\(\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx\) [1414]

Optimal result

Integrand size = 37, antiderivative size = 496 \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b \sqrt {d} f}-\frac {g^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b \sqrt {d} f}+\frac {g^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} \left (\sqrt {d}+\sqrt {d} \tan (e+f x)\right )}\right )}{\sqrt {2} b \sqrt {d} f}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}+\frac {g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{a f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \] Output:

1/2*g^(3/2)*arctan(1-2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/d^(1/2)/(g*cos(f 
*x+e))^(1/2))*2^(1/2)/b/d^(1/2)/f-1/2*g^(3/2)*arctan(1+2^(1/2)*g^(1/2)*(d* 
sin(f*x+e))^(1/2)/d^(1/2)/(g*cos(f*x+e))^(1/2))*2^(1/2)/b/d^(1/2)/f+1/2*g^ 
(3/2)*arctanh(2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos(f*x+e))^(1/2)/(d 
^(1/2)+d^(1/2)*tan(f*x+e)))*2^(1/2)/b/d^(1/2)/f+2*2^(1/2)*(-a^2+b^2)^(1/2) 
*g^2*cos(f*x+e)^(1/2)*EllipticPi((d*sin(f*x+e))^(1/2)/d^(1/2)/(1+cos(f*x+e 
))^(1/2),-a/(b-(-a^2+b^2)^(1/2)),I)/a/b/d^(1/2)/f/(g*cos(f*x+e))^(1/2)-2*2 
^(1/2)*(-a^2+b^2)^(1/2)*g^2*cos(f*x+e)^(1/2)*EllipticPi((d*sin(f*x+e))^(1/ 
2)/d^(1/2)/(1+cos(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)),I)/a/b/d^(1/2)/f/( 
g*cos(f*x+e))^(1/2)+g^2*InverseJacobiAM(e-1/4*Pi+f*x,2^(1/2))*sin(2*f*x+2* 
e)^(1/2)/a/f/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 15.54 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.36 \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {2 \left (b \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )-a \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) (g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)} \left (a+b \sqrt {\sin ^2(e+f x)}\right )}{5 \left (a^2-b^2\right ) d f g \sqrt [4]{\sin ^2(e+f x)} (a+b \sin (e+f x))} \] Input:

Integrate[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x] 
)),x]
 

Output:

(2*(b*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^ 
2 + b^2)] - a*AppellF1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x] 
^2)/(-a^2 + b^2)])*(g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]*(a + b*Sqrt 
[Sin[e + f*x]^2]))/(5*(a^2 - b^2)*d*f*g*(Sin[e + f*x]^2)^(1/4)*(a + b*Sin[ 
e + f*x]))
 

Rubi [A] (warning: unable to verify)

Time = 2.42 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.22, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.568, Rules used = {3042, 3379, 3042, 3317, 3042, 3053, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 3120, 3387, 3042, 3386, 1542}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))}dx\)

\(\Big \downarrow \) 3379

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3317

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (b \int \frac {1}{\sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}dx-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3053

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {a \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}dx}{d}\right )}{a b}\)

\(\Big \downarrow \) 3054

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \int \frac {d \tan (e+f x)}{g \left (\tan ^2(e+f x) d^2+d^2\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{f}\right )}{a b}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\int \frac {\tan (e+f x) d+d}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}+\frac {\int \frac {1}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int \frac {1}{-\frac {d \tan (e+f x)}{g}-1}d\left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\int \frac {d-d \tan (e+f x)}{\tan ^2(e+f x) d^2+d^2}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}\right )}{\sqrt {g} \left (\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}\right )}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-\frac {2 \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}-\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {2} \sqrt {d} g}+\frac {\int \frac {\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{\frac {\tan (e+f x) d}{g}+\frac {d}{g}+\frac {\sqrt {2} \sqrt {d \sin (e+f x)} \sqrt {d}}{\sqrt {g} \sqrt {g \cos (e+f x)}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}}{2 \sqrt {d} g}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \int \frac {1}{\sqrt {\sin (2 e+2 f x)}}dx}{\sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 3387

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \int \frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)} (a+b \sin (e+f x))}dx}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 3386

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} d \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \int \frac {1}{\left (\left (b-\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}+\frac {2 \sqrt {2} d \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \int \frac {1}{\left (\left (b+\sqrt {b^2-a^2}\right ) d+\frac {a \sin (e+f x) d}{\cos (e+f x)+1}\right ) \sqrt {1-\frac {\sin ^2(e+f x)}{(\cos (e+f x)+1)^2}}}d\frac {\sqrt {d \sin (e+f x)}}{\sqrt {\cos (e+f x)+1}}}{f}\right )}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \left (\frac {2 \sqrt {2} \sqrt {d} \left (1-\frac {b}{\sqrt {b^2-a^2}}\right ) \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (b-\sqrt {b^2-a^2}\right )}+\frac {2 \sqrt {2} \sqrt {d} \left (\frac {b}{\sqrt {b^2-a^2}}+1\right ) \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{f \left (\sqrt {b^2-a^2}+b\right )}\right )}{a b d \sqrt {g \cos (e+f x)}}+\frac {g^2 \left (\frac {b \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 a g \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt {d} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+d \tan (e+f x)+d\right )}{2 \sqrt {2} \sqrt {d} \sqrt {g}}}{2 g}\right )}{f}\right )}{a b}\)

Input:

Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]
 

Output:

((a^2 - b^2)*g^2*Sqrt[Cos[e + f*x]]*((2*Sqrt[2]*(1 - b/Sqrt[-a^2 + b^2])*S 
qrt[d]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]] 
/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/((b - Sqrt[-a^2 + b^2])*f) + (2*S 
qrt[2]*(1 + b/Sqrt[-a^2 + b^2])*Sqrt[d]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^ 
2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/ 
((b + Sqrt[-a^2 + b^2])*f)))/(a*b*d*Sqrt[g*Cos[e + f*x]]) + (g^2*((-2*a*g* 
((-(ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[ 
e + f*x]])]/(Sqrt[2]*Sqrt[d]*Sqrt[g])) + ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[ 
d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])]/(Sqrt[2]*Sqrt[d]*Sqrt[g]) 
)/(2*g) - (-1/2*Log[d - (Sqrt[2]*Sqrt[d]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqr 
t[g*Cos[e + f*x]] + d*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d]*Sqrt[g]) + Log[d + (S 
qrt[2]*Sqrt[d]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + d*Tan[ 
e + f*x]]/(2*Sqrt[2]*Sqrt[d]*Sqrt[g]))/(2*g)))/f + (b*EllipticF[e - Pi/4 + 
 f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f* 
x]])))/(a*b)
 

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 826

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

rule 3042

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

rule 3053

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ 
)]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b 
*Cos[e + f*x]])   Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f 
}, x]
 

rule 3054

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k*a*(b/f)   Subst[Int[x^(k 
*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos[e + 
 f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] 
&& LtQ[m, 1]
 

rule 3120

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

rule 3317

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a   Int[(g*Co 
s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d   Int[(g*Cos[e + f*x])^ 
p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
 

rule 3379

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/(a*b) 
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(b - a*Sin[e + f*x]), x], x 
] + Simp[g^2*((a^2 - b^2)/(a*b*d))   Int[(g*Cos[e + f*x])^(p - 2)*((d*Sin[e 
 + f*x])^(n + 1)/(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, 
 x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (LtQ[n, -1] 
 || (EqQ[p, 3/2] && EqQ[n, -2^(-1)]))
 

rule 3386

Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_ 
) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[-a^2 + b^2, 
2]}, Simp[2*Sqrt[2]*d*((b + q)/(f*q))   Subst[Int[1/((d*(b + q) + a*x^2)*Sq 
rt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - 
 Simp[2*Sqrt[2]*d*((b - q)/(f*q))   Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 
 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x]] /; F 
reeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

rule 3387

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

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (406 ) = 812\).

Time = 2.94 (sec) , antiderivative size = 1174, normalized size of antiderivative = 2.37

method	result	size
default	\(\text {Expression too large to display}\)	\(1174\)

Input:

int((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x,method=_R 
ETURNVERBOSE)
 

Output:

(1/2+1/2*I)/f*(a-b)*(-I*(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e) 
+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a+2*I*(-a^2+b^2)^(1/2)*El 
lipticF((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/2))*b-I*(-a^2+b^2)^(1/2)* 
EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2 
^(1/2))*b-I*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1 
/2)+a),1/2*2^(1/2))*a^2-I*(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+ 
e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b-I*EllipticPi((csc(f*x 
+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^2+I*Ellip 
ticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2 
))*a^2-I*(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b 
+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a+I*EllipticPi((csc(f*x+e)-cot(f*x+e)+1) 
^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b^2+2*I*(-a^2+b^2)^(1/2)*Ell 
ipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*a-2*(-a^2+b 
^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2) 
)*a-2*(-a^2+b^2)^(1/2)*EllipticF((csc(f*x+e)-cot(f*x+e)+1)^(1/2),1/2*2^(1/ 
2))*b+(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),a/(-b+(- 
a^2+b^2)^(1/2)+a),1/2*2^(1/2))*a+(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-c 
ot(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b+(-a^2+b^2)^(1/ 
2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/ 
2*2^(1/2))*a+(-a^2+b^2)^(1/2)*EllipticPi((csc(f*x+e)-cot(f*x+e)+1)^(1/2...
 

Fricas [F(-1)]

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

integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, al 
gorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((g*cos(e + f*x))**(3/2)/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x) 
)), x)
 

Maxima [F]

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

integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, al 
gorithm="maxima")
 

Output:

integrate((g*cos(f*x + e))^(3/2)/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) 
)), x)
 

Giac [F]

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

integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, al 
gorithm="giac")
 

Output:

integrate((g*cos(f*x + e))^(3/2)/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) 
)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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