\(\int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx\) [517]

Optimal result

Integrand size = 27, antiderivative size = 333 \[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (3 c^2+20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a (c-d)^3 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(3 c+5 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}} \] Output:

-1/3*d*(3*c+5*d)*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(3/2)-cos(f 
*x+e)/(c-d)/f/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2)-1/3*d*(3*c^2+20*c*d+ 
9*d^2)*cos(f*x+e)/a/(c-d)^3/(c+d)^2/f/(c+d*sin(f*x+e))^(1/2)+1/3*(3*c^2+20 
*c*d+9*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/(c-d)^3/(c+d)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1 
/3*(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/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 3.58 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\frac {\left (3 c^2+20 c d+9 d^2\right ) (c+d \sin (e+f x))+d \left (15 c^2+12 c d+5 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+\left (3 c^2+20 c d+9 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c+d)^2}+2 (c+d \sin (e+f x)) \left (\frac {3 \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}-\frac {3 c^2+13 c d+6 d^2+\frac {d^2 \cos (e+f x) \left (8 c^2+3 c d-d^2+d (7 c+3 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^2}}{(c+d)^2}\right )\right )}{3 a (c-d)^3 f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \] Input:

Integrate[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]
 

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(((3*c^2 + 20*c*d + 9*d^2)*(c + d 
*Sin[e + f*x]) + d*(15*c^2 + 12*c*d + 5*d^2)*EllipticF[(-2*e + Pi - 2*f*x) 
/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (3*c^2 + 20*c*d + 
9*d^2)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*Ellipt 
icF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*Sqrt[(c + d*Sin[e + f*x])/(c + 
d)])/(c + d)^2 + 2*(c + d*Sin[e + f*x])*((3*Sin[(e + f*x)/2])/(Cos[(e + f* 
x)/2] + Sin[(e + f*x)/2]) - (3*c^2 + 13*c*d + 6*d^2 + (d^2*Cos[e + f*x]*(8 
*c^2 + 3*c*d - d^2 + d*(7*c + 3*d)*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2)/ 
(c + d)^2)))/(3*a*(c - d)^3*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3247, 27, 3042, 3233, 27, 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.

\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a) (c+d \sin (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a \sin (e+f x)+a) (c+d \sin (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 3247

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle -\frac {d \left (\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {2 \int -\frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \left (\frac {\frac {\int \frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d \left (\frac {\frac {\int \frac {a \left (15 c^2+12 d c+5 d^2\right )+a \left (3 c^2+20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d \left (\frac {\frac {\frac {a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {d \left (\frac {\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {d \left (\frac {\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {d \left (\frac {\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}+\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}+\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {d \left (\frac {2 a (3 c+5 d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}+\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\frac {2 a \left (3 c^2+20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a (3 c+5 d) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}\right )}{2 a^2 (c-d)}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}\)

Input:

Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]
 

Output:

-(Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) 
) - (d*((2*a*(3*c + 5*d)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x 
])^(3/2)) + ((2*a*(3*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x])/((c^2 - d^2)*f*Sq 
rt[c + d*Sin[e + f*x]]) + ((2*a*(3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi 
/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin 
[e + f*x])/(c + d)]) - (2*a*(3*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)])/(d*f*Sqrt[c + d 
*Sin[e + f*x]]))/(c^2 - d^2))/(3*(c^2 - d^2))))/(2*a^2*(c - d))
 

Defintions of rubi rules used

rule 27

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

rule 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 3247

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1284\) vs. \(2(316)=632\).

Time = 1.60 (sec) , antiderivative size = 1285, normalized size of antiderivative = 3.86

method	result	size
default	\(\text {Expression too large to display}\)	\(1285\)

Input:

int(1/(a+sin(f*x+e)*a)/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/a*(1/(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)))^(1/2)-2*d/(2*c-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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c 
+d))^(1/2))-d/(c-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*(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)*(2/3/(c^2-d^2)/d*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e) 
+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2) 
^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 2098, normalized size of antiderivative = 6.30 \[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
 

Output:

-1/9*((6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48*c*d^4 - 15*d^5 - (6* 
c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^3 - (12*c^4*d - 4*c^ 
3*d^2 - 41*c^2*d^3 - 48*c*d^4 - 15*d^5)*cos(f*x + e)^2 + (6*c^5 - 5*c^4*d 
- 12*c^3*d^2 - 20*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e) + (6*c^5 + 7*c 
^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48*c*d^4 - 15*d^5 - (6*c^3*d^2 - 5*c^2*d^ 
3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^2 + 2*(6*c^4*d - 5*c^3*d^2 - 18*c^2*d^ 
3 - 15*c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassPInvers 
e(-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) + (6*c^5 + 7*c^4*d - 22*c^3*d^2 
- 56*c^2*d^3 - 48*c*d^4 - 15*d^5 - (6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15* 
d^5)*cos(f*x + e)^3 - (12*c^4*d - 4*c^3*d^2 - 41*c^2*d^3 - 48*c*d^4 - 15*d 
^5)*cos(f*x + e)^2 + (6*c^5 - 5*c^4*d - 12*c^3*d^2 - 20*c^2*d^3 - 18*c*d^4 
 - 15*d^5)*cos(f*x + e) + (6*c^5 + 7*c^4*d - 22*c^3*d^2 - 56*c^2*d^3 - 48* 
c*d^4 - 15*d^5 - (6*c^3*d^2 - 5*c^2*d^3 - 18*c*d^4 - 15*d^5)*cos(f*x + e)^ 
2 + 2*(6*c^4*d - 5*c^3*d^2 - 18*c^2*d^3 - 15*c*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) - 3*(-3*I*c^4*d - 26*I*c^3*d^2 - 52*I*c^2*d^3 - 38*I*c*d^4 - 9*I*d 
^5 + (3*I*c^2*d^3 + 20*I*c*d^4 + 9*I*d^5)*cos(f*x + e)^3 + (6*I*c^3*d^2 + 
43*I*c^2*d^3 + 38*I*c*d^4 + 9*I*d^5)*cos(f*x + e)^2 + (-3*I*c^4*d - 20*...
 

Sympy [F]

\[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\int \frac {1}{c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a} \] Input:

integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))**(5/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
 

Output:

integrate(1/((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:

int(1/((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)),x)
 

Output:

int(1/((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{4} d^{3}+3 \sin \left (f x +e \right )^{3} c \,d^{2}+\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c^{2} d +3 \sin \left (f x +e \right )^{2} c \,d^{2}+\sin \left (f x +e \right ) c^{3}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x}{a} \] Input:

int(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x)
 

Output:

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