\(\int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx\) [523]

Optimal result

Integrand size = 27, antiderivative size = 405 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac {(c-7 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac {\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac {d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^4 (c+d)^2 f \sqrt {c+d \sin (e+f x)}}-\frac {(c+3 d) \left (c^2-10 c d-7 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a^2 (c-d)^4 (c+d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (c^2-7 c d-10 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a^2 (c-d)^3 (c+d) f \sqrt {c+d \sin (e+f x)}} \] Output:

-1/3*d*(c^2-7*c*d-10*d^2)*cos(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sin(f*x+e))^ 
(3/2)-1/3*(c-7*d)*cos(f*x+e)/a^2/(c-d)^2/f/(1+sin(f*x+e))/(c+d*sin(f*x+e)) 
^(3/2)-1/3*cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2)-1/ 
3*d*(c+3*d)*(c^2-10*c*d-7*d^2)*cos(f*x+e)/a^2/(c-d)^4/(c+d)^2/f/(c+d*sin(f 
*x+e))^(1/2)+1/3*(c+3*d)*(c^2-10*c*d-7*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)^4/(c+d)^2/ 
f/((c+d*sin(f*x+e))/(c+d))^(1/2)+1/3*(c^2-7*c*d-10*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^2/(c-d)^3/(c+d)/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 10.53 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sqrt {c+d \sin (e+f x)} \left (-\frac {2 \left (c^3-7 c^2 d-27 c d^2-15 d^3\right )}{3 (c-d)^4 (c+d)^2}+\frac {2 \sin \left (\frac {1}{2} (e+f x)\right )}{3 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {1}{3 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {2 \left (c \sin \left (\frac {1}{2} (e+f x)\right )-9 d \sin \left (\frac {1}{2} (e+f x)\right )\right )}{3 (c-d)^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {2 d^3 \cos (e+f x)}{3 (c-d)^3 (c+d) (c+d \sin (e+f x))^2}+\frac {4 \left (5 c d^3 \cos (e+f x)+3 d^4 \cos (e+f x)\right )}{3 (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}\right )}{f (a+a \sin (e+f x))^2}+\frac {d \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \left (-\frac {2 \left (26 c^2 d+28 c d^2+10 d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{\sqrt {c+d \sin (e+f x)}}+\frac {2 \left (c^3-7 c^2 d-37 c d^2-21 d^3\right ) \cos ^2(e+f x) \sqrt {c+d \sin (e+f x)}}{d \left (1-\sin ^2(e+f x)\right )}-\frac {\left (-c^3+7 c^2 d+37 c d^2+21 d^3\right ) \left (\frac {2 (c+d) E\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{\sqrt {c+d \sin (e+f x)}}-\frac {2 c \operatorname {EllipticF}\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{\sqrt {c+d \sin (e+f x)}}\right )}{d}\right )}{6 (c-d)^4 (c+d)^2 f (a+a \sin (e+f x))^2} \] Input:

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

Output:

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*Sqrt[c + d*Sin[e + f*x]]*((-2*(c^ 
3 - 7*c^2*d - 27*c*d^2 - 15*d^3))/(3*(c - d)^4*(c + d)^2) + (2*Sin[(e + f* 
x)/2])/(3*(c - d)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3) - 1/(3*(c - d 
)^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (2*(c*Sin[(e + f*x)/2] - 9* 
d*Sin[(e + f*x)/2]))/(3*(c - d)^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + 
 (2*d^3*Cos[e + f*x])/(3*(c - d)^3*(c + d)*(c + d*Sin[e + f*x])^2) + (4*(5 
*c*d^3*Cos[e + f*x] + 3*d^4*Cos[e + f*x]))/(3*(c - d)^4*(c + d)^2*(c + d*S 
in[e + f*x]))))/(f*(a + a*Sin[e + f*x])^2) + (d*(Cos[(e + f*x)/2] + Sin[(e 
 + f*x)/2])^4*((-2*(26*c^2*d + 28*c*d^2 + 10*d^3)*EllipticF[(-e + Pi/2 - f 
*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e 
 + f*x]] + (2*(c^3 - 7*c^2*d - 37*c*d^2 - 21*d^3)*Cos[e + f*x]^2*Sqrt[c + 
d*Sin[e + f*x]])/(d*(1 - Sin[e + f*x]^2)) - ((-c^3 + 7*c^2*d + 37*c*d^2 + 
21*d^3)*((2*(c + d)*EllipticE[(-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c 
+ d*Sin[e + f*x])/(c + d)])/Sqrt[c + d*Sin[e + f*x]] - (2*c*EllipticF[(-e 
+ Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/Sqrt[c 
 + d*Sin[e + f*x]]))/d))/(6*(c - d)^4*(c + d)^2*f*(a + a*Sin[e + f*x])^2)
 

Rubi [A] (verified)

Time = 2.37 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.07, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3245, 27, 3042, 3457, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3245

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a (2 c-9 d)+5 a d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{5/2}}dx}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (2 c-9 d)+5 a d \sin (e+f x)}{(\sin (e+f x) a+a) (c+d \sin (e+f x))^{5/2}}dx}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3457

\(\displaystyle \frac {-\frac {\int -\frac {3 \left (10 d^2 a^2+(c-7 d) d \sin (e+f x) a^2\right )}{(c+d \sin (e+f x))^{5/2}}dx}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {10 d^2 a^2+(c-7 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^{5/2}}dx}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \int \frac {10 d^2 a^2+(c-7 d) d \sin (e+f x) a^2}{(c+d \sin (e+f x))^{5/2}}dx}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {3 \left (-\frac {2 \int -\frac {3 d^2 (9 c+7 d) a^2+d \left (c^2-7 d c-10 d^2\right ) \sin (e+f x) a^2}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {3 d^2 (9 c+7 d) a^2+d \left (c^2-7 d c-10 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {3 d^2 (9 c+7 d) a^2+d \left (c^2-7 d c-10 d^2\right ) \sin (e+f x) a^2}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {3 \left (\frac {-\frac {2 \int -\frac {2 a^2 d^2 \left (13 c^2+14 d c+5 d^2\right )-a^2 d (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {\int \frac {2 a^2 d^2 \left (13 c^2+14 d c+5 d^2\right )-a^2 d (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {\int \frac {2 a^2 d^2 \left (13 c^2+14 d c+5 d^2\right )-a^2 d (c+3 d) \left (c^2-10 d c-7 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {\frac {a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {\frac {2 a^2 \left (c^2-7 c d-10 d^2\right ) \left (c^2-d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^2 (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a^2 d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\right )}{a^2 (c-d)}-\frac {2 (c-7 d) \cos (e+f x)}{f (c-d) (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}}{6 a^2 (c-d)}-\frac {\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}}\)

Input:

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

Output:

-1/3*Cos[e + f*x]/((c - d)*f*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^( 
3/2)) + ((-2*(c - 7*d)*Cos[e + f*x])/((c - d)*f*(1 + Sin[e + f*x])*(c + d* 
Sin[e + f*x])^(3/2)) + (3*((-2*a^2*d*(c^2 - 7*c*d - 10*d^2)*Cos[e + f*x])/ 
(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((-2*a^2*d*(c + 3*d)*(c^2 - 
 10*c*d - 7*d^2)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + 
((-2*a^2*(c + 3*d)*(c^2 - 10*c*d - 7*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^2*(c^2 - 7*c*d - 10*d^2)*(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*Sqrt[c + d*Sin[e 
 + f*x]]))/(c^2 - d^2))/(3*(c^2 - d^2))))/(a^2*(c - d)))/(6*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 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] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1749\) vs. \(2(382)=764\).

Time = 2.34 (sec) , antiderivative size = 1750, normalized size of antiderivative = 4.32

method	result	size
default	\(\text {Expression too large to display}\)	\(1750\)

Input:

int(1/(a+sin(f*x+e)*a)^2/(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^2*(1/(c-d)^2*(-1/3/(c-d)*(-(-c-d 
*sin(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*s 
in(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 
*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^2/(c-d)^2*(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))/(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))+8/3* 
c*d/(c^2-d^2)^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*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))))...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 3145, normalized size of antiderivative = 7.77 \[ \int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

integrate(1/(a+a*sin(f*x+e))**2/(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)**2 + 2*c**2*sqrt(c 
+ d*sin(e + f*x))*sin(e + f*x) + c**2*sqrt(c + d*sin(e + f*x)) + 2*c*d*sqr 
t(c + d*sin(e + f*x))*sin(e + f*x)**3 + 4*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)**4 + 2*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**2
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{5} d^{3}+3 \sin \left (f x +e \right )^{4} c \,d^{2}+2 \sin \left (f x +e \right )^{4} d^{3}+3 \sin \left (f x +e \right )^{3} c^{2} d +6 \sin \left (f x +e \right )^{3} c \,d^{2}+\sin \left (f x +e \right )^{3} d^{3}+\sin \left (f x +e \right )^{2} c^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d +3 \sin \left (f x +e \right )^{2} c \,d^{2}+2 \sin \left (f x +e \right ) c^{3}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x}{a^{2}} \] Input:

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

Output:

int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**5*d**3 + 3*sin(e + f*x)**4*c*d 
**2 + 2*sin(e + f*x)**4*d**3 + 3*sin(e + f*x)**3*c**2*d + 6*sin(e + f*x)** 
3*c*d**2 + sin(e + f*x)**3*d**3 + sin(e + f*x)**2*c**3 + 6*sin(e + f*x)**2 
*c**2*d + 3*sin(e + f*x)**2*c*d**2 + 2*sin(e + f*x)*c**3 + 3*sin(e + f*x)* 
c**2*d + c**3),x)/a**2