3.15.31 \(\int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx\) [1431]

3.15.31.1 Optimal result
3.15.31.2 Mathematica [C] (warning: unable to verify)
3.15.31.3 Rubi [A] (verified)
3.15.31.4 Maple [B] (warning: unable to verify)
3.15.31.5 Fricas [F(-1)]
3.15.31.6 Sympy [F]
3.15.31.7 Maxima [F]
3.15.31.8 Giac [F]
3.15.31.9 Mupad [F(-1)]

3.15.31.1 Optimal result

Integrand size = 37, antiderivative size = 508 \[ \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=-\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b f \sqrt {g}}+\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b f \sqrt {g}}+\frac {2 \sqrt {2} a d^{3/2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b \sqrt {-a^2+b^2} f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} a d^{3/2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{b \sqrt {-a^2+b^2} f \sqrt {g \cos (e+f x)}}+\frac {d^{3/2} \log \left (\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b f \sqrt {g}}-\frac {d^{3/2} \log \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b f \sqrt {g}} \]

output
-1/2*d^(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))/b/f*2^(1/2)/g^(1/2)+1/2*d^(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))/b/f*2^(1/2)/g^(1/2)+1/4*d 
^(3/2)*ln(d^(1/2)-2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos(f*x+e))^(1/2 
)+d^(1/2)*tan(f*x+e))/b/f*2^(1/2)/g^(1/2)-1/4*d^(3/2)*ln(d^(1/2)+2^(1/2)*g 
^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))/b/f*2 
^(1/2)/g^(1/2)+2*a*d^(3/2)*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)*2^(1/2)*cos(f*x+e)^(1/2)/b/f/(-a^ 
2+b^2)^(1/2)/(g*cos(f*x+e))^(1/2)-2*a*d^(3/2)*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)*2^(1/2)*cos(f* 
x+e)^(1/2)/b/f/(-a^2+b^2)^(1/2)/(g*cos(f*x+e))^(1/2)
 
3.15.31.2 Mathematica [C] (warning: unable to verify)

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

Time = 11.80 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.02 \[ \int \frac {(d \sin (e+f x))^{3/2}}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {10 \left (a^2-b^2\right ) \cot (e+f x) (d \sin (e+f x))^{3/2} \left (a+b \sqrt {\sin ^2(e+f x)}\right ) \left (\frac {a \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{4},1,\frac {5}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )}{5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{4},1,\frac {5}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (-4 b^2 \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{4},2,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^2(e+f x)}+\frac {b \operatorname {AppellF1}\left (\frac {1}{4},-\frac {3}{4},1,\frac {5}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \sqrt {\sin ^2(e+f x)}}{-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {3}{4},1,\frac {5}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+\left (4 b^2 \operatorname {AppellF1}\left (\frac {5}{4},-\frac {3}{4},2,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+3 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^2(e+f x)}\right )}{f \sqrt {g \cos (e+f x)} (-a+b \sin (e+f x)) (a+b \sin (e+f x))^2} \]

input
Integrate[(d*Sin[e + f*x])^(3/2)/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x] 
)),x]
 
output
(10*(a^2 - b^2)*Cot[e + f*x]*(d*Sin[e + f*x])^(3/2)*(a + b*Sqrt[Sin[e + f* 
x]^2])*((a*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2 
)/(-a^2 + b^2)])/(5*(a^2 - b^2)*AppellF1[1/4, -1/4, 1, 5/4, Cos[e + f*x]^2 
, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-4*b^2*AppellF1[5/4, -1/4, 2, 9/4, 
 Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF 
1[5/4, 3/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Co 
s[e + f*x]^2) + (b*AppellF1[1/4, -3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e 
+ f*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[e + f*x]^2])/(-5*(a^2 - b^2)*AppellF1[1/4 
, -3/4, 1, 5/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] + (4*b^ 
2*AppellF1[5/4, -3/4, 2, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + 
 b^2)] + 3*(a^2 - b^2)*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos 
[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^2)))/(f*Sqrt[g*Cos[e + f*x]]*(-a 
+ b*Sin[e + f*x])*(a + b*Sin[e + f*x])^2)
 
3.15.31.3 Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {3042, 3388, 3042, 3054, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3388

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3054

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

\(\Big \downarrow \) 3387

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3386

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

\(\Big \downarrow \) 1542

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

input
Int[(d*Sin[e + f*x])^(3/2)/(Sqrt[g*Cos[e + f*x]]*(a + b*Sin[e + f*x])),x]
 
output
-((a*d*Sqrt[Cos[e + f*x]]*((2*Sqrt[2]*(1 - b/Sqrt[-a^2 + b^2])*Sqrt[d]*Ell 
ipticPi[-(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*Sqrt[2]*(1 
+ b/Sqrt[-a^2 + b^2])*Sqrt[d]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcS 
in[Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/((b + Sqrt 
[-a^2 + b^2])*f)))/(b*Sqrt[g*Cos[e + f*x]])) + (2*d^2*g*((-(ArcTan[1 - (Sq 
rt[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*Lo 
g[d - (Sqrt[2]*Sqrt[d]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] 
+ d*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d]*Sqrt[g]) + Log[d + (Sqrt[2]*Sqrt[d]*Sqr 
t[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)))/(b*f)
 

3.15.31.3.1 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 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 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]
 

rule 3388
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( 
n_))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d/b   Int[(g 
*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n - 1), x], x] - Simp[a*(d/b)   Int[(g*C 
os[e + f*x])^p*((d*Sin[e + f*x])^(n - 1)/(a + b*Sin[e + f*x])), x], x] /; F 
reeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && 
LtQ[-1, p, 1] && GtQ[n, 0]
 
3.15.31.4 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. 884 vs. \(2 (401 ) = 802\).

Time = 1.89 (sec) , antiderivative size = 885, normalized size of antiderivative = 1.74

method result size
default \(-\frac {\left (\frac {d \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1}\right )^{\frac {3}{2}} \left (i \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) a \sqrt {-a^{2}+b^{2}}-i \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) b \sqrt {-a^{2}+b^{2}}-i \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) a \sqrt {-a^{2}+b^{2}}+i \Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) b \sqrt {-a^{2}+b^{2}}-\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a^{2}+b^{2}}\, a +\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) b \sqrt {-a^{2}+b^{2}}-\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) a \sqrt {-a^{2}+b^{2}}+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) b \sqrt {-a^{2}+b^{2}}+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a^{2}+b^{2}}\, a +\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right ) a^{2}-\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {a}{-b +\sqrt {-a^{2}+b^{2}}+a}, \frac {\sqrt {2}}{2}\right ) a b +\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a^{2}+b^{2}}\, a -\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) a^{2}+\Pi \left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, -\frac {a}{b +\sqrt {-a^{2}+b^{2}}-a}, \frac {\sqrt {2}}{2}\right ) a b \right ) \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, \sqrt {2+2 \cot \left (f x +e \right )-2 \csc \left (f x +e \right )}\, \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2}+\sin ^{2}\left (f x +e \right )\right ) \sqrt {2}\, a}{f \left (1-\cos \left (f x +e \right )\right )^{2} \sqrt {-\frac {g \left (\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )-1\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \left (\csc ^{2}\left (f x +e \right )\right )+1}}\, b \sqrt {-a^{2}+b^{2}}\, \left (-b +\sqrt {-a^{2}+b^{2}}+a \right ) \left (b +\sqrt {-a^{2}+b^{2}}-a \right )}\) \(885\)

input
int((d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))/(g*cos(f*x+e))^(1/2),x,method=_R 
ETURNVERBOSE)
 
output
-1/f*(d/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(3/2)/( 
1-cos(f*x+e))^2*(I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1 
/2*2^(1/2))*a*(-a^2+b^2)^(1/2)-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/ 
2),1/2-1/2*I,1/2*2^(1/2))*b*(-a^2+b^2)^(1/2)-I*EllipticPi((-cot(f*x+e)+csc 
(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*a*(-a^2+b^2)^(1/2)+I*EllipticPi((- 
cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*b*(-a^2+b^2)^(1/2)-E 
llipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^ 
2)^(1/2)*a+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/ 
2))*b*(-a^2+b^2)^(1/2)-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2 
*I,1/2*2^(1/2))*a*(-a^2+b^2)^(1/2)+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^( 
1/2),1/2+1/2*I,1/2*2^(1/2))*b*(-a^2+b^2)^(1/2)+EllipticPi((-cot(f*x+e)+csc 
(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a 
+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2 
*2^(1/2))*a^2-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2) 
^(1/2)+a),1/2*2^(1/2))*a*b+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/ 
(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-EllipticPi((-cot(f* 
x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2+Ellipt 
icPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2 
))*a*b)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2) 
*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)/(-g*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)...
 
3.15.31.5 Fricas [F(-1)]

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

input
integrate((d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))/(g*cos(f*x+e))^(1/2),x, al 
gorithm="fricas")
 
output
Timed out
 
3.15.31.6 Sympy [F]

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

input
integrate((d*sin(f*x+e))**(3/2)/(a+b*sin(f*x+e))/(g*cos(f*x+e))**(1/2),x)
 
output
Integral((d*sin(e + f*x))**(3/2)/(sqrt(g*cos(e + f*x))*(a + b*sin(e + f*x) 
)), x)
 
3.15.31.7 Maxima [F]

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

input
integrate((d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))/(g*cos(f*x+e))^(1/2),x, al 
gorithm="maxima")
 
output
integrate((d*sin(f*x + e))^(3/2)/(sqrt(g*cos(f*x + e))*(b*sin(f*x + e) + a 
)), x)
 
3.15.31.8 Giac [F]

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

input
integrate((d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))/(g*cos(f*x+e))^(1/2),x, al 
gorithm="giac")
 
output
integrate((d*sin(f*x + e))^(3/2)/(sqrt(g*cos(f*x + e))*(b*sin(f*x + e) + a 
)), x)
 
3.15.31.9 Mupad [F(-1)]

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

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