\(\int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx\) [729]

Optimal result

Integrand size = 25, antiderivative size = 669 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=-\frac {b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (b c-a d)^5 f}-\frac {d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^5 \left (c^2-d^2\right )^{5/2} f}-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))} \] Output:

-b^3*(10*a^3*b*c*d-4*a*b^3*c*d-20*a^4*d^2-a^2*b^2*(2*c^2-29*d^2)-b^4*(c^2+ 
12*d^2))*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)/ 
(-a*d+b*c)^5/f-d^3*(a^2*d^2*(2*c^2+d^2)-a*b*(10*c^3*d-4*c*d^3)+b^2*(20*c^4 
-29*c^2*d^2+12*d^4))*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(-a* 
d+b*c)^5/(c^2-d^2)^(5/2)/f-1/2*d*(a^4*d^3-b^4*d*(5*c^2-6*d^2)+2*a^2*b^2*d* 
(4*c^2-5*d^2)-3*a*b^3*c*(c^2-d^2))*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^3/(c^ 
2-d^2)/f/(c+d*sin(f*x+e))^2+1/2*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b 
*sin(f*x+e))^2/(c+d*sin(f*x+e))^2+1/2*b^2*(-7*a^2*d+3*a*b*c+4*b^2*d)*cos(f 
*x+e)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^2+3/2*d 
*(a^5*c*d^4-2*a^3*b^2*c*d^4+a*b^4*c*(c^4-2*c^2*d^2+2*d^4)+b^5*d*(2*c^4-7*c 
^2*d^2+4*d^4)-a^2*b^3*d*(3*c^4-12*c^2*d^2+7*d^4)-a^4*b*(3*c^2*d^3-2*d^5))* 
cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^4/(c^2-d^2)^2/f/(c+d*sin(f*x+e))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1815\) vs. \(2(669)=1338\).

Time = 8.81 (sec) , antiderivative size = 1815, normalized size of antiderivative = 2.71 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]
 

Output:

-((b^3*(2*a^2*b^2*c^2 + b^4*c^2 - 10*a^3*b*c*d + 4*a*b^3*c*d + 20*a^4*d^2 
- 29*a^2*b^2*d^2 + 12*b^4*d^2)*ArcTan[(Sec[(e + f*x)/2]*(b*Cos[(e + f*x)/2 
] + a*Sin[(e + f*x)/2]))/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(-(b*c) + a* 
d)^5*f)) - (d^3*(20*b^2*c^4 - 10*a*b*c^3*d + 2*a^2*c^2*d^2 - 29*b^2*c^2*d^ 
2 + 4*a*b*c*d^3 + a^2*d^4 + 12*b^2*d^4)*ArcTan[(Sec[(e + f*x)/2]*(d*Cos[(e 
 + f*x)/2] + c*Sin[(e + f*x)/2]))/Sqrt[c^2 - d^2]])/((b*c - a*d)^5*(c^2 - 
d^2)^(5/2)*f) + (32*a^2*b^5*c^7*Cos[e + f*x] - 8*b^7*c^7*Cos[e + f*x] - 80 
*a^3*b^4*c^6*d*Cos[e + f*x] + 68*a*b^6*c^6*d*Cos[e + f*x] - 92*a^2*b^5*c^5 
*d^2*Cos[e + f*x] + 38*b^7*c^5*d^2*Cos[e + f*x] + 140*a^3*b^4*c^4*d^3*Cos[ 
e + f*x] - 122*a*b^6*c^4*d^3*Cos[e + f*x] - 80*a^6*b*c^3*d^4*Cos[e + f*x] 
+ 140*a^4*b^3*c^3*d^4*Cos[e + f*x] + 48*a^2*b^5*c^3*d^4*Cos[e + f*x] - 72* 
b^7*c^3*d^4*Cos[e + f*x] + 32*a^7*c^2*d^5*Cos[e + f*x] - 92*a^5*b^2*c^2*d^ 
5*Cos[e + f*x] + 48*a^3*b^4*c^2*d^5*Cos[e + f*x] + 12*a*b^6*c^2*d^5*Cos[e 
+ f*x] + 68*a^6*b*c*d^6*Cos[e + f*x] - 122*a^4*b^3*c*d^6*Cos[e + f*x] + 12 
*a^2*b^5*c*d^6*Cos[e + f*x] + 36*b^7*c*d^6*Cos[e + f*x] - 8*a^7*d^7*Cos[e 
+ f*x] + 38*a^5*b^2*d^7*Cos[e + f*x] - 72*a^3*b^4*d^7*Cos[e + f*x] + 36*a* 
b^6*d^7*Cos[e + f*x] - 12*a*b^6*c^6*d*Cos[3*(e + f*x)] + 28*a^2*b^5*c^5*d^ 
2*Cos[3*(e + f*x)] - 22*b^7*c^5*d^2*Cos[3*(e + f*x)] + 20*a^3*b^4*c^4*d^3* 
Cos[3*(e + f*x)] + 10*a*b^6*c^4*d^3*Cos[3*(e + f*x)] + 20*a^4*b^3*c^3*d^4* 
Cos[3*(e + f*x)] - 96*a^2*b^5*c^3*d^4*Cos[3*(e + f*x)] + 64*b^7*c^3*d^4...
 

Rubi [A] (verified)

Time = 5.15 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.15, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3281, 25, 3042, 3534, 3042, 3534, 27, 3042, 3534, 3042, 3480, 3042, 3139, 1083, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3}dx\)

\(\Big \downarrow \) 3281

\(\displaystyle \frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {\int -\frac {-3 b^2 d \sin ^2(e+f x)-b (b c-2 a d) \sin (e+f x)+2 \left (-d a^2+b c a+2 b^2 d\right )}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3}dx}{2 \left (a^2-b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-3 b^2 d \sin ^2(e+f x)-b (b c-2 a d) \sin (e+f x)+2 \left (-d a^2+b c a+2 b^2 d\right )}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3}dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {-3 b^2 d \sin (e+f x)^2-b (b c-2 a d) \sin (e+f x)+2 \left (-d a^2+b c a+2 b^2 d\right )}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3}dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\int \frac {-2 d^2 a^4+4 b c d a^3-2 b^2 \left (c^2-10 d^2\right ) a^2-7 b^3 c d a+2 b^2 d \left (-7 d a^2+3 b c a+4 b^2 d\right ) \sin ^2(e+f x)-b^4 \left (c^2+12 d^2\right )-b d \left (-4 d a^3+b^2 d a+3 b^3 c\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3}dx}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\int \frac {-2 d^2 a^4+4 b c d a^3-2 b^2 \left (c^2-10 d^2\right ) a^2-7 b^3 c d a+2 b^2 d \left (-7 d a^2+3 b c a+4 b^2 d\right ) \sin (e+f x)^2-b^4 \left (c^2+12 d^2\right )-b d \left (-4 d a^3+b^2 d a+3 b^3 c\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3}dx}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {\int \frac {2 \left (2 c d^3 a^5-6 b d^2 \left (c^2-d^2\right ) a^4+2 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (2 c^4-23 d^2 c^2+21 d^4\right ) a^2-2 b^4 c d \left (3 c^2-4 d^2\right ) a-b d \left (d^3 a^4+2 b^2 d \left (4 c^2-5 d^2\right ) a^2-3 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (5 c^2-6 d^2\right )\right ) \sin ^2(e+f x)-b^5 \left (c^4+11 d^2 c^2-12 d^4\right )+d \left (-d^3 a^5+2 b c d^2 a^4+2 b^2 d \left (3 c^2-2 d^2\right ) a^3-b^3 c \left (c^2+3 d^2\right ) a^2-b^4 d \left (3 c^2-2 d^2\right ) a-2 b^5 c \left (c^2-2 d^2\right )\right ) \sin (e+f x)\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right ) (b c-a d)}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {\int \frac {2 c d^3 a^5-6 b d^2 \left (c^2-d^2\right ) a^4+2 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (2 c^4-23 d^2 c^2+21 d^4\right ) a^2-2 b^4 c d \left (3 c^2-4 d^2\right ) a-b d \left (d^3 a^4+2 b^2 d \left (4 c^2-5 d^2\right ) a^2-3 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (5 c^2-6 d^2\right )\right ) \sin ^2(e+f x)-b^5 \left (c^4+11 d^2 c^2-12 d^4\right )+d \left (-d^3 a^5+2 b c d^2 a^4+2 b^2 d \left (3 c^2-2 d^2\right ) a^3-b^3 c \left (c^2+3 d^2\right ) a^2-b^4 d \left (3 c^2-2 d^2\right ) a-2 b^5 c \left (c^2-2 d^2\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {\int \frac {2 c d^3 a^5-6 b d^2 \left (c^2-d^2\right ) a^4+2 b^2 c d \left (3 c^2-5 d^2\right ) a^3-b^3 \left (2 c^4-23 d^2 c^2+21 d^4\right ) a^2-2 b^4 c d \left (3 c^2-4 d^2\right ) a-b d \left (d^3 a^4+2 b^2 d \left (4 c^2-5 d^2\right ) a^2-3 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (5 c^2-6 d^2\right )\right ) \sin (e+f x)^2-b^5 \left (c^4+11 d^2 c^2-12 d^4\right )+d \left (-d^3 a^5+2 b c d^2 a^4+2 b^2 d \left (3 c^2-2 d^2\right ) a^3-b^3 c \left (c^2+3 d^2\right ) a^2-b^4 d \left (3 c^2-2 d^2\right ) a-2 b^5 c \left (c^2-2 d^2\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\cos (e+f x) b^2}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {\frac {b^2 \left (-7 d a^2+3 b c a+4 b^2 d\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {d \left (d^3 a^4+2 b^2 d \left (4 c^2-5 d^2\right ) a^2-3 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {\frac {\int \frac {-\left (\left (d^6+2 c^2 d^4\right ) a^6\right )+b c d^3 \left (8 c^2-5 d^2\right ) a^5-2 b^2 d^2 \left (6 c^4-14 d^2 c^2+5 d^4\right ) a^4+2 b^3 c d \left (4 c^4-16 d^2 c^2+9 d^4\right ) a^3-b^4 \left (2 c^6-28 d^2 c^4+52 d^4 c^2-23 d^6\right ) a^2-b^5 c d \left (5 c^4-18 d^2 c^2+10 d^4\right ) a-b^6 \left (c^2-d^2\right )^2 \left (c^2+12 d^2\right )-b d (b c+a d) \left (\left (d^4+2 c^2 d^2\right ) a^4-b \left (10 c^3 d-4 c d^3\right ) a^3+2 b^2 \left (c^4+4 d^2 c^2-5 d^4\right ) a^2+2 b^3 c d \left (2 c^2+d^2\right ) a+b^4 \left (c^4-10 d^2 c^2+6 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))}dx}{(b c-a d) \left (c^2-d^2\right )}-\frac {3 d \left (c d^4 a^5-b \left (3 c^2 d^3-2 d^5\right ) a^4-2 b^2 c d^4 a^3-b^3 d \left (3 c^4-12 d^2 c^2+7 d^4\right ) a^2+b^4 c \left (c^4-2 d^2 c^2+2 d^4\right ) a+b^5 d \left (2 c^4-7 d^2 c^2+4 d^4\right )\right ) \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))}}{(b c-a d) \left (c^2-d^2\right )}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cos (e+f x) b^2}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {\frac {b^2 \left (-7 d a^2+3 b c a+4 b^2 d\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {d \left (d^3 a^4+2 b^2 d \left (4 c^2-5 d^2\right ) a^2-3 b^3 c \left (c^2-d^2\right ) a-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {\frac {\int \frac {-\left (\left (d^6+2 c^2 d^4\right ) a^6\right )+b c d^3 \left (8 c^2-5 d^2\right ) a^5-2 b^2 d^2 \left (6 c^4-14 d^2 c^2+5 d^4\right ) a^4+2 b^3 c d \left (4 c^4-16 d^2 c^2+9 d^4\right ) a^3-b^4 \left (2 c^6-28 d^2 c^4+52 d^4 c^2-23 d^6\right ) a^2-b^5 c d \left (5 c^4-18 d^2 c^2+10 d^4\right ) a-b^6 \left (c^2-d^2\right )^2 \left (c^2+12 d^2\right )-b d (b c+a d) \left (\left (d^4+2 c^2 d^2\right ) a^4-b \left (10 c^3 d-4 c d^3\right ) a^3+2 b^2 \left (c^4+4 d^2 c^2-5 d^4\right ) a^2+2 b^3 c d \left (2 c^2+d^2\right ) a+b^4 \left (c^4-10 d^2 c^2+6 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))}dx}{(b c-a d) \left (c^2-d^2\right )}-\frac {3 d \left (c d^4 a^5-b \left (3 c^2 d^3-2 d^5\right ) a^4-2 b^2 c d^4 a^3-b^3 d \left (3 c^4-12 d^2 c^2+7 d^4\right ) a^2+b^4 c \left (c^4-2 d^2 c^2+2 d^4\right ) a+b^5 d \left (2 c^4-7 d^2 c^2+4 d^4\right )\right ) \cos (e+f x)}{(b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))}}{(b c-a d) \left (c^2-d^2\right )}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}\)

\(\Big \downarrow \) 3480

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {\frac {\frac {d^3 \left (a^2-b^2\right )^2 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{b c-a d}+\frac {b^3 \left (c^2-d^2\right )^2 \left (-20 a^4 d^2+10 a^3 b c d-a^2 b^2 \left (2 c^2-29 d^2\right )-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{a+b \sin (e+f x)}dx}{b c-a d}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {3 d \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {\frac {\frac {d^3 \left (a^2-b^2\right )^2 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{b c-a d}+\frac {b^3 \left (c^2-d^2\right )^2 \left (-20 a^4 d^2+10 a^3 b c d-a^2 b^2 \left (2 c^2-29 d^2\right )-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{a+b \sin (e+f x)}dx}{b c-a d}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {3 d \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {\frac {\frac {2 d^3 \left (a^2-b^2\right )^2 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f (b c-a d)}+\frac {2 b^3 \left (c^2-d^2\right )^2 \left (-20 a^4 d^2+10 a^3 b c d-a^2 b^2 \left (2 c^2-29 d^2\right )-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 b \tan \left (\frac {1}{2} (e+f x)\right )+a}d\tan \left (\frac {1}{2} (e+f x)\right )}{f (b c-a d)}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {3 d \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {\frac {-\frac {4 d^3 \left (a^2-b^2\right )^2 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b c-a d)}-\frac {4 b^3 \left (c^2-d^2\right )^2 \left (-20 a^4 d^2+10 a^3 b c d-a^2 b^2 \left (2 c^2-29 d^2\right )-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b c-a d)}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {3 d \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}}{\left (c^2-d^2\right ) (b c-a d)}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}+\frac {\frac {\frac {2 d^3 \left (a^2-b^2\right )^2 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \sqrt {c^2-d^2} (b c-a d)}+\frac {2 b^3 \left (c^2-d^2\right )^2 \left (-20 a^4 d^2+10 a^3 b c d-a^2 b^2 \left (2 c^2-29 d^2\right )-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (e+f x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{f \sqrt {a^2-b^2} (b c-a d)}}{\left (c^2-d^2\right ) (b c-a d)}-\frac {3 d \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}}{\left (c^2-d^2\right ) (b c-a d)}}{\left (a^2-b^2\right ) (b c-a d)}}{2 \left (a^2-b^2\right ) (b c-a d)}\)

Input:

Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]
 

Output:

(b^2*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c 
+ d*Sin[e + f*x])^2) + ((b^2*(3*a*b*c - 7*a^2*d + 4*b^2*d)*Cos[e + f*x])/( 
(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - ( 
(d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b^ 
3*c*(c^2 - d^2))*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f 
*x])^2) + (((2*b^3*(c^2 - d^2)^2*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 
- a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTan[(2*b + 2*a*Tan[(e 
+ f*x)/2])/(2*Sqrt[a^2 - b^2])])/(Sqrt[a^2 - b^2]*(b*c - a*d)*f) + (2*(a^2 
 - b^2)^2*d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*(10*c^3*d - 4*c*d^3) + b^2*(20* 
c^4 - 29*c^2*d^2 + 12*d^4))*ArcTan[(2*d + 2*c*Tan[(e + f*x)/2])/(2*Sqrt[c^ 
2 - d^2])])/((b*c - a*d)*Sqrt[c^2 - d^2]*f))/((b*c - a*d)*(c^2 - d^2)) - ( 
3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5 
*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - 
a^4*b*(3*c^2*d^3 - 2*d^5))*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*(c + d 
*Sin[e + f*x])))/((b*c - a*d)*(c^2 - d^2)))/((a^2 - b^2)*(b*c - a*d)))/(2* 
(a^2 - b^2)*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 3281

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 
))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n 
 + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2* 
n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))
 

rule 3480

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b 
- a*B)/(b*c - a*d)   Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ 
(b*c - a*d)   Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f 
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

Maple [A] (verified)

Time = 26.21 (sec) , antiderivative size = 1157, normalized size of antiderivative = 1.73

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1157\)
default	\(\text {Expression too large to display}\)	\(1157\)
risch	\(\text {Expression too large to display}\)	\(6257\)

Input:

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)
 

Output:

1/f*(2*d^3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(a*d-b*c)^2*((1/2 
*d^2*(5*a^2*c^2*d^2-2*a^2*d^4-16*a*b*c^3*d+10*a*b*c*d^3+11*b^2*c^4-8*b^2*c 
^2*d^2)/(c^4-2*c^2*d^2+d^4)/c*tan(1/2*f*x+1/2*e)^3+1/2*d*(4*a^2*c^4*d^2+7* 
a^2*c^2*d^4-2*a^2*d^6-14*a*b*c^5*d-20*a*b*c^3*d^3+16*a*b*c*d^5+10*b^2*c^6+ 
13*b^2*c^4*d^2-14*b^2*c^2*d^4)/(c^4-2*c^2*d^2+d^4)/c^2*tan(1/2*f*x+1/2*e)^ 
2+1/2*d^2*(11*a^2*c^2*d^2-2*a^2*d^4-40*a*b*c^3*d+22*a*b*c*d^3+29*b^2*c^4-2 
0*b^2*c^2*d^2)/c/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e)+1/2*d*(4*a^2*c^2*d 
^2-a^2*d^4-14*a*b*c^3*d+8*a*b*c*d^3+10*b^2*c^4-7*b^2*c^2*d^2)/(c^4-2*c^2*d 
^2+d^4))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)^2+1/2*(2*a^2*c^ 
2*d^2+a^2*d^4-10*a*b*c^3*d+4*a*b*c*d^3+20*b^2*c^4-29*b^2*c^2*d^2+12*b^2*d^ 
4)/(c^4-2*c^2*d^2+d^4)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+ 
2*d)/(c^2-d^2)^(1/2)))-2*b^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)^3*((1/2 
*b^2*(11*a^4*d^2-16*a^3*b*c*d+5*a^2*b^2*c^2-8*a^2*b^2*d^2+10*a*b^3*c*d-2*b 
^4*c^2)/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3+1/2*b*(10*a^6*d^2-14*a^ 
5*b*c*d+4*a^4*b^2*c^2+13*a^4*b^2*d^2-20*a^3*b^3*c*d+7*a^2*b^4*c^2-14*a^2*b 
^4*d^2+16*a*b^5*c*d-2*b^6*c^2)/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^ 
2+1/2*b^2*(29*a^4*d^2-40*a^3*b*c*d+11*a^2*b^2*c^2-20*a^2*b^2*d^2+22*a*b^3* 
c*d-2*b^4*c^2)/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)+1/2*b*(10*a^4*d^2- 
14*a^3*b*c*d+4*a^2*b^2*c^2-7*a^2*b^2*d^2+8*a*b^3*c*d-b^4*c^2)/(a^4-2*a^2*b 
^2+b^4))/(tan(1/2*f*x+1/2*e)^2*a+2*b*tan(1/2*f*x+1/2*e)+a)^2+1/2*(20*a^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6885 vs. \(2 (651) = 1302\).

Time = 2.59 (sec) , antiderivative size = 6885, normalized size of antiderivative = 10.29 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="giac")
 

Output:

((2*a^2*b^5*c^2 + b^7*c^2 - 10*a^3*b^4*c*d + 4*a*b^6*c*d + 20*a^4*b^3*d^2 
- 29*a^2*b^5*d^2 + 12*b^7*d^2)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + 
arctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^4*b^5*c^5 - 2*a^ 
2*b^7*c^5 + b^9*c^5 - 5*a^5*b^4*c^4*d + 10*a^3*b^6*c^4*d - 5*a*b^8*c^4*d + 
 10*a^6*b^3*c^3*d^2 - 20*a^4*b^5*c^3*d^2 + 10*a^2*b^7*c^3*d^2 - 10*a^7*b^2 
*c^2*d^3 + 20*a^5*b^4*c^2*d^3 - 10*a^3*b^6*c^2*d^3 + 5*a^8*b*c*d^4 - 10*a^ 
6*b^3*c*d^4 + 5*a^4*b^5*c*d^4 - a^9*d^5 + 2*a^7*b^2*d^5 - a^5*b^4*d^5)*sqr 
t(a^2 - b^2)) - (20*b^2*c^4*d^3 - 10*a*b*c^3*d^4 + 2*a^2*c^2*d^5 - 29*b^2* 
c^2*d^5 + 4*a*b*c*d^6 + a^2*d^7 + 12*b^2*d^7)*(pi*floor(1/2*(f*x + e)/pi + 
 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((b^5 
*c^9 - 5*a*b^4*c^8*d + 10*a^2*b^3*c^7*d^2 - 2*b^5*c^7*d^2 - 10*a^3*b^2*c^6 
*d^3 + 10*a*b^4*c^6*d^3 + 5*a^4*b*c^5*d^4 - 20*a^2*b^3*c^5*d^4 + b^5*c^5*d 
^4 - a^5*c^4*d^5 + 20*a^3*b^2*c^4*d^5 - 5*a*b^4*c^4*d^5 - 10*a^4*b*c^3*d^6 
 + 10*a^2*b^3*c^3*d^6 + 2*a^5*c^2*d^7 - 10*a^3*b^2*c^2*d^7 + 5*a^4*b*c*d^8 
 - a^5*d^9)*sqrt(c^2 - d^2)) + (5*a^3*b^6*c^9*tan(1/2*f*x + 1/2*e)^7 - 2*a 
*b^8*c^9*tan(1/2*f*x + 1/2*e)^7 - 11*a^4*b^5*c^8*d*tan(1/2*f*x + 1/2*e)^7 
+ 8*a^2*b^7*c^8*d*tan(1/2*f*x + 1/2*e)^7 - 10*a^3*b^6*c^7*d^2*tan(1/2*f*x 
+ 1/2*e)^7 + 4*a*b^8*c^7*d^2*tan(1/2*f*x + 1/2*e)^7 + 22*a^4*b^5*c^6*d^3*t 
an(1/2*f*x + 1/2*e)^7 - 16*a^2*b^7*c^6*d^3*tan(1/2*f*x + 1/2*e)^7 + 5*a^3* 
b^6*c^5*d^4*tan(1/2*f*x + 1/2*e)^7 - 2*a*b^8*c^5*d^4*tan(1/2*f*x + 1/2*...
 

Mupad [B] (verification not implemented)

Time = 67.81 (sec) , antiderivative size = 571173, normalized size of antiderivative = 853.77 \[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

int(1/((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^3),x)
 

Output:

(atan((((((4*a^24*d^24 + 4*b^24*c^24 + 16*a^2*b^22*c^24 + 16*a^4*b^20*c^24 
 - 1152*a^10*b^14*d^24 + 5568*a^12*b^12*d^24 - 10568*a^14*b^10*d^24 + 9460 
*a^16*b^8*d^24 - 3560*a^18*b^6*d^24 + 136*a^20*b^4*d^24 + 76*a^22*b^2*d^24 
 + 16*a^24*c^2*d^22 + 16*a^24*c^4*d^20 - 1152*b^24*c^10*d^14 + 5568*b^24*c 
^12*d^12 - 10568*b^24*c^14*d^10 + 9460*b^24*c^16*d^8 - 3560*b^24*c^18*d^6 
+ 136*b^24*c^20*d^4 + 76*b^24*c^22*d^2 + 11520*a*b^23*c^9*d^15 - 56448*a*b 
^23*c^11*d^13 + 109456*a*b^23*c^13*d^11 - 101240*a*b^23*c^15*d^9 + 40720*a 
*b^23*c^17*d^7 - 2960*a*b^23*c^19*d^5 - 536*a*b^23*c^21*d^3 - 176*a^3*b^21 
*c^23*d - 320*a^5*b^19*c^23*d + 11520*a^9*b^15*c*d^23 - 56448*a^11*b^13*c* 
d^23 + 109456*a^13*b^11*c*d^23 - 101240*a^15*b^9*c*d^23 + 40720*a^17*b^7*c 
*d^23 - 2960*a^19*b^5*c*d^23 - 536*a^21*b^3*c*d^23 - 176*a^23*b*c^3*d^21 - 
 320*a^23*b*c^5*d^19 - 51840*a^2*b^22*c^8*d^16 + 263808*a^2*b^22*c^10*d^14 
 - 541208*a^2*b^22*c^12*d^12 + 547088*a^2*b^22*c^14*d^10 - 263320*a^2*b^22 
*c^16*d^8 + 44120*a^2*b^22*c^18*d^6 - 1564*a^2*b^22*c^20*d^4 - 196*a^2*b^2 
2*c^22*d^2 + 138240*a^3*b^21*c^7*d^17 - 758400*a^3*b^21*c^9*d^15 + 1720736 
*a^3*b^21*c^11*d^13 - 2002728*a^3*b^21*c^13*d^11 + 1210560*a^3*b^21*c^15*d 
^9 - 335040*a^3*b^21*c^17*d^7 + 37680*a^3*b^21*c^19*d^5 - 288*a^3*b^21*c^2 
1*d^3 - 241920*a^4*b^20*c^6*d^18 + 1512000*a^4*b^20*c^8*d^16 - 3975688*a^4 
*b^20*c^10*d^14 + 5501328*a^4*b^20*c^12*d^12 - 4147952*a^4*b^20*c^14*d^10 
+ 1586920*a^4*b^20*c^16*d^8 - 276020*a^4*b^20*c^18*d^6 + 21124*a^4*b^20...
 

Reduce [F]

\[ \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx=\int \frac {1}{\left (\sin \left (f x +e \right ) b +a \right )^{3} \left (\sin \left (f x +e \right ) d +c \right )^{3}}d x \] Input:

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x)
 

Output:

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x)