\(\int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx\) [745]

Optimal result

Integrand size = 27, antiderivative size = 496 \[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d^2 f}+\frac {2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac {2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{315 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{315 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:

-2/315*(189*a^2*b*c*d^2+105*a^3*d^3-9*a*b^2*d*(6*c^2-25*d^2)+b^3*(8*c^3+39 
*c*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f+2/315*b*(54*a*b*c*d-189*a 
^2*d^2-b^2*(8*c^2+49*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f+8/63*b^ 
2*(-5*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d^2/f-2/9*b^2*cos(f*x+e)* 
(a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2)/d/f-2/315*(420*a^3*c*d^3+189*a^2*b 
*d^2*(c^2+3*d^2)-a*b^2*(54*c^3*d-738*c*d^3)+b^3*(8*c^4+33*c^2*d^2+147*d^4) 
)*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/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-2/315*(c^2-d^2)*(189*a^2* 
b*c*d^2+105*a^3*d^3-9*a*b^2*d*(6*c^2-25*d^2)+b^3*(8*c^3+39*c*d^2))*Inverse 
JacobiAM(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 = 6.12 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.83 \[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\frac {-8 \left (d^2 \left (756 a^2 b c d^2+105 a^3 d \left (3 c^2+d^2\right )+9 a b^2 d \left (51 c^2+25 d^2\right )+2 b^3 \left (c^3+93 c d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )+a b^2 \left (-54 c^3 d+738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (-2 \left (1512 a^2 b c d^2+420 a^3 d^3+9 a b^2 d \left (12 c^2+115 d^2\right )+b^3 \left (-16 c^3+402 c d^2\right )\right ) \cos (e+f x)+b d \left (10 b d (10 b c+27 a d) \cos (3 (e+f x))-2 \left (432 a b c d+378 a^2 d^2+b^2 \left (6 c^2+133 d^2\right )-35 b^2 d^2 \cos (2 (e+f x))\right ) \sin (2 (e+f x))\right )\right )}{1260 d^3 f \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

(-8*(d^2*(756*a^2*b*c*d^2 + 105*a^3*d*(3*c^2 + d^2) + 9*a*b^2*d*(51*c^2 + 
25*d^2) + 2*b^3*(c^3 + 93*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/( 
c + d)] + (420*a^3*c*d^3 + 189*a^2*b*d^2*(c^2 + 3*d^2) + a*b^2*(-54*c^3*d 
+ 738*c*d^3) + b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))*((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)] + d*(c + d*Sin[e + f*x])* 
(-2*(1512*a^2*b*c*d^2 + 420*a^3*d^3 + 9*a*b^2*d*(12*c^2 + 115*d^2) + b^3*( 
-16*c^3 + 402*c*d^2))*Cos[e + f*x] + b*d*(10*b*d*(10*b*c + 27*a*d)*Cos[3*( 
e + f*x)] - 2*(432*a*b*c*d + 378*a^2*d^2 + b^2*(6*c^2 + 133*d^2) - 35*b^2* 
d^2*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])))/(1260*d^3*f*Sqrt[c + d*Sin[e + f 
*x]])
 

Rubi [A] (verified)

Time = 2.63 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.03, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3272, 27, 3042, 3502, 27, 3042, 3232, 27, 3042, 3232, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3272

\(\displaystyle \frac {2 \int \frac {1}{2} (c+d \sin (e+f x))^{3/2} \left (9 d a^3+5 b^2 d a-4 b^2 (b c-5 a d) \sin ^2(e+f x)+2 b^3 c-b \left (-27 d a^2+2 b c a-7 b^2 d\right ) \sin (e+f x)\right )dx}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (c+d \sin (e+f x))^{3/2} \left (9 d a^3+5 b^2 d a-4 b^2 (b c-5 a d) \sin ^2(e+f x)+2 b^3 c-b \left (-27 d a^2+2 b c a-7 b^2 d\right ) \sin (e+f x)\right )dx}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (c+d \sin (e+f x))^{3/2} \left (9 d a^3+5 b^2 d a-4 b^2 (b c-5 a d) \sin (e+f x)^2+2 b^3 c-b \left (-27 d a^2+2 b c a-7 b^2 d\right ) \sin (e+f x)\right )dx}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {2 \int -\frac {1}{2} (c+d \sin (e+f x))^{3/2} \left (3 d \left (-21 d a^3-45 b^2 d a+2 b^3 c\right )+b \left (-\left (\left (8 c^2+49 d^2\right ) b^2\right )+54 a c d b-189 a^2 d^2\right ) \sin (e+f x)\right )dx}{7 d}+\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\int (c+d \sin (e+f x))^{3/2} \left (3 d \left (-21 d a^3-45 b^2 d a+2 b^3 c\right )+b \left (-\left (\left (8 c^2+49 d^2\right ) b^2\right )+54 a c d b-189 a^2 d^2\right ) \sin (e+f x)\right )dx}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\int (c+d \sin (e+f x))^{3/2} \left (3 d \left (-21 d a^3-45 b^2 d a+2 b^3 c\right )+b \left (-\left (\left (8 c^2+49 d^2\right ) b^2\right )+54 a c d b-189 a^2 d^2\right ) \sin (e+f x)\right )dx}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {\frac {2}{5} \int -\frac {3}{2} \sqrt {c+d \sin (e+f x)} \left (d \left (105 c d a^3+189 b d^2 a^2+171 b^2 c d a-b^3 \left (2 c^2-49 d^2\right )\right )+\left (\left (8 c^3+39 d^2 c\right ) b^3-9 a d \left (6 c^2-25 d^2\right ) b^2+189 a^2 c d^2 b+105 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (105 c d a^3+189 b d^2 a^2+171 b^2 c d a-b^3 \left (2 c^2-49 d^2\right )\right )+\left (\left (8 c^3+39 d^2 c\right ) b^3-9 a d \left (6 c^2-25 d^2\right ) b^2+189 a^2 c d^2 b+105 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (105 c d a^3+189 b d^2 a^2+171 b^2 c d a-b^3 \left (2 c^2-49 d^2\right )\right )+\left (\left (8 c^3+39 d^2 c\right ) b^3-9 a d \left (6 c^2-25 d^2\right ) b^2+189 a^2 c d^2 b+105 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {2}{3} \int \frac {d \left (105 d \left (3 c^2+d^2\right ) a^3+756 b c d^2 a^2+9 b^2 d \left (51 c^2+25 d^2\right ) a+2 b^3 \left (c^3+93 d^2 c\right )\right )+\left (\left (8 c^4+33 d^2 c^2+147 d^4\right ) b^3-a \left (54 c^3 d-738 c d^3\right ) b^2+189 a^2 d^2 \left (c^2+3 d^2\right ) b+420 a^3 c d^3\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \int \frac {d \left (105 d \left (3 c^2+d^2\right ) a^3+756 b c d^2 a^2+9 b^2 d \left (51 c^2+25 d^2\right ) a+2 b^3 \left (c^3+93 d^2 c\right )\right )+\left (\left (8 c^4+33 d^2 c^2+147 d^4\right ) b^3-a \left (54 c^3 d-738 c d^3\right ) b^2+189 a^2 d^2 \left (c^2+3 d^2\right ) b+420 a^3 c d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \int \frac {d \left (105 d \left (3 c^2+d^2\right ) a^3+756 b c d^2 a^2+9 b^2 d \left (51 c^2+25 d^2\right ) a+2 b^3 \left (c^3+93 d^2 c\right )\right )+\left (\left (8 c^4+33 d^2 c^2+147 d^4\right ) b^3-a \left (54 c^3 d-738 c d^3\right ) b^2+189 a^2 d^2 \left (c^2+3 d^2\right ) b+420 a^3 c d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}-\frac {-\frac {2 b \left (-189 a^2 d^2+54 a b c d-\left (b^2 \left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (105 a^3 d^3+189 a^2 b c d^2-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )}{7 d}}{9 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}\)

Input:

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

Output:

(-2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2))/(9*d 
*f) + ((8*b^2*(b*c - 5*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d* 
f) - ((-2*b*(54*a*b*c*d - 189*a^2*d^2 - b^2*(8*c^2 + 49*d^2))*Cos[e + f*x] 
*(c + d*Sin[e + f*x])^(3/2))/(5*f) - (3*((-2*(189*a^2*b*c*d^2 + 105*a^3*d^ 
3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*Cos[e + f*x]*Sqrt 
[c + d*Sin[e + f*x]])/(3*f) + ((2*(420*a^3*c*d^3 + 189*a^2*b*d^2*(c^2 + 3* 
d^2) - a*b^2*(54*c^3*d - 738*c*d^3) + b^3*(8*c^4 + 33*c^2*d^2 + 147*d^4))* 
EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d* 
f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(189*a^2*b*c*d^2 + 
105*a^3*d^3 - 9*a*b^2*d*(6*c^2 - 25*d^2) + b^3*(8*c^3 + 39*c*d^2))*Ellipti 
cF[(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))/5)/(7*d))/(9*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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

rule 3134

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

rule 3140

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

rule 3142

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

rule 3231

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

rule 3232

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

rule 3272

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

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. \(2089\) vs. \(2(473)=946\).

Time = 10.60 (sec) , antiderivative size = 2090, normalized size of antiderivative = 4.21

method	result	size
default	\(\text {Expression too large to display}\)	\(2090\)
parts	\(\text {Expression too large to display}\)	\(3590\)

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)*(2*a^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))+b^2*d*(3*a*d+2*b*c)*(-2/7/d*sin(f*x+e)^2* 
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+12/35*c/d^2*sin(f*x+e)*(-(-c-d*sin 
(f*x+e))*cos(f*x+e)^2)^(1/2)-2/3*(5/7+24/35*c^2/d^2)/d*(-(-c-d*sin(f*x+e)) 
*cos(f*x+e)^2)^(1/2)+2*(-4/35*c^2/d^2+5/21)*(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))+2/105*(-48*c^3-44*c*d^2)/d^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-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))))+2*a^2*c*(2*a*d+3*b*c)*(c/d-1)*((c+d*s 
in(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*(3*a^2*d^2+6*a*b*c*d+b^2*c^2)*(- 
2/5/d*sin(f*x+e)*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+8/15*c/d^2*(-(-c- 
d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+4/15*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.75 \[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/945*((16*b^3*c^5 - 108*a*b^2*c^4*d + 6*(63*a^2*b + 10*b^3)*c^3*d^2 - 3* 
(35*a^3 - 33*a*b^2)*c^2*d^3 - 6*(189*a^2*b + 44*b^3)*c*d^4 - 45*(7*a^3 + 1 
5*a*b^2)*d^5)*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 - 108*a*b^2*c^4*d + 6*(63*a^2*b + 10*b^3)*c^3* 
d^2 - 3*(35*a^3 - 33*a*b^2)*c^2*d^3 - 6*(189*a^2*b + 44*b^3)*c*d^4 - 45*(7 
*a^3 + 15*a*b^2)*d^5)*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*s 
in(f*x + e) + 2*I*c)/d) + 3*(8*I*b^3*c^4*d - 54*I*a*b^2*c^3*d^2 + 3*I*(63* 
a^2*b + 11*b^3)*c^2*d^3 + 6*I*(70*a^3 + 123*a*b^2)*c*d^4 + 21*I*(27*a^2*b 
+ 7*b^3)*d^5)*sqrt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/2 
7*(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 + 54*I*a*b^2*c^3*d^2 - 3*I*(63*a^2*b + 
 11*b^3)*c^2*d^3 - 6*I*(70*a^3 + 123*a*b^2)*c*d^4 - 21*I*(27*a^2*b + 7*b^3 
)*d^5)*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, -8/2 
7*(-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*(5*(10*b^3*c*d^4 + 27*a*b^2*d^5)*cos(f*x + e)^3 + (4*b^3*c 
^3*d^2 - 27*a*b^2*c^2*d^3 - 6*(63*a^2*b + 23*b^3)*c*d^4 - 15*(7*a^3 + 2...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

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