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)
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]])
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)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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]
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])))
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]
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...
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...
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \] 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)
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] 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)
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] 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)
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)
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx=\left (\int \sqrt {\sin \left (f x +e \right ) d +c}d x \right ) a^{3} c +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}d x \right ) b^{3} d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) a \,b^{2} d +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) b^{3} c +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) a^{2} b d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) a \,b^{2} c +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) a^{3} d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) a^{2} b c \] 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