\(\int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx\) [522]

Optimal result

Integrand size = 27, antiderivative size = 326 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^3 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {(c-5 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}}-\frac {\left (c^2-5 c d-12 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a^2 (c-d)^3 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c-5 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a^2 (c-d)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

-1/3*d*(c^2-5*c*d-12*d^2)*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sin(f*x+e))^ 
(1/2)-1/3*(c-5*d)*cos(f*x+e)/a^2/(c-d)^2/f/(1+sin(f*x+e))/(c+d*sin(f*x+e)) 
^(1/2)-1/3*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2)+1/ 
3*(c^2-5*c*d-12*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^2/(c-d)^3/(c+d)/f/((c+d*sin(f*x+e))/(c+d) 
)^(1/2)+1/3*(c-5*d)*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^2/(c-d)^2/f/(c+d*sin(f*x+e))^(1/2 
)
 

Mathematica [A] (verified)

Time = 7.78 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left ((c+d \sin (e+f x)) \left (-\frac {2 \left (c^2-5 c d-9 d^2\right )}{c+d}+\frac {2 (c-d) \sin \left (\frac {1}{2} (e+f x)\right )}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {-c+d}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {2 (c-6 d) \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}+\frac {6 d^3 \cos (e+f x)}{(c+d) (c+d \sin (e+f x))}\right )+\frac {\left (c^2-5 c d-12 d^2\right ) (c+d \sin (e+f x))-d^2 (11 c+5 d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (c^2-5 c d-12 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{c+d}\right )}{3 a^2 (c-d)^3 f (1+\sin (e+f x))^2 \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*((c + d*Sin[e + f*x])*((-2*(c^2 - 
 5*c*d - 9*d^2))/(c + d) + (2*(c - d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] 
+ Sin[(e + f*x)/2])^3 + (-c + d)/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 
 (2*(c - 6*d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + (6 
*d^3*Cos[e + f*x])/((c + d)*(c + d*Sin[e + f*x]))) + ((c^2 - 5*c*d - 12*d^ 
2)*(c + d*Sin[e + f*x]) - d^2*(11*c + 5*d)*EllipticF[(-2*e + Pi - 2*f*x)/4 
, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c^2 - 5*c*d - 12*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)]) 
/(c + d)))/(3*a^2*(c - d)^3*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x] 
])
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.06, 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, 3233, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3245

\(\displaystyle -\frac {\int -\frac {a (2 c-7 d)+3 a d \sin (e+f x)}{2 (\sin (e+f x) a+a) (c+d \sin (e+f x))^{3/2}}dx}{3 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (2 c-7 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{3/2}}dx}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (2 c-7 d)+3 a d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{3/2}}dx}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {12 d^2 a^2+(c-5 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^{3/2}}dx}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {12 d^2 a^2+(c-5 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^{3/2}}dx}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {12 d^2 a^2+(c-5 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^{3/2}}dx}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {a^2 d^2 (11 c+5 d)-a^2 d \left (c^2-5 d c-12 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {a^2 d^2 (11 c+5 d)-a^2 d \left (c^2-5 d c-12 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {a^2 d^2 (11 c+5 d)-a^2 d \left (c^2-5 d c-12 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\frac {a^2 (c-5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a^2 \left (c^2-5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {\frac {a^2 (c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (c^2-5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {a^2 (c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (c^2-5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {\frac {2 a^2 (c-5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 \left (c^2-5 c d-12 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d \left (c^2-5 c d-12 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{a^2 (c-d)}-\frac {2 (c-5 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) \sqrt {c+d \sin (e+f x)}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 \sqrt {c+d \sin (e+f x)}}\)

Input:

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

Output:

-1/3*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x 
]]) + ((-2*(c - 5*d)*Cos[e + f*x])/((c - d)*f*(1 + Sin[e + f*x])*Sqrt[c + 
d*Sin[e + f*x]]) + ((-2*a^2*d*(c^2 - 5*c*d - 12*d^2)*Cos[e + f*x])/((c^2 - 
 d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*a^2*(c^2 - 5*c*d - 12*d^2)*Ellipt 
icE[(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^2*(c - 5*d)*(c^2 - d^2)*EllipticF[(e 
- Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(f*Sqr 
t[c + d*Sin[e + f*x]]))/(c^2 - d^2))/(a^2*(c - d)))/(6*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 3233

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1292\) vs. \(2(307)=614\).

Time = 1.61 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1293\)

Input:

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

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/a^2*(1/(c-d)*(-1/3/(c-d)*(-(-c-d*s 
in(f*x+e))*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))^2-1/3*(-sin(f*x+e)^2*d-c*sin 
(f*x+e)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d)/((1+sin(f*x+e))*(-1+sin(f*x+e))*(- 
c-d*sin(f*x+e)))^(1/2)+2*d^2/(3*c^2-6*c*d+3*d^2)*(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))-1/3*d*(c-3*d)/(c-d)^2*(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*s 
in(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))))+d^2/(c-d)^2*(2*d*cos(f*x+e)^2/(c^2-d^2) 
/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2*c/(c^2-d^2)*(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))+2/(c^2-d^2)*d*(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))))-d/(c-d)^2*(-(-sin(f*x+e)^2*d-c*sin(f*x+e) 
+d*sin(f*x+e)+c)/(c-d)/((1+sin(f*x+e))*(-1+sin(f*x+e))*(-c-d*sin(f*x+e)...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 1942, normalized size of antiderivative = 5.96 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/9*((4*c^4 - 16*c^3*d - 2*c^2*d^2 + 48*c*d^3 + 30*d^4 - (2*c^3*d - 10*c^ 
2*d^2 + 9*c*d^3 + 15*d^4)*cos(f*x + e)^3 - (2*c^4 - 6*c^3*d - 11*c^2*d^2 + 
 33*c*d^3 + 30*d^4)*cos(f*x + e)^2 + (2*c^4 - 8*c^3*d - c^2*d^2 + 24*c*d^3 
 + 15*d^4)*cos(f*x + e) + (4*c^4 - 16*c^3*d - 2*c^2*d^2 + 48*c*d^3 + 30*d^ 
4 - (2*c^3*d - 10*c^2*d^2 + 9*c*d^3 + 15*d^4)*cos(f*x + e)^2 + (2*c^4 - 8* 
c^3*d - c^2*d^2 + 24*c*d^3 + 15*d^4)*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) + (4*c^4 
- 16*c^3*d - 2*c^2*d^2 + 48*c*d^3 + 30*d^4 - (2*c^3*d - 10*c^2*d^2 + 9*c*d 
^3 + 15*d^4)*cos(f*x + e)^3 - (2*c^4 - 6*c^3*d - 11*c^2*d^2 + 33*c*d^3 + 3 
0*d^4)*cos(f*x + e)^2 + (2*c^4 - 8*c^3*d - c^2*d^2 + 24*c*d^3 + 15*d^4)*co 
s(f*x + e) + (4*c^4 - 16*c^3*d - 2*c^2*d^2 + 48*c*d^3 + 30*d^4 - (2*c^3*d 
- 10*c^2*d^2 + 9*c*d^3 + 15*d^4)*cos(f*x + e)^2 + (2*c^4 - 8*c^3*d - c^2*d 
^2 + 24*c*d^3 + 15*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-1/2*I*d)*weierst 
rassPInverse(-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*(-2*I*c^3*d + 8* 
I*c^2*d^2 + 34*I*c*d^3 + 24*I*d^4 + (I*c^2*d^2 - 5*I*c*d^3 - 12*I*d^4)*cos 
(f*x + e)^3 + (I*c^3*d - 3*I*c^2*d^2 - 22*I*c*d^3 - 24*I*d^4)*cos(f*x + e) 
^2 + (-I*c^3*d + 4*I*c^2*d^2 + 17*I*c*d^3 + 12*I*d^4)*cos(f*x + e) + (-2*I 
*c^3*d + 8*I*c^2*d^2 + 34*I*c*d^3 + 24*I*d^4 + (I*c^2*d^2 - 5*I*c*d^3 -...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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