@xD x +
10
1
ãArcSin@xD x3 -
10
3
ãArcSin@xD
1 - x2 -
10
3
ãArcSin@xD x2
1 - x2
10
Sec@xD
IntB
, xF
1 + Sin@xD
11
Int@Hx * Log@1 + x ^ 2D * Log@x + Sqrt@1 + x ^ 2DDL  Sqrt@1 + x ^ 2D, xD
4 x - 2 ArcTan@xD - x LogA1 + x2 E - 2
1 + x2 LogBx +
1 + x2 F +
Printed by Wolfram Mathematica Student Edition
1 + x2 LogA1 + x2 E LogBx +
1 + x2 F
3
4
mma_rubi4.nb
12
Int@ArcTan@x + Sqrt@1 - x ^ 2DD, xD
ArcSin@xD
-
1 - 2 x2
1
+
2
3 ArcTanB
4
3
-
F+
ä-
3
ä+
3
1-x2
12
1
12
J3 ä -
J3 ä +
3 N ArcTanB
3 N ArcTanB
x
-
ä- 3
1 - x2
ä+ 3
-
ä- 3
x
ä+ 3
1-
x2
+
3
-
1
F
x
ArcTanB
F+
ä-
3
ä+
3
x
ArcTanB
1-x2
3
F + x ArcTanBx +
1 - x2 F -
F
1
8
-
LogA1 - x2 + x4 E
13
Int@x * ArcTan@x + Sqrt@1 - x ^ 2DD  Sqrt@1 - x ^ 2D, xD
ArcTanB
ArcSin@xD
-
1 - 2 x2
1
+
2
3 ArcTanB
4
3
-
F+
2
ä-
3
ä+
3
1-x2
12
1
12
J3 ä -
J3 ä +
3 N ArcTanB
3 N ArcTanB
x
-
ä- 3
1 - x2
ä+ 3
-
ä- 3
x
ä+ 3
1-
x2
F-
F+
-
3
-
1
F
x
ä-
3
ä+
3
x
ArcTanB
1-x2
2
3
1 - x2 ArcTanBx +
F
+
1 - x2 F +
1
8
LogA1 - x2 + x4 E
14
Int@ArcSin@xD  H1 + Sqrt@1 - x ^ 2DL, xD
1 - x2 ArcSin@xD
ArcSin@xD
-
+
x
ArcSin@xD2
+
x
- ArcTanhB
2
1 - x2 F - Log@xD
Printed by Wolfram Mathematica Student Edition
mma_rubi4.nb
15
Int@Log@x + Sqrt@1 + x ^ 2DD  H1 - x ^ 2L ^ H3  2L, xD
1
2
ArcSinAx2 E +
1 + x2 F
x LogBx +
1 - x2
16
Int@ArcSin@xD  H1 + x ^ 2L ^ H3  2L, xD
x ArcSin@xD
-
ArcSinAx2 E
2
1 + x2
17
Int@Log@x + Sqrt@x ^ 2 - 1DD  H1 + x ^ 2L ^ H3  2L, xD
1
2
ArcCoshAx2 E +
x LogBx +
- 1 + x2 F
1 + x2
18
Int@Log@xD  Hx ^ 2 * Sqrt@x ^ 2 - 1DL, xD
- 1 + x2
x
- ArcTanhB
x
- 1 + x2
F+
- 1 + x2 Log@xD
x
19
Int@Sqrt@1 + x ^ 3D  x, xD
2
1 + x3
2
3
ArcTanhB
3
1 + x3 F
20
Int@x * Log@x + Sqrt@x ^ 2 - 1DD  Sqrt@x ^ 2 - 1D, xD
-x +
- 1 + x2 LogBx +
- 1 + x2 F
Printed by Wolfram Mathematica Student Edition
5