\(\int \frac {1}{(a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx\) [527]

Optimal result

Integrand size = 27, antiderivative size = 344 \[ \int \frac {1}{(a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 (c-d) f (a+a \sin (e+f x))^3}-\frac {2 (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 a (c-d)^2 f (a+a \sin (e+f x))^2}-\frac {\left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{30 (c-d)^3 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (4 c^2-15 c d+27 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{30 a^3 (c-d)^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (4 c^2-11 c d+15 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{30 a^3 (c-d)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

-1/5*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(c-d)/f/(a+a*sin(f*x+e))^3-2/15*(c- 
3*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/a/(c-d)^2/f/(a+a*sin(f*x+e))^2-1/30 
*(4*c^2-15*c*d+27*d^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(c-d)^3/f/(a^3+a^ 
3*sin(f*x+e))+1/30*(4*c^2-15*c*d+27*d^2)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f* 
x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/a^3/(c-d)^3/f/((c+d*sin 
(f*x+e))/(c+d))^(1/2)+1/30*(4*c^2-11*c*d+15*d^2)*InverseJacobiAM(1/2*e-1/4 
*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/a^3/(c 
-d)^2/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 11.10 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \left (-\left (\left (4 c^2-15 c d+27 d^2\right ) (c+d \sin (e+f x))\right )+\frac {2 \left (6 (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right )-3 (c-d)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 (c-3 d) (c-d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 (c-3 d) (c-d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+\left (4 c^2-15 c d+27 d^2\right ) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4\right ) (c+d \sin (e+f x))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+d^2 (c+15 d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (4 c^2-15 c d+27 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{30 a^3 (c-d)^3 f (1+\sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-((4*c^2 - 15*c*d + 27*d^2)*(c + 
 d*Sin[e + f*x])) + (2*(6*(c - d)^2*Sin[(e + f*x)/2] - 3*(c - d)^2*(Cos[(e 
 + f*x)/2] + Sin[(e + f*x)/2]) + 4*(c - 3*d)*(c - d)*Sin[(e + f*x)/2]*(Cos 
[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 2*(c - 3*d)*(c - d)*(Cos[(e + f*x)/2 
] + Sin[(e + f*x)/2])^3 + (4*c^2 - 15*c*d + 27*d^2)*Sin[(e + f*x)/2]*(Cos[ 
(e + f*x)/2] + Sin[(e + f*x)/2])^4)*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2 
] + Sin[(e + f*x)/2])^5 + d^2*(c + 15*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, 
(2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (4*c^2 - 15*c*d + 27*d 
^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF 
[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + d)] 
))/(30*a^3*(c - d)^3*f*(1 + Sin[e + f*x])^3*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 2.00 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.08, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3245, 27, 3042, 3457, 25, 3042, 3457, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3245

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (4 c-9 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (4 c-9 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a)^2 \sqrt {c+d \sin (e+f x)}}dx}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {\left (4 c^2-13 d c+21 d^2\right ) a^2+2 (c-3 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-13 d c+21 d^2\right ) a^2+2 (c-3 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\left (4 c^2-13 d c+21 d^2\right ) a^2+2 (c-3 d) d \sin (e+f x) a^2}{(\sin (e+f x) a+a) \sqrt {c+d \sin (e+f x)}}dx}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {\frac {-\frac {\int \frac {d^2 (c+15 d) a^3+d \left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx}{a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\int \frac {d^2 (c+15 d) a^3+d \left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\int \frac {d^2 (c+15 d) a^3+d \left (4 c^2-15 d c+27 d^2\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {-\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {-\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {-\frac {a^2 \left (4 c^2-15 c d+27 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}-\frac {\frac {2 a^3 \left (4 c^2-15 c d+27 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^3 (c-d) \left (4 c^2-11 c d+15 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{2 a^2 (c-d)}}{3 a^2 (c-d)}-\frac {4 a (c-3 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2}}{10 a^2 (c-d)}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{5 f (c-d) (a \sin (e+f x)+a)^3}\)

Input:

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

Output:

-1/5*(Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x 
])^3) + ((-4*a*(c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*(c - d) 
*f*(a + a*Sin[e + f*x])^2) + (-((a^2*(4*c^2 - 15*c*d + 27*d^2)*Cos[e + f*x 
]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x]))) - ((2*a^3*(4 
*c^2 - 15*c*d + 27*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[ 
c + d*Sin[e + f*x]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*a^3*(c - 
d)*(4*c^2 - 11*c*d + 15*d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]* 
Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e + f*x]]))/(2*a^2*( 
c - d)))/(3*a^2*(c - d)))/(10*a^2*(c - d))
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 3042

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

rule 3132

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

rule 3134

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

rule 3140

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

rule 3142

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

rule 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*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]
 

rule 3245

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

rule 3457

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

Maple [A] (warning: unable to verify)

Time = 0.83 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.72

method	result	size
default	\(\frac {\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \left (-\frac {\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{5 \left (c -d \right ) \left (1+\sin \left (f x +e \right )\right )^{3}}-\frac {2 \left (c -3 d \right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{15 \left (c -d \right )^{2} \left (1+\sin \left (f x +e \right )\right )^{2}}-\frac {\left (-\sin \left (f x +e \right )^{2} d -c \sin \left (f x +e \right )+d \sin \left (f x +e \right )+c \right ) \left (4 c^{2}-15 c d +27 d^{2}\right )}{30 \left (c -d \right )^{3} \sqrt {\left (1+\sin \left (f x +e \right )\right ) \left (-1+\sin \left (f x +e \right )\right ) \left (-c -d \sin \left (f x +e \right )\right )}}+\frac {2 \left (-c \,d^{2}-15 d^{3}\right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (60 c^{3}-180 c^{2} d +180 c \,d^{2}-60 d^{3}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}-\frac {d \left (4 c^{2}-15 c d +27 d^{2}\right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{30 \left (c -d \right )^{3} \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{a^{3} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(591\)

Input:

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

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/a^3*(-1/5/(c-d)*(-(-c-d*sin(f*x+e) 
)*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^3-2/15*(c-3*d)/(c-d)^2*(-(-c-d*sin(f* 
x+e))*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^2-1/30*(-sin(f*x+e)^2*d-c*sin(f*x 
+e)+d*sin(f*x+e)+c)/(c-d)^3*(4*c^2-15*c*d+27*d^2)/((1+sin(f*x+e))*(-1+sin( 
f*x+e))*(-c-d*sin(f*x+e)))^(1/2)+2*(-c*d^2-15*d^3)/(60*c^3-180*c^2*d+180*c 
*d^2-60*d^3)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d 
))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2) 
^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-1/30* 
d*(4*c^2-15*c*d+27*d^2)/(c-d)^3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d* 
(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f* 
x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/ 
2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c 
+d))^(1/2))))/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 1640, normalized size of antiderivative = 4.77 \[ \int \frac {1}{(a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/90*(((8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^3 - 32*c^3 + 12 
0*c^2*d - 204*c*d^2 + 180*d^3 + 3*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*c 
os(f*x + e)^2 - 2*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e) - (3 
2*c^3 - 120*c^2*d + 204*c*d^2 - 180*d^3 - (8*c^3 - 30*c^2*d + 51*c*d^2 - 4 
5*d^3)*cos(f*x + e)^2 + 2*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + 
 e))*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/ 
d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f* 
x + e) - 2*I*c)/d) + ((8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*cos(f*x + e)^ 
3 - 32*c^3 + 120*c^2*d - 204*c*d^2 + 180*d^3 + 3*(8*c^3 - 30*c^2*d + 51*c* 
d^2 - 45*d^3)*cos(f*x + e)^2 - 2*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45*d^3)*co 
s(f*x + e) - (32*c^3 - 120*c^2*d + 204*c*d^2 - 180*d^3 - (8*c^3 - 30*c^2*d 
 + 51*c*d^2 - 45*d^3)*cos(f*x + e)^2 + 2*(8*c^3 - 30*c^2*d + 51*c*d^2 - 45 
*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassPInverse(-4/3* 
(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + 
e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*((4*I*c^2*d - 15*I*c*d^2 + 27*I*d^ 
3)*cos(f*x + e)^3 - 16*I*c^2*d + 60*I*c*d^2 - 108*I*d^3 + 3*(4*I*c^2*d - 1 
5*I*c*d^2 + 27*I*d^3)*cos(f*x + e)^2 + 2*(-4*I*c^2*d + 15*I*c*d^2 - 27*I*d 
^3)*cos(f*x + e) + (-16*I*c^2*d + 60*I*c*d^2 - 108*I*d^3 + (4*I*c^2*d - 15 
*I*c*d^2 + 27*I*d^3)*cos(f*x + e)^2 + 2*(-4*I*c^2*d + 15*I*c*d^2 - 27*I*d^ 
3)*cos(f*x + e))*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^...
 

Sympy [F]

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

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

Output:

Integral(1/(sqrt(c + d*sin(e + f*x))*sin(e + f*x)**3 + 3*sqrt(c + d*sin(e 
+ f*x))*sin(e + f*x)**2 + 3*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + sqrt(c 
 + d*sin(e + f*x))), x)/a**3
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(a+a \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{4} d +\sin \left (f x +e \right )^{3} c +3 \sin \left (f x +e \right )^{3} d +3 \sin \left (f x +e \right )^{2} c +3 \sin \left (f x +e \right )^{2} d +3 \sin \left (f x +e \right ) c +\sin \left (f x +e \right ) d +c}d x}{a^{3}} \] Input:

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

Output:

int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**4*d + sin(e + f*x)**3*c + 3*si 
n(e + f*x)**3*d + 3*sin(e + f*x)**2*c + 3*sin(e + f*x)**2*d + 3*sin(e + f* 
x)*c + sin(e + f*x)*d + c),x)/a**3