\(\int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx\) [86]

Optimal result

Integrand size = 25, antiderivative size = 611 \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=-\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}+\frac {5 b^{3/2} \left (7 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{13/4} d e^{3/2}}-\frac {b}{2 \left (a^2-b^2\right ) d e (a+b \cos (c+d x))^2 \sqrt {e \sin (c+d x)}}-\frac {9 a b}{4 \left (a^2-b^2\right )^2 d e (a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}}+\frac {5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)}{4 \left (a^2-b^2\right )^3 d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {5 a b \left (7 a^2+2 b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{8 \left (a^2-b^2\right )^3 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \sin (c+d x)}}-\frac {a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{4 \left (a^2-b^2\right )^3 d e^2 \sqrt {\sin (c+d x)}} \] Output:

-5/8*b^(3/2)*(7*a^2+2*b^2)*arctan(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^ 
(1/4)/e^(1/2))/(-a^2+b^2)^(13/4)/d/e^(3/2)+5/8*b^(3/2)*(7*a^2+2*b^2)*arcta 
nh(b^(1/2)*(e*sin(d*x+c))^(1/2)/(-a^2+b^2)^(1/4)/e^(1/2))/(-a^2+b^2)^(13/4 
)/d/e^(3/2)-1/2*b/(a^2-b^2)/d/e/(a+b*cos(d*x+c))^2/(e*sin(d*x+c))^(1/2)-9/ 
4*a*b/(a^2-b^2)^2/d/e/(a+b*cos(d*x+c))/(e*sin(d*x+c))^(1/2)+1/4*(5*b*(7*a^ 
2+2*b^2)-a*(8*a^2+37*b^2)*cos(d*x+c))/(a^2-b^2)^3/d/e/(e*sin(d*x+c))^(1/2) 
+5/8*a*b*(7*a^2+2*b^2)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d*x),2*b/(b-(-a^2+b 
^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^3/(b-(-a^2+b^2)^(1/2))/d/e/ 
(e*sin(d*x+c))^(1/2)+5/8*a*b*(7*a^2+2*b^2)*EllipticPi(cos(1/2*c+1/4*Pi+1/2 
*d*x),2*b/(b+(-a^2+b^2)^(1/2)),2^(1/2))*sin(d*x+c)^(1/2)/(a^2-b^2)^3/(b+(- 
a^2+b^2)^(1/2))/d/e/(e*sin(d*x+c))^(1/2)+1/4*a*(8*a^2+37*b^2)*EllipticE(co 
s(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/(a^2-b^2)^3/d/e^2/si 
n(d*x+c)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 7.25 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2)),x]
 

Output:

(Sin[c + d*x]^2*((-2*(-3*a^2*b - b^3 + a^3*Cos[c + d*x] + 3*a*b^2*Cos[c + 
d*x])*Csc[c + d*x])/(a^2 - b^2)^3 + (b^3*Sin[c + d*x])/(2*(a^2 - b^2)^2*(a 
 + b*Cos[c + d*x])^2) + (13*a*b^3*Sin[c + d*x])/(4*(a^2 - b^2)^3*(a + b*Co 
s[c + d*x]))))/(d*(e*Sin[c + d*x])^(3/2)) - (Sin[c + d*x]^(3/2)*(((8*a^3*b 
 + 37*a*b^3)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - ( 
Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqr 
t[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] 
- Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + 
 Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x] 
] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^ 
2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - 
Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c 
 + d*x]^2)) + (2*(8*a^4 + 72*a^2*b^2 + 10*b^4)*Cos[c + d*x]*(((1/8 + I/8)* 
(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2 
*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log 
[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] 
+ I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^ 
(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4) 
) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a 
^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c +...
 

Rubi [A] (warning: unable to verify)

Time = 2.88 (sec) , antiderivative size = 575, normalized size of antiderivative = 0.94, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 3173, 27, 3042, 3343, 27, 3042, 3345, 27, 3042, 3346, 3042, 3121, 3042, 3119, 3180, 266, 827, 218, 221, 3042, 3286, 3042, 3284}

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.

\(\displaystyle \int \frac {1}{(e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\)

\(\Big \downarrow \) 3173

\(\displaystyle -\frac {\int -\frac {4 a-5 b \cos (c+d x)}{2 (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}dx}{2 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {4 a-5 b \cos (c+d x)}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}dx}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {4 a+5 b \sin \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3343

\(\displaystyle \frac {-\frac {\int -\frac {8 a^2-27 b \cos (c+d x) a+10 b^2}{2 (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}dx}{a^2-b^2}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {2 \left (4 a^2+5 b^2\right )-27 a b \cos (c+d x)}{(a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}dx}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {2 \left (4 a^2+5 b^2\right )+27 a b \sin \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )}dx}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3345

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {2 \int \frac {\left (8 a^4+72 b^2 a^2+b \left (8 a^2+37 b^2\right ) \cos (c+d x) a+10 b^4\right ) \sqrt {e \sin (c+d x)}}{2 (a+b \cos (c+d x))}dx}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {\int \frac {\left (2 \left (4 a^4+36 b^2 a^2+5 b^4\right )+a b \left (8 a^2+37 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {\int \frac {\sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )} \left (2 \left (4 a^4+36 b^2 a^2+5 b^4\right )+a b \left (8 a^2+37 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3346

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a \left (8 a^2+37 b^2\right ) \int \sqrt {e \sin (c+d x)}dx+5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \sin (c+d x)}}{a+b \cos (c+d x)}dx}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {a \left (8 a^2+37 b^2\right ) \int \sqrt {e \sin (c+d x)}dx+5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {\frac {a \left (8 a^2+37 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )}}{a-b \sin \left (c+d x-\frac {\pi }{2}\right )}dx+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3180

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {b e \int \frac {\sqrt {e \sin (c+d x)}}{b^2 \sin ^2(c+d x) e^2+\left (a^2-b^2\right ) e^2}d(e \sin (c+d x))}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {2 b e \int \frac {e^2 \sin ^2(c+d x)}{b^2 e^4 \sin ^4(c+d x)+\left (a^2-b^2\right ) e^2}d\sqrt {e \sin (c+d x)}}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {2 b e \left (\frac {\int \frac {1}{b e^2 \sin ^2(c+d x)+\sqrt {b^2-a^2} e}d\sqrt {e \sin (c+d x)}}{2 b}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\int \frac {1}{\sqrt {b^2-a^2} e-b e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}}{2 b}\right )}{d}-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b}+\frac {a e \int \frac {1}{\sqrt {e \sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {b^2-a^2}-b \sin (c+d x)\right )}dx}{2 b \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)} \left (b \sin (c+d x)+\sqrt {b^2-a^2}\right )}dx}{2 b \sqrt {e \sin (c+d x)}}-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\frac {\frac {2 \left (5 b \left (7 a^2+2 b^2\right )-a \left (8 a^2+37 b^2\right ) \cos (c+d x)\right )}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)}}-\frac {5 b^2 \left (7 a^2+2 b^2\right ) \left (-\frac {2 b e \left (\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e} \sin (c+d x)}{\sqrt [4]{b^2-a^2}}\right )}{2 b^{3/2} \sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \sin (c+d x)}}+\frac {a e \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \sin (c+d x)}}\right )+\frac {2 a \left (8 a^2+37 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}}{e^2 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {9 a b}{d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))}}{4 \left (a^2-b^2\right )}-\frac {b}{2 d e \left (a^2-b^2\right ) \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^2}\)

Input:

Int[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2)),x]
 

Output:

-1/2*b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]]) + ((- 
9*a*b)/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]) + ((2*( 
5*b*(7*a^2 + 2*b^2) - a*(8*a^2 + 37*b^2)*Cos[c + d*x]))/((a^2 - b^2)*d*e*S 
qrt[e*Sin[c + d*x]]) - ((2*a*(8*a^2 + 37*b^2)*EllipticE[(c - Pi/2 + d*x)/2 
, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + 5*b^2*(7*a^2 + 2*b^2)* 
((-2*b*e*(ArcTan[(Sqrt[b]*Sqrt[e]*Sin[c + d*x])/(-a^2 + b^2)^(1/4)]/(2*b^( 
3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e]) - ArcTanh[(Sqrt[b]*Sqrt[e]*Sin[c + d*x])/ 
(-a^2 + b^2)^(1/4)]/(2*b^(3/2)*(-a^2 + b^2)^(1/4)*Sqrt[e])))/d + (a*e*Elli 
pticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d 
*x]])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*e*EllipticPi[ 
(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/( 
b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]])))/((a^2 - b^2)*e^2))/(2*( 
a^2 - b^2)))/(4*(a^2 - b^2))
 

Defintions of rubi rules used

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 218

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

rule 221

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

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 3042

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

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3121

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 

rule 3173

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + 
f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) 
   Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + 
p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b 
^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
 

rule 3180

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_ 
)]), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Simp[a*(g/(2*b))   Int[1/(S 
qrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (-Simp[a*(g/(2*b))   In 
t[1/(Sqrt[g*Cos[e + f*x]]*(q - b*Cos[e + f*x])), x], x] + Simp[b*(g/f)   Su 
bst[Int[Sqrt[x]/(g^2*(a^2 - b^2) + b^2*x^2), x], x, g*Cos[e + f*x]], x])] / 
; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
 

rule 3284

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

rule 3286

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

rule 3343

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c 
 - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - 
 b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1))   Int[(g*Cos[e + f*x])^p 
*(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p 
 + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ 
[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
 

rule 3345

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co 
s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* 
Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 
 1))   Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 
 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], 
x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt 
Q[p, -1] && IntegerQ[2*m]
 

rule 3346

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)* 
(x_)]))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b   Int 
[(g*Cos[e + f*x])^p, x], x] + Simp[(b*c - a*d)/b   Int[(g*Cos[e + f*x])^p/( 
a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - 
 b^2, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2835\) vs. \(2(541)=1082\).

Time = 4.20 (sec) , antiderivative size = 2836, normalized size of antiderivative = 4.64

method	result	size
default	\(\text {Expression too large to display}\)	\(2836\)

Input:

int(1/(a+cos(d*x+c)*b)^3/(e*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

(2*e^3*b*(b^2/e^4/(a-b)^3/(a+b)^3*(1/8*(e*sin(d*x+c))^(3/2)*e^2*(-11*cos(d 
*x+c)^2*a^2*b^2-2*b^4*cos(d*x+c)^2+15*a^4-2*a^2*b^2)/(-b^2*cos(d*x+c)^2*e^ 
2+a^2*e^2)^2+1/8*(35/8*a^2+5/4*b^2)/b^2/(e^2*(a^2-b^2)/b^2)^(1/4)*2^(1/2)* 
(ln((e*sin(d*x+c)-(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)*2^(1/2)+( 
e^2*(a^2-b^2)/b^2)^(1/2))/(e*sin(d*x+c)+(e^2*(a^2-b^2)/b^2)^(1/4)*(e*sin(d 
*x+c))^(1/2)*2^(1/2)+(e^2*(a^2-b^2)/b^2)^(1/2)))+2*arctan(2^(1/2)/(e^2*(a^ 
2-b^2)/b^2)^(1/4)*(e*sin(d*x+c))^(1/2)+1)+2*arctan(2^(1/2)/(e^2*(a^2-b^2)/ 
b^2)^(1/4)*(e*sin(d*x+c))^(1/2)-1)))-(-3*a^2-b^2)/e^4/(a^2-b^2)^3/(e*sin(d 
*x+c))^(1/2))-(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/e*a*(-(a^2+3*b^2)/(a^2-b^2 
)^3*(2*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*Ellipt 
icE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c) 
)^(1/2)*sin(d*x+c)^(1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-2*cos 
(d*x+c)^2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)+b^2*(a^2+3*b^2)/(a-b)^2/(a+b) 
^2*(1/2*b^2/e/a^2/(a^2-b^2)*sin(d*x+c)*(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)/( 
-cos(d*x+c)^2*b^2+a^2)-1/2/a^2/(a^2-b^2)*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x 
+c))^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*EllipticE((1 
-sin(d*x+c))^(1/2),1/2*2^(1/2))+1/4/a^2/(a^2-b^2)*(1-sin(d*x+c))^(1/2)*(2+ 
2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c))^(1/2)*Ell 
ipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-3/8/(a^2-b^2)/b^2*(1-sin(d*x+c))^ 
(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(cos(d*x+c)^2*e*sin(d*x+c...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cos(d*x+c))**3/(e*sin(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((b*cos(d*x + c) + a)^3*(e*sin(d*x + c))^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:

int(1/((e*sin(c + d*x))^(3/2)*(a + b*cos(c + d*x))^3),x)
 

Output:

int(1/((e*sin(c + d*x))^(3/2)*(a + b*cos(c + d*x))^3), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )}}{\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} b^{3}+3 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2} b +\sin \left (d x +c \right )^{2} a^{3}}d x \right )}{e^{2}} \] Input:

int(1/(a+b*cos(d*x+c))^3/(e*sin(d*x+c))^(3/2),x)
 

Output:

(sqrt(e)*int(sqrt(sin(c + d*x))/(cos(c + d*x)**3*sin(c + d*x)**2*b**3 + 3* 
cos(c + d*x)**2*sin(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*sin(c + d*x)**2*a* 
*2*b + sin(c + d*x)**2*a**3),x))/e**2