\(\int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx\) [748]

Optimal result

Integrand size = 27, antiderivative size = 361 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b \left (6 a b c d-3 a^2 d^2-b^2 \left (4 c^2-d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d^2 \left (c^2-d^2\right ) f}-\frac {2 \left (9 a^2 b c d^2-3 a^3 d^3-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 \left (c^2-d^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 b \left (18 a b c d-9 a^2 d^2-b^2 \left (8 c^2+d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:

2*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^ 
(1/2)+2/3*b*(6*a*b*c*d-3*a^2*d^2-b^2*(4*c^2-d^2))*cos(f*x+e)*(c+d*sin(f*x+ 
e))^(1/2)/d^2/(c^2-d^2)/f+2/3*(9*a^2*b*c*d^2-3*a^3*d^3-9*a*b^2*d*(2*c^2-d^ 
2)+b^3*(8*c^3-5*c*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)/d^3/(c^2-d^2)/f/((c+d*sin(f*x+e))/(c+d)) 
^(1/2)-2/3*b*(18*a*b*c*d-9*a^2*d^2-b^2*(8*c^2+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)/d^3 
/f/(c+d*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 3.71 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \left (\frac {\left (d^2 \left (-3 a^3 c d-9 a b^2 c d+9 a^2 b d^2+b^3 \left (2 c^2+d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (9 a^2 b c d^2-3 a^3 d^3+9 a b^2 d \left (-2 c^2+d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d) (c+d)}-\frac {d \cos (e+f x) \left (9 a b^2 c^2 d-9 a^2 b c d^2+3 a^3 d^3+b^3 \left (-4 c^3+c d^2\right )+b^3 d \left (-c^2+d^2\right ) \sin (e+f x)\right )}{-c^2+d^2}\right )}{3 d^3 f \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

(2*(((d^2*(-3*a^3*c*d - 9*a*b^2*c*d + 9*a^2*b*d^2 + b^3*(2*c^2 + d^2))*Ell 
ipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (9*a^2*b*c*d^2 - 3*a^3*d^3 
+ 9*a*b^2*d*(-2*c^2 + d^2) + b^3*(8*c^3 - 5*c*d^2))*((c + d)*EllipticE[(-2 
*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2 
*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)*(c + d)) - (d* 
Cos[e + f*x]*(9*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 3*a^3*d^3 + b^3*(-4*c^3 + c* 
d^2) + b^3*d*(-c^2 + d^2)*Sin[e + f*x]))/(-c^2 + d^2)))/(3*d^3*f*Sqrt[c + 
d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 1.87 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3271, 27, 3042, 3502, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3271

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\frac {2 \int -\frac {d \left (3 c d a^3-9 b d^2 a^2+9 b^2 c d a-b^3 \left (2 c^2+d^2\right )\right )-\left (\left (8 c^3-5 c d^2\right ) b^3-9 a d \left (2 c^2-d^2\right ) b^2+9 a^2 c d^2 b-3 a^3 d^3\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\int \frac {d \left (3 c d a^3-9 b d^2 a^2+9 b^2 c d a-b^3 \left (2 c^2+d^2\right )\right )-\left (\left (8 c^3-5 c d^2\right ) b^3-9 a d \left (2 c^2-d^2\right ) b^2+9 a^2 c d^2 b-3 a^3 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\int \frac {d \left (3 c d a^3-9 b d^2 a^2+9 b^2 c d a-b^3 \left (2 c^2+d^2\right )\right )-\left (\left (8 c^3-5 c d^2\right ) b^3-9 a d \left (2 c^2-d^2\right ) b^2+9 a^2 c d^2 b-3 a^3 d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {-\frac {b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {-\frac {b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {-\frac {b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {-\frac {b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {-\frac {b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 \left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {-\frac {b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {-\frac {b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}}{d \left (c^2-d^2\right )}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {2 b \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (4 c^2-d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}-\frac {-\frac {2 b \left (c^2-d^2\right ) \left (-9 a^2 d^2+18 a b c d-\left (b^2 \left (8 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-3 a^3 d^3+9 a^2 b c d^2-9 a b^2 d \left (2 c^2-d^2\right )+b^3 \left (8 c^3-5 c d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{3 d}}{d \left (c^2-d^2\right )}\)

Input:

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

Output:

(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(d*(c^2 - d^2)*f*Sqrt[ 
c + d*Sin[e + f*x]]) - ((-2*b*(6*a*b*c*d - 3*a^2*d^2 - b^2*(4*c^2 - d^2))* 
Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) - ((-2*(9*a^2*b*c*d^2 - 3*a 
^3*d^3 - 9*a*b^2*d*(2*c^2 - d^2) + b^3*(8*c^3 - 5*c*d^2))*EllipticE[(e - P 
i/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Si 
n[e + f*x])/(c + d)]) - (2*b*(c^2 - d^2)*(18*a*b*c*d - 9*a^2*d^2 - b^2*(8* 
c^2 + d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e 
 + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(3*d))/(d*(c^2 - d^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 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 3271

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co 
s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* 
(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin 
[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 
2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 
+ b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - 
 d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*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] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || 
IntegersQ[2*m, 2*n])
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1385\) vs. \(2(348)=696\).

Time = 7.76 (sec) , antiderivative size = 1386, normalized size of antiderivative = 3.84

method	result	size
default	\(\text {Expression too large to display}\)	\(1386\)
parts	\(\text {Expression too large to display}\)	\(2576\)

Input:

int((a+b*sin(f*x+e))^3/(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)*(b/d^3*(2*b^2*c^2*(c/d-1)*((c+d*si 
n(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*b*d*(3*a*d-b*c)*(c/d-1)*((c+d*si 
n(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*b^2*(-2/3/d*(-(-c-d*sin(f*x+e)) 
*cos(f*x+e)^2)^(1/2)+2/3*(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))*c 
os(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^ 
(1/2))-4/3*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c 
+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^ 
2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^ 
(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+6*a 
^2*d^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/ 
2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2) 
*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))-6*a*b*c*d*( 
c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*( 
1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*Elli...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 1034, normalized size of antiderivative = 2.86 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/9*((16*b^3*c^5 - 36*a*b^2*c^4*d + 2*(9*a^2*b - 8*b^3)*c^3*d^2 + 3*(a^3 + 
 15*a*b^2)*c^2*d^3 - 3*(9*a^2*b + b^3)*c*d^4 + (16*b^3*c^4*d - 36*a*b^2*c^ 
3*d^2 + 2*(9*a^2*b - 8*b^3)*c^2*d^3 + 3*(a^3 + 15*a*b^2)*c*d^4 - 3*(9*a^2* 
b + b^3)*d^5)*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 
- 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I 
*d*sin(f*x + e) - 2*I*c)/d) + (16*b^3*c^5 - 36*a*b^2*c^4*d + 2*(9*a^2*b - 
8*b^3)*c^3*d^2 + 3*(a^3 + 15*a*b^2)*c^2*d^3 - 3*(9*a^2*b + b^3)*c*d^4 + (1 
6*b^3*c^4*d - 36*a*b^2*c^3*d^2 + 2*(9*a^2*b - 8*b^3)*c^2*d^3 + 3*(a^3 + 15 
*a*b^2)*c*d^4 - 3*(9*a^2*b + b^3)*d^5)*sin(f*x + e))*sqrt(-1/2*I*d)*weiers 
trassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 
1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(8*I*b^3*c^4*d 
- 18*I*a*b^2*c^3*d^2 + I*(9*a^2*b - 5*b^3)*c^2*d^3 - 3*I*(a^3 - 3*a*b^2)*c 
*d^4 + (8*I*b^3*c^3*d^2 - 18*I*a*b^2*c^2*d^3 + I*(9*a^2*b - 5*b^3)*c*d^4 - 
 3*I*(a^3 - 3*a*b^2)*d^5)*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassZeta(-4/3 
*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse 
(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f 
*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(-8*I*b^3*c^4*d + 18*I*a*b^2 
*c^3*d^2 - I*(9*a^2*b - 5*b^3)*c^2*d^3 + 3*I*(a^3 - 3*a*b^2)*c*d^4 + (-8*I 
*b^3*c^3*d^2 + 18*I*a*b^2*c^2*d^3 - I*(9*a^2*b - 5*b^3)*c*d^4 + 3*I*(a^3 - 
 3*a*b^2)*d^5)*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassZeta(-4/3*(4*c^2...
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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