Trig Function Integration Problem 1

1
ⅆx
1 + Sin[a + b x]
◼ The Rubi results are simple and symmetric:
Int
1
1 + Sin[a + b x]
, x
Cos[a + b x]
-
b 1 + Sin[a + b x]
Int
1
1 - Sin[a + b x]
, x
Cos[a + b x]
b 1 - Sin[a + b x]
◼ The Mathematica results are symmetric but more complicated:
1

1 + Sin[a + b x]
ⅆx
2 Sin 1 (a + b x)
2
b
Cos 1
2
(a + b x) + Sin 1 (a + b x)
2
1

1 - Sin[a + b x]
ⅆx
2 Sin 1 (a + b x)
2
b
Cos 1
2
(a + b x) - Sin 1 (a + b x)
2
◼ The Maple results are simple and symmetric:
int 1  1 + sin (a + b * x), x;
2
b 1 +
Tan 1
2
(a + b x)
int 1  1 - sin (a + b * x), x;
-
2
b - 1 + Tan 1 (a + b x)
2
Trig Function Integration Problem 2

1
ⅆx
1 + Sec[a + b x]
◼ The Rubi result is simple:
Int
x-
1
1 + Sec[a + b x]
, x
Sin[a + b x]
b 1 + Cos[a + b x]
◼ The Mathematica result is unnecessarily complicated:
1

1 + Sec[a + b x]
2 Cos
1
2
ⅆx
(a + b x) Sec[a + b x] (a + b x) Cos
1
2
(a + b x) - Sin
1
2
(a + b x)
 b 1 + Sec[a + b x]
◼ The Maple result includes a term whose derivative is 1, the same as the derivative of x:
int 1  1 + sec (a + b * x), x;
Tan 1 (a + b x)
-
2
b
2 ArcTanTan 1 (a + b x)
+
2
b
Trig Function Integration Problem 3
 Sin[x] Tan[n x] ⅆ x
◼ The Rubi results are simple, expressed in trigonometric form and grow modestly with n:
Int[Sin[x] Tan[x], x]
ArcTanh[Sin[x]] - Sin[x]
Int[Sin[x] Tan[2 x], x]
ArcTanh
2 Sin[x]
- Sin[x]
2
Int[Sin[x] Tan[3 x], x]
1
3
ArcTanh[Sin[x]] +
1
3
ArcTanh[2 Sin[x]] - Sin[x]
Int[Sin[x] Tan[4 x], x]
1
2-
4
2
ArcTanh
2 Sin[x]
2-
+
2
1
2+
4
2
ArcTanh
2 Sin[x]
2+
 - Sin[x]
2
◼ The Mathematica results grow unpredictably, are expressed in terms of logarithms and involve RootSum
when n is 4:
 Sin[x] Tan[x] ⅆ x
x
x
x
x
- LogCos  - Sin  + LogCos  + Sin  - Sin[x]
2
2
2
2
 Sin[x] Tan[2 x] ⅆ x
1
-2 ⅈ
8
2 ArcTan
Cos x  - - 1 +
2
1 +
2ⅈ
2 ArcTan
2
2
2
Cos x 
2
Cos x  - 1 +
- 1 +
2 Log4 + 2
2
2  Sin x 
2  Sin x 
2
Cos x 
2
2 Cos[x] - 2
-
- Sin x 
2
- Sin x 
+2
2 Log
2
2 Sin[x] -
2 Log- 2 - 2 +
 Sin[x] Tan[3 x] ⅆ x
-
1
2 + 2 Sin[x] -
x
x
1
x
x
LogCos  - Sin  + LogCos  + Sin  3
2
2
3
2
2
1
1
Log[- 1 + 2 Sin[x]] + Log[1 + 2 Sin[x]] - Sin[x]
6
6
2 Cos[x] +
2 Sin[x] - 8 Sin[x]
 Sin[x] Tan[4 x] ⅆ x
1
16
RootSum1 + #18 &,
2 ArcTan
1
7
2 ArcTan
#1
Sin[x]
Cos[x] - #1
Sin[x]
Cos[x] - #1
 - ⅈ Log1 - 2 Cos[x] #1 + #12  +
 #16 - ⅈ Log1 - 2 Cos[x] #1 + #12  #16 & - Sin[x]
◼ The Maple results are simple, but expressed in terms of logarithms when n is odd:
int (sin (x) * tan (x), x);
Log[Sec[x] + Tan[x]] - Sin[x]
int sin (x) * tan 2 * x, x;
ArcTanh
2 Sin[x]
- Sin[x]
2
int sin (x) * tan 3 * x, x;
1
-
6
Log[- 1 + Sin[x]] +
1
6
Log[1 + Sin[x]] -
1
6
Log[- 1 + 2 Sin[x]] +
1
6
Log[1 + 2 Sin[x]] - Sin[x]
int (sin (x) * tan (4 * x), x);
1
4
2-
2
ArcTanh
2 Sin[x]
2-
2
+
1
4
2+
2
ArcTanh
2 Sin[x]
2+
 - Sin[x]
2
Note that these systems give similar results to the above for the cosine function.
Trig Function Integration Problem 4

a + b Sin[x] ⅆ x
◼ Rubi instantaneously computes the general case, and the special cases are relatively simple:
Int
a + b Sin[x] , x
2 EllipticE 1 - π + 2 x,
4
2b

a+b
a + b Sin[x]
a+b Sin[x]
a+b
Int
1 + Sin[x] , x
2 Cos[x]
-
1 + Sin[x]
Int
1 - Sin[x] , x
2 Cos[x]
1 - Sin[x]
◼ Mathematica requires several seconds to compute the general case, and the special cases are more
complicated:

a + b Sin[x] ⅆ x
2 EllipticE 1 π - 2 x,
4
-
2b

a+b
a + b Sin[x]
a+b Sin[x]
a+b

1 + Sin[x] ⅆ x
2 - Cos x  + Sin x 
2
1 + Sin[x]
2
Cos x  + Sin x 
2

2
1 - Sin[x] ⅆ x
2 Cos x  + Sin x 
2
2
Cos x 
2
1 - Sin[x]
- Sin x 
2
◼ The Maple result for the general case is more complicated (note that Maple defines EllipticE and F
differently than Mathmatica):
int (sqrt (a + b * sin (x)), x);
2 (a - b)
a + b Sin[x]
b 1 - Sin[x]
a-b
a+b
(a + b) EllipticF
a + b Sin[x]
a + b Sin[x] 
b Cos[x]
int sqrt 1 + sin (x), x;
2 Sin[x] - 1
1 + Sin[x]
Cos[x]
int sqrt 1 - sin (x), x;
2 1 - Sin[x] 1 + Sin[x]
Cos[x]
1 - Sin[x]
a-b
,
-
a-b
a+b
b 1 + Sin[x]
a-b
 - (a + b) EllipticE
a + b Sin[x]
a-b
,
a-b
a+b


Trig Function Integration Problem 5
 Sec[a + b x] ⅆ x & 
Sec[a + b x]2 ⅆ x
◼ The Rubi results are symmetric and simple:
Int[Sec[a + b x], x]
ArcTanh[Sin[a + b x]]
b
Int
Sec[a + b x]2 , x
ArcSinh[Tan[a + b x]]
b
◼ The Mathematica results are symmetric but not simple:
 Sec[a + b x] ⅆ x
LogCos a +
2
-
- Sin a +
2
bx

2
b
LogCos a +
+
2
bx

2
+ Sin a +
2
bx

2
b
Sec[a + b x]2 ⅆ x

-
bx

2
1
b
Cos[a + b x]
LogCos
1
2
(a + b x) - Sin
1
2
(a + b x) - LogCos
1
2
(a + b x) + Sin
1
2
◼ The Maple results are relatively simple but not symmetric:
int (sec (a + b * x), x);
Log[Sec[a + b x] + Tan[a + b x]]
b
int sqrt sec (a + b * x) ^ 2, x;
-
1
b
2 ArcTanh- 1 + Cos[a + b x] Csc[a + b x] Cos[a + b x]
Sec[a + b x]2
(a + b x)
Sec[a + b x]2
Trig Function Integration Problem 6

Sec[x]2 ⅆ x & 
Csc[x]2 ⅆ x
◼ The Rubi results are simple and symmetric:
Int
Sec[x]2 , x
ArcSinh[Tan[x]]
Int
Csc[x]2 , x
- ArcSinh[Cot[x]]
◼ The Mathematica results are symmetric but more complicated:

Sec[x]2 ⅆ x
x
x
x
x
Cos[x] - LogCos  - Sin  + LogCos  + Sin 
2
2
2
2

Csc[x]2 ⅆ x
Csc[x]2
x
x
- LogCos  + LogSin  Sin[x]
2
2
◼ The Maple results are not symmetric and more complicated:
int sqrt sec (x) ^ 2, x;
- 2 ArcTanh- 1 + Cos[x] Csc[x] Cos[x]
Sec[x]2
int sqrt csc (x) ^ 2, x;
1
2
4
1
1 - Cos[x]2
Sin[x] Log
1 - Cos[x]
Sin[x]

Sec[x]2
Trig Function Integration Problem 7
1 + Sec[x] ⅆ x & 

1 + Csc[x] ⅆ x
◼ The Rubi results are simple, expressed in terms of elementary functions and symmetric:
Int
1 + Sec[x] , x
Tan[x]
2 ArcTan

1 + Sec[x]
Int
1 + Csc[x] , x
Cot[x]
- 2 ArcTan

1 + Csc[x]
◼ The Mathematica results are large, not expressed in terms of elementary functions and not symmetric:

1 + Sec[x] ⅆ x
4
2 3 + 2
2
-1 +
2 - - 2 +
2  Cos x 
2
1-
1 + Cos x 
2 + - 2 +
2
-1 +
2 + - 2 +
x
2  Cos 
2
2 EllipticPi- 3 + 2
2
EllipticFArcSin
2 , - ArcSin

-
1 + Sec[x]

4
3-2
Tan x 
4
3-2
x
Sec  Sec[x]
2
Tan x 
- -1 +
, 17 - 12
x
2  Cos 
2
, 17 - 12
2 +
2
2
2
2 + - 2 +
x
x 2
2  Cos  Sec 
2
4
1 + Csc[x] ⅆ x
2 ArcTan
- 1 + Csc[x]  Cot[x]
- 1 + Csc[x]
1 + Csc[x]
◼ The Maple results are expressed in terms of elementary functions, but large and not symmetric:
int sqrt 1 + sec (x), x;
- 2 ArcTanh
-
Cos[x]
1 + Cos[x]
Tan[x]
int sqrt 1 + csc (x), x;
-
Cos[x]
1 + Cos[x]
1 + Cos[x]
Cos[x]
-
1
2 - 1 + Cos[x] - Sin[x]
1 - Cos[x]
Sin[x]
Sin[x]
Log - 1 + Cos[x] -
- 1 + Cos[x] +
4 ArcTan
1 + Sin[x]
2
1 - Cos[x]
Sin[x]
1 - Cos[x]
Sin[x]
- 1 + Cos[x] -
Sin[x]
1 - Cos[x]
Sin[x]
2 Sin[x] - Sin[x] 
2 Sin[x] - Sin[x]  + 4 ArcTan
2 - 1 + Log - 1 + Cos[x] +
1 - Cos[x]
Sin[x]
2 Sin[x] - Sin[x] 
1 - Cos[x]
Sin[x]
1 - Cos[x]
Sin[x]
2 + 1 +
2 Sin[x] - Sin[x] 
Trig Function Integration Problem 8

a Cos[x] + b Sin[x] ⅆ x
◼ Rubi returns a relatively simple result by using the identity a cos(z)+b sin(z) = sqrt(a^2+b^2) cos(zarctan(a,b)):
Int
c * Cos[x - d] , x
c Cos[d - x] EllipticE d-x , 2
2
2
Cos[d - x]
Int
a Cos[x] + b Sin[x] , x
2 EllipticE
1
2
a Cos[x] + b Sin[x]
(x - ArcTan[a, b]), 2

a Cos[x] + b Sin[x]
a2 + b2
◼ Mathematica knows the identity, but instead returns a complicated expression involving a hypergeometric
function:
c * Cos[x - d] ⅆ x

c Cos[d - x] EllipticE d-x , 2
2
2
Cos[d - x]

a Cos[x] + b Sin[x] ⅆ x
1
- b a2 + b2  HypergeometricPFQ-
b 2
Sinx - ArcTan 
a
2 a2 b
ab
1+
b2
a2
1+
b2
a2
2 a3
2
,-
1+
1
3
b 2
b
,  , Cosx - ArcTan   Sinx - ArcTan  +
4
4
a
a
b2
a2
b
Cos[x] - 2 a a2 + b2  Cosx - ArcTan  +
a
b
b
Sin[x] + a2 b Sinx - ArcTan  + b3 Sinx - ArcTan 
a
a
a Cos[x] + b Sin[x]

b 2
Sinx - ArcTan 
a
◼ The Maple results are more complicated (note that Maple defines EllipticE and F differently than
Mathmatica):
int (sqrt (c * cos (x - d)), x);
-
2c
c
1 - Cos
d-x
2
- 1 + 2 Cos

2
d-x
2
1 - 2 Cos

2
Sin
d-x
2
d-x
2

2
EllipticECos

leafcount (int (sqrt (a * cos (x) + b * sin (x)), x));
"Large expression involving EllipticE and EllipticF."
d-x
2
,
2 
Trig Function Integration Problem 9
Sin[x]

a+b
ⅆx
Sin[x]2
◼ The Rubi results are simple and expressed in trigonometric form:
Sin[x]
Int
a+b
ArcTan
, x
Sin[x]2
b Cos[x]

a+b-b Cos[x]2
-
b
Sin[x]
Int
1
- ArcSin
, x
+ Sin[x]2
Cos[x]

2
Sin[x]
Int
, x
1 - Sin[x]2
-
Cos[x] Log[Cos[x]]
Cos[x]2
◼ The Mathematica results are simple but can involve the imaginary unit:
Sin[x]
ⅆx

a + b Sin[x]2
-
Log
2
- b Cos[x] +
2 a + b - b Cos[2 x] 
-b
Sin[x]

ⅆx
1 + Sin[x]2
ⅈ Logⅈ
2 Cos[x] +
Sin[x]

1
-
3 - Cos[2 x] 
ⅆx
- Sin[x]2
Cos[x] Log[Cos[x]]
Cos[x]2
◼ The Maple results are more complicated:
int sin (x) / sqrt a + b * sin (x) ^ 2, x;
2 b Sin[x]2 + a - b
- a + b Sin[x]2  - 1 + Sin[x]2  ArcTan
2
2
b Cos[x]
b
a + b Sin[x]2
int sin (x) / sqrt 1 + sin (x) ^ 2, x;
Cos[x]2 1 + Sin[x]2  ArcSinSin[x]2 
2 Cos[x]
1 + Sin[x]2
int sin (x) / sqrt 1 - sin (x) ^ 2, x;
- 1 + Sin[x]2  Log[Sin[x] - 1] + Log[1 + Sin[x]]
2 Cos[x]
1 - Sin[x]2
- a + b Sin[x]2  - 1 + Sin[x]2 
 
Trig Function Integration Problem 10
Cot[x]

a+b
Tan[x]2
+c
ⅆx
Tan[x]4
◼ Rubi is able to integrate the expression:
Cot[x]
Int
a+b
ArcTanh 2
Tan[x]2
+c
, x
Tan[x]4
a+b Tan[x]2 +c Tan[x]4
2 a+b Tan[x]2
a
2
ArcTanh 2

+
a-b+c
a+b Tan[x]2 +c Tan[x]4
2 a-b+(b-2 c) Tan[x]2
2
a
a-b+c
◼ Mathematica is unable to integrate the expression in 60 seconds:
Cot[x]
ⅆx

a+b
Tan[x]2
+c
Tan[x]4
$Aborted
◼ Maple is unable to integrate the expression:
int cot (x) / sqrt a + b * tan (x) ^ 2 + c * tan (x) ^ 4, x;
Cot[x]

a + b Tan[x]2 + c Tan[x]4
ⅆx
