\(\int \frac {1}{(a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx\) [770]

Optimal result

Integrand size = 27, antiderivative size = 503 \[ \int \frac {1}{(a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx=\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2}+\frac {3 b^2 \left (2 a b c-3 a^2 d+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x))}+\frac {3 b \left (2 a b c-3 a^2 d+b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 (b c-a d)^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (6 a b c-7 a^2 d+b^2 d\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 \left (a^2-b^2\right )^2 (b c-a d) f \sqrt {c+d \sin (e+f x)}}-\frac {\left (20 a^3 b c d+4 a b^3 c d-15 a^4 d^2-2 a^2 b^2 \left (4 c^2-3 d^2\right )-b^4 \left (4 c^2+3 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 (a+b)^3 (b c-a d)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

1/2*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin( 
f*x+e))^2+3/4*b^2*(-3*a^2*d+2*a*b*c+b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/ 
2)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))-3/4*b*(-3*a^2*d+2*a*b*c+b^2 
*d)*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-b^2)^2/(-a*d+b*c)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/ 
4*(-7*a^2*d+6*a*b*c+b^2*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-b^2)^2/(-a*d+b*c)/f/(c+ 
d*sin(f*x+e))^(1/2)+1/4*(20*a^3*b*c*d+4*a*b^3*c*d-15*a^4*d^2-2*a^2*b^2*(4* 
c^2-3*d^2)-b^4*(4*c^2+3*d^2))*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2*b/(a+ 
b),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/(a-b)^2/(a+b)^3 
/(-a*d+b*c)^2/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.16 (sec) , antiderivative size = 1069, normalized size of antiderivative = 2.13 \[ \int \frac {1}{(a+b \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[c + d*Sin[e + f*x]]*(-1/2*(b^2*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + 
a*d)*(a + b*Sin[e + f*x])^2) + (3*(2*a*b^3*c*Cos[e + f*x] - 3*a^2*b^2*d*Co 
s[e + f*x] + b^4*d*Cos[e + f*x]))/(4*(a^2 - b^2)^2*(-(b*c) + a*d)^2*(a + b 
*Sin[e + f*x]))))/f + ((-2*(16*a^2*b^2*c^2 + 8*b^4*c^2 - 32*a^3*b*c*d + 2* 
a*b^3*c*d + 16*a^4*d^2 - 19*a^2*b^2*d^2 + 9*b^4*d^2)*EllipticPi[(2*b)/(a + 
 b), (-e + Pi/2 - f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d) 
])/((a + b)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(20*a^2*b^2*c*d + 4*b^4*c*d 
 - 32*a^3*b*d^2 + 8*a*b^3*d^2)*Cos[e + f*x]*((b*c - a*d)*EllipticF[I*ArcSi 
nh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + a*d*E 
llipticPi[(b*(c + d))/(b*c - a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + 
d*Sin[e + f*x]]], (c + d)/(c - d)])*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqr 
t[-((d + d*Sin[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])) 
)/(b*d^2*Sqrt[-(c + d)^(-1)]*(b*c - a*d)*(a + b*Sin[e + f*x])*Sqrt[1 - Sin 
[e + f*x]^2]*Sqrt[-((c^2 - d^2 - 2*c*(c + d*Sin[e + f*x]) + (c + d*Sin[e + 
 f*x])^2)/d^2)]) - ((2*I)*(-6*a*b^3*c*d + 9*a^2*b^2*d^2 - 3*b^4*d^2)*Cos[e 
 + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*EllipticE[I*ArcSinh[Sqrt 
[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)/(c - d)] + d*(-2*(a + b 
)*(-(b*c) + a*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e 
+ f*x]]], (c + d)/(c - d)] + (2*a^2 - b^2)*d*EllipticPi[(b*(c + d))/(b*c - 
 a*d), I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c + d)...
 

Rubi [A] (verified)

Time = 4.13 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3281, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3281

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\int \frac {-8 d^2 a^4+16 b c d a^3-b^2 \left (8 c^2-5 d^2\right ) a^2+2 b^3 c d a-3 b^2 d \left (-3 d a^2+2 b c a+b^2 d\right ) \sin ^2(e+f x)-b^4 \left (4 c^2+3 d^2\right )-2 b d \left (-8 d a^3+5 b c a^2+2 b^2 d a+b^3 c\right ) \sin (e+f x)}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\int \frac {-8 d^2 a^4+16 b c d a^3-b^2 \left (8 c^2-5 d^2\right ) a^2+2 b^3 c d a-3 b^2 d \left (-3 d a^2+2 b c a+b^2 d\right ) \sin ^2(e+f x)-b^4 \left (4 c^2+3 d^2\right )-2 b d \left (-8 d a^3+5 b c a^2+2 b^2 d a+b^3 c\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\int \frac {-8 d^2 a^4+16 b c d a^3-b^2 \left (8 c^2-5 d^2\right ) a^2+2 b^3 c d a-3 b^2 d \left (-3 d a^2+2 b c a+b^2 d\right ) \sin (e+f x)^2-b^4 \left (4 c^2+3 d^2\right )-2 b d \left (-8 d a^3+5 b c a^2+2 b^2 d a+b^3 c\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {-3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \int \sqrt {c+d \sin (e+f x)}dx-\frac {\int -\frac {d (b c-a d) \left (-7 d a^2+6 b c a+b^2 d\right ) \sin (e+f x) b^2+d \left (-8 d^2 a^4+7 b c d a^3-b^2 \left (2 c^2-5 d^2\right ) a^2+5 b^3 c d a-b^4 \left (4 c^2+3 d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {d (b c-a d) \left (-7 d a^2+6 b c a+b^2 d\right ) \sin (e+f x) b^2+d \left (-8 d^2 a^4+7 b c d a^3-b^2 \left (2 c^2-5 d^2\right ) a^2+5 b^3 c d a-b^4 \left (4 c^2+3 d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \int \sqrt {c+d \sin (e+f x)}dx}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {d (b c-a d) \left (-7 d a^2+6 b c a+b^2 d\right ) \sin (e+f x) b^2+d \left (-8 d^2 a^4+7 b c d a^3-b^2 \left (2 c^2-5 d^2\right ) a^2+5 b^3 c d a-b^4 \left (4 c^2+3 d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \int \sqrt {c+d \sin (e+f x)}dx}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {d (b c-a d) \left (-7 d a^2+6 b c a+b^2 d\right ) \sin (e+f x) b^2+d \left (-8 d^2 a^4+7 b c d a^3-b^2 \left (2 c^2-5 d^2\right ) a^2+5 b^3 c d a-b^4 \left (4 c^2+3 d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {d (b c-a d) \left (-7 d a^2+6 b c a+b^2 d\right ) \sin (e+f x) b^2+d \left (-8 d^2 a^4+7 b c d a^3-b^2 \left (2 c^2-5 d^2\right ) a^2+5 b^3 c d a-b^4 \left (4 c^2+3 d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {3 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{\sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {d (b c-a d) \left (-7 d a^2+6 b c a+b^2 d\right ) \sin (e+f x) b^2+d \left (-8 d^2 a^4+7 b c d a^3-b^2 \left (2 c^2-5 d^2\right ) a^2+5 b^3 c d a-b^4 \left (4 c^2+3 d^2\right )\right ) b}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx+b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx+b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\frac {b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\frac {b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx+\frac {2 b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\frac {b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\frac {b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{\sqrt {c+d \sin (e+f x)}}+\frac {2 b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}+\frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {b^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2}+\frac {\frac {3 b^2 \left (-3 a^2 d+2 a b c+b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac {\frac {\frac {2 b d (b c-a d) \left (-7 a^2 d+6 a b c+b^2 d\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f \sqrt {c+d \sin (e+f x)}}+\frac {2 b d \left (-15 a^4 d^2+20 a^3 b c d-2 a^2 b^2 \left (4 c^2-3 d^2\right )+4 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {6 b \left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}}{4 \left (a^2-b^2\right ) (b c-a d)}\)

Input:

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

Output:

(b^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*(b*c - a*d)*f*( 
a + b*Sin[e + f*x])^2) + ((3*b^2*(2*a*b*c - 3*a^2*d + b^2*d)*Cos[e + f*x]* 
Sqrt[c + d*Sin[e + f*x]])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])) 
 - ((-6*b*(2*a*b*c - 3*a^2*d + b^2*d)*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*b*d*(b*c - a*d)*(6*a*b*c - 7*a^2*d + b^2*d)*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*b*d*(20*a^3*b*c*d + 4*a*b^3*c*d - 15*a^4*d^2 - 2*a^2*b^2* 
(4*c^2 - 3*d^2) - b^4*(4*c^2 + 3*d^2))*EllipticPi[(2*b)/(a + b), (e - Pi/2 
 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((a + b)*f*S 
qrt[c + d*Sin[e + f*x]]))/(b*d))/(2*(a^2 - b^2)*(b*c - a*d)))/(4*(a^2 - b^ 
2)*(b*c - a*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 3281

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 + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 
))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n 
 + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2* 
n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))
 

rule 3284

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

rule 3286

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

rule 3481

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

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x] 
, x] - Simp[1/(b*d)   Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 
] && NeQ[c^2 - d^2, 0]
 

Maple [A] (warning: unable to verify)

Time = 1.51 (sec) , antiderivative size = 864, 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 {b^{2} \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{2 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right ) \left (a +b \sin \left (f x +e \right )\right )^{2}}-\frac {3 b^{2} \left (3 a^{2} d -2 a b c -b^{2} d \right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} \left (a +b \sin \left (f x +e \right )\right )}-\frac {d \left (7 a^{3} d -4 a^{2} b c -a \,b^{2} d -2 b^{3} c \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 )}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}-\frac {3 b d \left (3 a^{2} d -2 a b c -b^{2} d \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 )}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {\left (15 a^{4} d^{2}-20 a^{3} b c d +8 a^{2} b^{2} c^{2}-6 a^{2} b^{2} d^{2}-4 a \,b^{3} c d +4 b^{4} c^{2}+3 b^{4} 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}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \frac {-\frac {c}{d}+1}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {c -d}{c +d}}\right )}{4 \left (a^{3} d -a^{2} b c -a \,b^{2} d +b^{3} c \right )^{2} b \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(864\)

Input:

int(1/(a+b*sin(f*x+e))^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)*(-1/2*b^2/(a^3*d-a^2*b*c-a*b^2*d+b 
^3*c)*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))^2-3/4*b^2*( 
3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2*(-(-c-d*sin(f*x+e)) 
*cos(f*x+e)^2)^(1/2)/(a+b*sin(f*x+e))-1/4*d*(7*a^3*d-4*a^2*b*c-a*b^2*d-2*b 
^3*c)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^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))-3/4*b*d*(3*a^2*d-2*a*b*c-b^2*d)/(a^3*d-a^2*b*c-a*b^2*d 
+b^3*c)^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)))+1/4*(15*a 
^4*d^2-20*a^3*b*c*d+8*a^2*b^2*c^2-6*a^2*b^2*d^2-4*a*b^3*c*d+4*b^4*c^2+3*b^ 
4*d^2)/(a^3*d-a^2*b*c-a*b^2*d+b^3*c)^2/b*(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+a/b)*EllipticPi(((c+d*sin(f*x+e)) 
/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2)))/cos(f*x+e)/(c+d*si 
n(f*x+e))^(1/2)/f
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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