\(\int \frac {(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^3} \, dx\) [768]

Optimal result

Integrand size = 27, antiderivative size = 472 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^3} \, dx=\frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left (6 a b c-a^2 d-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{4 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {\left (6 a b c-a^2 d-5 b^2 d\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{4 b \left (a^2-b^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(b c-a d) \left (6 a b c+a^2 d-7 b^2 d\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 b^2 \left (a^2-b^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {\left (4 a^3 b c d+20 a b^3 c d+a^4 d^2-b^4 \left (4 c^2+3 d^2\right )-2 a^2 b^2 \left (4 c^2+5 d^2\right )\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{4 (a-b)^2 b^2 (a+b)^3 f \sqrt {c+d \sin (e+f x)}} \] Output:

1/2*(-a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2-b^2)/f/(a+b*sin(f*x+ 
e))^2+1/4*(-a^2*d+6*a*b*c-5*b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/(a^2- 
b^2)^2/f/(a+b*sin(f*x+e))-1/4*(-a^2*d+6*a*b*c-5*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)/b/(a^2-b^ 
2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/4*(-a*d+b*c)*(a^2*d+6*a*b*c-7*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)/b^2/(a^2-b^2)^2/f/(c+d*sin(f*x+e))^(1/2)+1/4*(4*a^3* 
b*c*d+20*a*b^3*c*d+a^4*d^2-b^4*(4*c^2+3*d^2)-2*a^2*b^2*(4*c^2+5*d^2))*Elli 
pticPi(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/b^2/(a+b)^3/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.02 (sec) , antiderivative size = 1001, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+b \sin (e+f x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[c + d*Sin[e + f*x]]*((b*c*Cos[e + f*x] - a*d*Cos[e + f*x])/(2*(a^2 - 
 b^2)*(a + b*Sin[e + f*x])^2) + (6*a*b*c*Cos[e + f*x] - a^2*d*Cos[e + f*x] 
 - 5*b^2*d*Cos[e + f*x])/(4*(a^2 - b^2)^2*(a + b*Sin[e + f*x]))))/f + ((-2 
*(16*a^2*c^2 + 8*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2 + b^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)*Sqrt[c + d*Sin[e + f*x]]) - ((2*I)*(20*a^2*c*d + 4*b^2 
*c*d - 24*a*b*d^2)*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*S 
in[e + f*x])/(c - d))]*(-(b*c) + a*d + b*(c + d*Sin[e + f*x])))/(b*d^2*Sqr 
t[-(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*c*d + a^2*d^2 + 5*b^2*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)*Ell 
ipticF[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[Sqr 
t[-(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) +...
 

Rubi [A] (verified)

Time = 4.07 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.741, Rules used = {3042, 3278, 27, 3042, 3534, 27, 3042, 3538, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3278

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {-\frac {\int -\frac {-d (b c-a d) \left (-d a^2+6 b c a-5 b^2 d\right ) \sin ^2(e+f x)-2 d (b c-a d) \left (5 c a^2-6 b d a+b^2 c\right ) \sin (e+f x)+(b c-a d) \left (-\left (\left (8 c^2+3 d^2\right ) a^2\right )+18 b c d a-b^2 \left (4 c^2+3 d^2\right )\right )}{2 (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{\left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {\int \frac {-d (b c-a d) \left (-d a^2+6 b c a-5 b^2 d\right ) \sin ^2(e+f x)-2 d (b c-a d) \left (5 c a^2-6 b d a+b^2 c\right ) \sin (e+f x)+(b c-a d) \left (-\left (\left (8 c^2+3 d^2\right ) a^2\right )+18 b c d a-b^2 \left (4 c^2+3 d^2\right )\right )}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {\int \frac {-d (b c-a d) \left (-d a^2+6 b c a-5 b^2 d\right ) \sin (e+f x)^2-2 d (b c-a d) \left (5 c a^2-6 b d a+b^2 c\right ) \sin (e+f x)+(b c-a d) \left (-\left (\left (8 c^2+3 d^2\right ) a^2\right )+18 b c d a-b^2 \left (4 c^2+3 d^2\right )\right )}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {(b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \int \sqrt {c+d \sin (e+f x)}dx}{b}-\frac {\int \frac {d (b c-a d) \left (c d a^3+b \left (2 c^2+3 d^2\right ) a^2-13 b^2 c d a+b^3 \left (4 c^2+3 d^2\right )\right )-d (b c-a d)^2 \left (d a^2+6 b c a-7 b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {(b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \int \sqrt {c+d \sin (e+f x)}dx}{b}-\frac {\int \frac {d (b c-a d) \left (c d a^3+b \left (2 c^2+3 d^2\right ) a^2-13 b^2 c d a+b^3 \left (4 c^2+3 d^2\right )\right )-d (b c-a d)^2 \left (d a^2+6 b c a-7 b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {(b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {d (b c-a d) \left (c d a^3+b \left (2 c^2+3 d^2\right ) a^2-13 b^2 c d a+b^3 \left (4 c^2+3 d^2\right )\right )-d (b c-a d)^2 \left (d a^2+6 b c a-7 b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {(b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{b \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\int \frac {d (b c-a d) \left (c d a^3+b \left (2 c^2+3 d^2\right ) a^2-13 b^2 c d a+b^3 \left (4 c^2+3 d^2\right )\right )-d (b c-a d)^2 \left (d a^2+6 b c a-7 b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {\int \frac {d (b c-a d) \left (c d a^3+b \left (2 c^2+3 d^2\right ) a^2-13 b^2 c d a+b^3 \left (4 c^2+3 d^2\right )\right )-d (b c-a d)^2 \left (d a^2+6 b c a-7 b^2 d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {-\frac {d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {-\frac {d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {-\frac {d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {-\frac {d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {-\frac {d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}dx}{b}-\frac {2 d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {-\frac {d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {-\frac {d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{b \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}}{b d}-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {(b c-a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}-\frac {\frac {-\frac {2 (b c-a d) \left (a^2 (-d)+6 a b c-5 b^2 d\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {-\frac {2 d \left (a^2 d+6 a b c-7 b^2 d\right ) (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (a^4 d^2+4 a^3 b c d-2 a^2 b^2 \left (4 c^2+5 d^2\right )+20 a b^3 c d-b^4 \left (4 c^2+3 d^2\right )\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{b f (a+b) \sqrt {c+d \sin (e+f x)}}}{b d}}{2 \left (a^2-b^2\right ) (b c-a d)}-\frac {\left (a^2 (-d)+6 a b c-5 b^2 d\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}}{4 \left (a^2-b^2\right )}\)

Input:

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

Output:

((b*c - a*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(2*(a^2 - b^2)*f*(a + 
b*Sin[e + f*x])^2) - (-(((6*a*b*c - a^2*d - 5*b^2*d)*Cos[e + f*x]*Sqrt[c + 
 d*Sin[e + f*x]])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))) + ((-2*(b*c - a*d) 
*(6*a*b*c - a^2*d - 5*b^2*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]* 
Sqrt[c + d*Sin[e + f*x]])/(b*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - ((-2* 
d*(b*c - a*d)^2*(6*a*b*c + a^2*d - 7*b^2*d)*EllipticF[(e - Pi/2 + f*x)/2, 
(2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*f*Sqrt[c + d*Sin[e + 
 f*x]]) - (2*d*(b*c - a*d)*(4*a^3*b*c*d + 20*a*b^3*c*d + a^4*d^2 - b^4*(4* 
c^2 + 3*d^2) - 2*a^2*b^2*(4*c^2 + 5*d^2))*EllipticPi[(2*b)/(a + b), (e - P 
i/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(b*(a + b 
)*f*Sqrt[c + d*Sin[e + f*x]]))/(b*d))/(2*(a^2 - b^2)*(b*c - a*d)))/(4*(a^2 
 - b^2))
 

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 3278

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*(a + b*Si 
n[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n - 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)*(c + 
d*Sin[e + f*x])^(n - 2)*Simp[c*(a*c - b*d)*(m + 1) + d*(b*c - a*d)*(n - 1) 
+ (d*(a*c - b*d)*(m + 1) - c*(b*c - a*d)*(m + 2))*Sin[e + f*x] - d*(b*c - a 
*d)*(m + n + 1)*Sin[e + f*x]^2, 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] && LtQ[m, -1 
] && LtQ[1, n, 2] && IntegersQ[2*m, 2*n]
 

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. \(1710\) vs. \(2(454)=908\).

Time = 4.38 (sec) , antiderivative size = 1711, normalized size of antiderivative = 3.62

method	result	size
default	\(\text {Expression too large to display}\)	\(1711\)

Input:

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

Output:

(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(2*d^2/b^3*(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*s 
in(f*x+e))/(c-d))^(1/2),(-c/d+1)/(-c/d+a/b),((c-d)/(c+d))^(1/2))+(a^2*d^2- 
2*a*b*c*d+b^2*c^2)/b^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)*Ell 
ipticE(((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))*co 
s(f*x+e)^2)^(1/2)/(-c/d+a/b)*EllipticPi(((c+d*sin(f*x+e))/(c-d))^(1/2),...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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