\(\int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx\) [763]

Optimal result

Integrand size = 27, antiderivative size = 449 \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx=\frac {d \left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}+\frac {\left (2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{\left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {b \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{\left (a^2-b^2\right ) (b c-a d) f \sqrt {c+d \sin (e+f x)}}+\frac {b \left (2 a b c-5 a^2 d+3 b^2 d\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a-b) (a+b)^2 (b c-a d)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:

d*(2*a^2*d^2+b^2*(c^2-3*d^2))*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)^2/(c^2-d^2)/ 
f/(c+d*sin(f*x+e))^(1/2)+b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f* 
x+e))/(c+d*sin(f*x+e))^(1/2)-(2*a^2*d^2+b^2*(c^2-3*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-b^ 
2)/(-a*d+b*c)^2/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-b*InverseJacobi 
AM(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)/(-a*d+b*c)/f/(c+d*sin(f*x+e))^(1/2)-b*(-5*a^2*d+2*a*b*c+3* 
b^2*d)*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)/(a+b)^2/(-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 = 7.10 (sec) , antiderivative size = 1057, normalized size of antiderivative = 2.35 \[ \int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[c + d*Sin[e + f*x]]*((b^3*Cos[e + f*x])/((a^2 - b^2)*(-(b*c) + a*d)^ 
2*(a + b*Sin[e + f*x])) + (2*d^3*Cos[e + f*x])/((b*c - a*d)^2*(c^2 - d^2)* 
(c + d*Sin[e + f*x]))))/f + ((-2*(4*a*b^2*c^3 - 8*a^2*b*c^2*d + 7*b^3*c^2* 
d + 4*a^3*c*d^2 - 8*a*b^2*c*d^2 + 10*a^2*b*d^3 - 9*b^3*d^3)*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)*(4*a*b^2*c^2*d + 4*a 
^2*b*c*d^2 - 4*b^3*c*d^2 + 4*a^3*d^3 - 8*a*b^2*d^3)*Cos[e + f*x]*((b*c - a 
*d)*EllipticF[I*ArcSinh[Sqrt[-(c + d)^(-1)]*Sqrt[c + d*Sin[e + f*x]]], (c 
+ d)/(c - d)] + a*d*EllipticPi[(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)]*Sqrt[-((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)*(-(b^3*c^2*d) - 2*a^2*b*d^3 
 + 3*b^3*d^3)*Cos[e + f*x]*Cos[2*(e + f*x)]*(2*b*(c - d)*(b*c - a*d)*Ellip 
ticE[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)/(c - d)]))*Sqrt[(d - d*Sin[e + f*x])/(c + d)]*Sqrt[-((...
 

Rubi [A] (verified)

Time = 3.96 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3281

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3481

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

\(\Big \downarrow \) 3286

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

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

Input:

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

Output:

(b^2*Cos[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*Sqrt[c 
+ d*Sin[e + f*x]]) + ((2*d*(2*a^2*d^2 + b^2*(c^2 - 3*d^2))*Cos[e + f*x])/( 
(b*c - a*d)*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((2*(2*a^2*d^2 + b^2 
*(c^2 - 3*d^2))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Si 
n[e + f*x]])/(f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((-2*b^2*d*(b*c - a* 
d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Si 
n[e + f*x])/(c + d)])/(f*Sqrt[c + d*Sin[e + f*x]]) + (2*b^2*d*(2*a*b*c - 5 
*a^2*d + 3*b^2*d)*(c^2 - 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*Sqrt[c + d 
*Sin[e + f*x]]))/(b*d))/((b*c - a*d)*(c^2 - d^2)))/(2*(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 [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1259\) vs. \(2(443)=886\).

Time = 2.88 (sec) , antiderivative size = 1260, normalized size of antiderivative = 2.81

method	result	size
default	\(\text {Expression too large to display}\)	\(1260\)

Input:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(1/(a+b*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*b**2*d**2 + 2*sin(e + f*x)** 
3*a*b*d**2 + 2*sin(e + f*x)**3*b**2*c*d + sin(e + f*x)**2*a**2*d**2 + 4*si 
n(e + f*x)**2*a*b*c*d + sin(e + f*x)**2*b**2*c**2 + 2*sin(e + f*x)*a**2*c* 
d + 2*sin(e + f*x)*a*b*c**2 + a**2*c**2),x)