Charlwood Integration Test Suite
The following is a list of the 50 example integration problems from Kevin Charlwood’s 2008 article Integration on Computer Algebra
Systems. Each integral along with its optimal antiderivative (that is, the best antiderivative found so far) is shown. These problems are
also available in machine readable form from the Rubi home page expressed in Axiom, Maple, Mathematica and Maxima syntax.
Ÿ Problem #1
à ArcSin@xD Log@xD â x Š - 2
1 - x2 + ArcTanhB
1 - x2 F - x ArcSin@xD H1 - Log@xDL +
1 - x2 Log@xD
Ÿ Problem #2
à
x ArcSin@xD
âx Š x 1-
1 - x2 ArcSin@xD
x2
Ÿ Problem #3
à ArcSinB
x+1 -
x F âx Š
J
x +3
1+x N
4
-x +
x
1+x
3
-
+ x ArcSinB
x -
8
2
1+x F
Ÿ Problem #4
à LogB1 + x
-2 x +
1 + x2 F â x Š
2 J1 +
5 N ArcTanB
-2 +
5
x+
Ÿ Problem #5
á
Cos@xD2
x
âx Š
Cos@xD4
+
Cos@xD2
+1
1
+
3
ArcTanB
3
1+
1 + x2
F-
2 J- 1 +
5 N ArcTanhB
Cos@xD I1 + Cos@xD2 M Sin@xD
Cos@xD2
1+
Cos@xD2
+
Cos@xD4
F
2+
5
x+
1 + x2
F + x LogB1 + x
1 + x2 F
Charlwood Integration Problems
2
Ÿ Problem #6
à Tan@xD
1
4
1 + Tan@xD
âx Š 2
ArcSinhATan@xD E -
1-Tan@xD2
ArcTanhB
2
1+Tan@xD4
2
2
F
1
+
1 + Tan@xD4
2
Ÿ Problem #7
á
Tan@xD
1 + Sec@xD3 F
2
âx Š -
ArcTanhB
3
1 + Sec@xD3
Ÿ Problem #8
à
Tan@xD2 + 2 Tan@xD + 2 â x Š ArcSinh@1 + Tan@xDD -
1
2
J1 +
5 N ArcTanB
2 J1 +
2 - J1 +
5 N
5 N Tan@xD
2 + 2 Tan@xD + Tan@xD2
F-
1
2
J- 1 +
5 N ArcTanhB
2 J- 1 +
2 - J1 5 N
5 N Tan@xD
2 + 2 Tan@xD + Tan@xD2
Ÿ Problem #9
à Sin@xD ArcTanB
Sec@xD - 1 F â x Š
1
ArcTanB
2
- 1 + Sec@xD F - ArcTanB
- 1 + Sec@xD F Cos@xD +
1
Cos@xD
- 1 + Sec@xD
2
Ÿ Problem #10
à
x3 ãArcSin@xD
1
âx Š
1-
ãArcSin@xD 3 x + x3 - 3
1 - x2 - 3 x2
1 - x2
10
x2
Ÿ Problem #11
á
x LogA1 + x2 E LogBx +
1+
x2
1 + x2 F
â x Š 4 x - 2 ArcTan@xD - 2
1 + x2 LogBx +
1 + x2 F + LogA1 + x2 E - x +
1 + x2 LogBx +
1 + x2 F
F
Charlwood Integration Problems
3
Ÿ Problem #12
1 - x2 F â x Š
à ArcTanBx +
ArcSin@xD
-
-1 +
1
+
2
3 x
3 ArcTanB
4
1 - x2
1 - x2 F -
x ArcTanBx +
1
ArcTanhBx
4
F+
1+
1
3 x
3 ArcTanB
4
1 - x2
1 - x2 F -
1
8
F-
1
F-
1
- 1 + 2 x2
3 ArcTanB
4
3
LogA1 - x2 + x4 E
F+
Ÿ Problem #13
á
x ArcTanBx +
1 - x2
1 - x2 F
ArcSin@xD
-
âx Š
-1 +
1
+
2
3 x
3 ArcTanB
4
1 - x2
1 - x2 F +
1 - x2 ArcTanBx +
1
F+
1+
1
4
ArcTanhBx
4
1 - x2
1 - x2 F +
1
8
ArcSin@xD
1+
1-
ArcSin@xD2
x ArcSin@xD
âx Š -
x2
+
1+
1-
- LogB1 +
2
x2
1 - x2 F
Ÿ Problem #15
á
1 + x2 F
LogBx +
I1 - x2 M
32
1
âx Š 2
ArcSinAx2 E +
Ÿ Problem #16
á
ArcSin@xD
I1 + x2 M
32
x ArcSin@xD
âx Š
1 + x2
x LogBx +
ArcSinAx2 E
2
1 + x2 F
1 - x2
- 1 + 2 x2
3 ArcTanB
4
LogA1 - x2 + x4 E
Ÿ Problem #14
à
3 x
3 ArcTanB
3
F-
Charlwood Integration Problems
4
Ÿ Problem #17
á
x2 - 1 F
LogBx +
I1 + x2 M
1
âx Š -
32
2
ArcCoshAx2 E +
- 1 + x2 F
x LogBx +
1 + x2
Ÿ Problem #18
à
- 1 + x2
Log@xD
x
âx Š
x2
- ArcTanhB
x
x2 - 1
- 1 + x2
F+
- 1 + x2 Log@xD
x
Ÿ Problem #19
á
1 + x3
2
1 + x3
âx Š
x
2
-
3
ArcTanhB
3
1 + x3 F
Ÿ Problem #20
á
x LogBx +
x2 - 1 F
âx Š -x +
- 1 + x2 LogBx +
x2 - 1
- 1 + x2 F
Ÿ Problem #21
à
x3 ArcSin@xD
1
âx Š
x
1 + x2 -
4
1 - x4
1
1 - x4 ArcSin@xD +
ArcSinh@xD
2
4
Ÿ Problem #22
à
x3 ArcSec@xD
- 1 + x4
1
âx Š -
+
- 1 + x4 ArcSec@xD +
2
x4 - 1
2
1-
1
x2
1-
1
1
x2
ArcTanhB
2
x
- 1 + x4
x
F
Ÿ Problem #23
á
x ArcTan@xD LogBx +
1 + x2
1 + x2 F
1
â x Š - x ArcTan@xD +
2
LogA1 + x2 E +
1 + x2 ArcTan@xD LogBx +
1 + x2 F -
1
2
1 + x2 F
2
LogBx +
Charlwood Integration Problems
5
Ÿ Problem #24
á
x LogB1 +
1-
1 - x2 F
âx Š
1 + x2 F
âx Š -x +
1 - x2 F
âx Š
1 - x2 F -
1 - x2 - LogB1 +
x2
1 - x2 LogB1 +
1 - x2 F
Ÿ Problem #25
á
x LogBx +
1+
1 + x2 LogBx +
x2
1 + x2 F
Ÿ Problem #26
á
x LogBx +
1-
ArcTanhB
2 xF
1 - x2 +
ArcTanhB
2
-
x2
2
2
1 - x2 F
-
1 - x2 LogBx +
Ÿ Problem #27
à
1 - x2
Log@xD
âx Š x2
1 - x2 Log@xD
- ArcSin@xD -
x
1 - x2
x
Ÿ Problem #28
à
x ArcTan@xD
â x Š - ArcSinh@xD +
1 + x2 ArcTan@xD
1 + x2
Ÿ Problem #29
à
1 - x2 ArcTan@xD
ArcTan@xD
âx Š x2
1 - x2
- ArcTanhB
x
1 - x2 F +
1 - x2
2 ArcTanhB
2
Ÿ Problem #30
à
x ArcTan@xD
â x Š - ArcSin@xD 1 - x2
1 - x2 ArcTan@xD +
2 x
2 ArcTanB
1 - x2
F
F
1 - x2 F
Charlwood Integration Problems
6
Ÿ Problem #31
à
âx Š x2
1 + x2 F
1 + x2 ArcTan@xD
ArcTan@xD
- ArcTanhB
x
1 + x2
Ÿ Problem #32
à
1 - x2 ArcSin@xD
ArcSin@xD
âx Š x2
+ Log@xD
x
1 - x2
Ÿ Problem #33
à
x Log@xD
âx Š x2
x2 - 1 + ArcTanB
-1
x2 - 1 F +
x2 - 1 Log@xD
Ÿ Problem #34
à
1 + x2
Log@xD
âx Š x2
1 + x2 Log@xD
+ ArcSinh@xD -
x
1 + x2
x
Ÿ Problem #35
à
x ArcSec@xD
âx Š
x2 - 1 ArcSec@xD -
x2 - 1
x Log@xD
x2
Ÿ Problem #36
à
x Log@xD
âx Š 1+
1 + x2 + ArcTanhB
x2
Ÿ Problem #37
à
ArcTanhB
Sin@xD
2
âx Š 1 + Sin@xD2
Cos@xD
2
F
1 + x2 F +
1 + x2 Log@xD
Charlwood Integration Problems
7
Ÿ Problem #38
2 x
ArcTanhB
á
1 + x2
I1 - x2 M
1+x4
âx Š
1 + x4
2
F
Ÿ Problem #39
2 x
ArcTanB
á
1I1 + x2 M
x2
1+x4
âx Š
1 + x4
2
F
Ÿ Problem #40
à
Log@Sin@xDD
Cos@xD Log@Sin@xDD
â x Š - x - ArcTanh@Cos@xDD -
1 + Sin@xD
1 + Sin@xD
Ÿ Problem #41
à Log@Sin@xDD
4 Cos@xD
1 + Sin@xD â x Š
2 Cos@xD Log@Sin@xDD
-
1 + Sin@xD
1 + Sin@xD
Ÿ Problem #42
á
Sec@xD4 -1
ArcTanhB
Sec@xD
2 Sec@xD Tan@xD
âx Š 2
Sec@xD4 - 1
F
Ÿ Problem #43
1-Tan@xD2
ArcTanhB
á
Tan@xD
1+Tan@xD4
2
âx Š 1 + Tan@xD4
2
2
F
Cos@xD
- 4 ArcTanhB
1 + Sin@xD
F
Charlwood Integration Problems
8
Ÿ Problem #44
3 Cos@xD I1+Sin@xD2 M
3
F
1 + Sec@xD
-1 +
ArcTanhB
á
Sin@xD
1-Sin@xD6
2
âx Š
2
6
1 - Sin@xD
Ÿ Problem #45
à
Sec@xD + 1 -
Sec@xD - 1
2 Cot@xD
- 1 + Sec@xD
âx Š
-2 + 2
2
2
ArcTanB
2
2+2
-
1+
2
J-
2
ArcTanB
-
2
2+2
-1 +
2
2
2 -
- 1 + Sec@xD +
-
1 + Sec@xD N
- 1 + Sec@xD +
1 + Sec@xD
- 1 + Sec@xD +
1 + Sec@xD
ArcTanhB
2 -
- 1 + Sec@xD +
1 + Sec@xD
F-
J-
-
2 -
- 1 + Sec@xD +
- 1 + Sec@xD +
1 + Sec@xD N
1 + Sec@xD
-2 + 2
1+
2
2
-
F
- 1 + Sec@xD +
1 + Sec@xD
ArcTanhB
2 -
- 1 + Sec@xD +
1 + Sec@xD
F+
F
Ÿ Problem #46
2
2
2
à x LogAx + 1E ArcTan@xD â x Š x ArcTan@xD I3 - LogA1 + x EM -
1
4
I6 - LogA1 + x2 EM LogA1 + x2 E -
1
2
ArcTan@xD2 I3 + x2 - I1 + x2 M LogA1 + x2 EM
Ÿ Problem #47
à ArcTanBx
x ArcTanBx
1 + x2 F â x Š
1 + x2 F +
1
ArcTanB
2
3 -2
1 + x2 F -
1
ArcTanB
3 +2
2
Ÿ Problem #48
à ArcTanB
x+1 -
x F âx Š
x
2
+ H1 + xL ArcTanB
1+x -
x F
1 + x2 F -
1
4
3 LogB2 + x2 -
3
1 + x2 F +
1
4
3 LogB2 + x2 +
3
1 + x2 F
Charlwood Integration Problems
9
Ÿ Problem #49 (Note: The problem was altered so as to have an answer involving only elementary functions and operators.)
à ArcSinB
x
1-
x2
F â x Š x ArcSinB
x
1-
x2
F + ArcTanB
1 - 2 x2 F
Ÿ Problem #50
à ArcTanBx
x ArcTanBx
1 - x2 F â x Š
1 - x2 F -
1
2
J1 +
5 N ArcTanB
1
2
J1 +
5 N
1 - x2 F +
1
2
J- 1 +
5 N ArcTanhB
1
2
J- 1 +
5 N
1 - x2 F