Integrand size = 27, antiderivative size = 642 \[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3465 d^3 f \sqrt {c+d \sin (e+f x)}} \] Output:
-2/3465*(1848*a^3*c*d^3+495*a^2*b*d^2*(3*c^2+5*d^2)-66*a*b^2*d*(5*c^3-57*c *d^2)+5*b^3*(8*c^4+57*c^2*d^2+135*d^4))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/ d^2/f-2/3465*(1485*a^2*b*c*d^2+693*a^3*d^3-33*a*b^2*d*(10*c^2-49*d^2)+5*b^ 3*(8*c^3+67*c*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f+2/693*b*(66*a* b*c*d-297*a^2*d^2-b^2*(8*c^2+81*d^2))*cos(f*x+e)*(c+d*sin(f*x+e))^(5/2)/d^ 2/f+8/99*b^2*(-6*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(7/2)/d^2/f-2/11*b^2 *cos(f*x+e)*(a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(7/2)/d/f-2/3465*(231*a^3*d^ 3*(23*c^2+9*d^2)+495*a^2*b*c*d^2*(3*c^2+29*d^2)-33*a*b^2*d*(10*c^4-279*c^2 *d^2-147*d^4)+5*b^3*(8*c^5+51*c^3*d^2+741*c*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*si n(f*x+e))/(c+d))^(1/2)-2/3465*(c^2-d^2)*(1848*a^3*c*d^3+495*a^2*b*d^2*(3*c ^2+5*d^2)-66*a*b^2*d*(5*c^3-57*c*d^2)+5*b^3*(8*c^4+57*c^2*d^2+135*d^4))*In verseJacobiAM(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 = 6.45 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.85 \[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\frac {-16 \left (d^2 \left (495 a^2 b d^2 \left (27 c^2+5 d^2\right )+231 a^3 c d \left (15 c^2+17 d^2\right )+33 a b^2 c d \left (155 c^2+261 d^2\right )+5 b^3 \left (2 c^4+663 c^2 d^2+135 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )+33 a b^2 d \left (-10 c^4+279 c^2 d^2+147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (2 \left (-20328 a^3 c d^3-990 a^2 b d^2 \left (36 c^2+23 d^2\right )-66 a b^2 c d \left (20 c^2+747 d^2\right )+5 b^3 \left (32 c^4-1866 c^2 d^2-1305 d^4\right )\right ) \cos (e+f x)+5 b d^2 \left (2508 a b c d+1188 a^2 d^2+b^2 \left (452 c^2+513 d^2\right )\right ) \cos (3 (e+f x))-315 b^3 d^4 \cos (5 (e+f x))-4 d \left (8910 a^2 b c d^2+1386 a^3 d^3+33 a b^2 d \left (150 c^2+133 d^2\right )+5 b^3 \left (6 c^3+619 c d^2\right )\right ) \sin (2 (e+f x))+70 b^2 d^3 (23 b c+33 a d) \sin (4 (e+f x))\right )}{27720 d^3 f \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(-16*(d^2*(495*a^2*b*d^2*(27*c^2 + 5*d^2) + 231*a^3*c*d*(15*c^2 + 17*d^2) + 33*a*b^2*c*d*(155*c^2 + 261*d^2) + 5*b^3*(2*c^4 + 663*c^2*d^2 + 135*d^4) )*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (231*a^3*d^3*(23*c^2 + 9*d^2) + 495*a^2*b*c*d^2*(3*c^2 + 29*d^2) + 33*a*b^2*d*(-10*c^4 + 279*c^2 *d^2 + 147*d^4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4))*((c + d)*Ellipti cE[(-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*(-20328*a^3*c*d^3 - 990*a^2*b*d^2*(36*c^2 + 23*d^2) - 66*a*b^2*c *d*(20*c^2 + 747*d^2) + 5*b^3*(32*c^4 - 1866*c^2*d^2 - 1305*d^4))*Cos[e + f*x] + 5*b*d^2*(2508*a*b*c*d + 1188*a^2*d^2 + b^2*(452*c^2 + 513*d^2))*Cos [3*(e + f*x)] - 315*b^3*d^4*Cos[5*(e + f*x)] - 4*d*(8910*a^2*b*c*d^2 + 138 6*a^3*d^3 + 33*a*b^2*d*(150*c^2 + 133*d^2) + 5*b^3*(6*c^3 + 619*c*d^2))*Si n[2*(e + f*x)] + 70*b^2*d^3*(23*b*c + 33*a*d)*Sin[4*(e + f*x)]))/(27720*d^ 3*f*Sqrt[c + d*Sin[e + f*x]])
Time = 3.50 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.03, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 3272, 27, 3042, 3502, 27, 3042, 3232, 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.
| \(\displaystyle \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3272 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (c+d \sin (e+f x))^{5/2} \left (11 d a^3+7 b^2 d a-4 b^2 (b c-6 a d) \sin ^2(e+f x)+2 b^3 c-b \left (-33 d a^2+2 b c a-9 b^2 d\right ) \sin (e+f x)\right )dx}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^{5/2} \left (11 d a^3+7 b^2 d a-4 b^2 (b c-6 a d) \sin ^2(e+f x)+2 b^3 c-b \left (-33 d a^2+2 b c a-9 b^2 d\right ) \sin (e+f x)\right )dx}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^{5/2} \left (11 d a^3+7 b^2 d a-4 b^2 (b c-6 a d) \sin (e+f x)^2+2 b^3 c-b \left (-33 d a^2+2 b c a-9 b^2 d\right ) \sin (e+f x)\right )dx}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int -\frac {1}{2} (c+d \sin (e+f x))^{5/2} \left (d \left (-99 d a^3-231 b^2 d a+10 b^3 c\right )+b \left (-\left (\left (8 c^2+81 d^2\right ) b^2\right )+66 a c d b-297 a^2 d^2\right ) \sin (e+f x)\right )dx}{9 d}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\int (c+d \sin (e+f x))^{5/2} \left (d \left (-99 d a^3-231 b^2 d a+10 b^3 c\right )+b \left (-\left (\left (8 c^2+81 d^2\right ) b^2\right )+66 a c d b-297 a^2 d^2\right ) \sin (e+f x)\right )dx}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\int (c+d \sin (e+f x))^{5/2} \left (d \left (-99 d a^3-231 b^2 d a+10 b^3 c\right )+b \left (-\left (\left (8 c^2+81 d^2\right ) b^2\right )+66 a c d b-297 a^2 d^2\right ) \sin (e+f x)\right )dx}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {2}{7} \int -\frac {1}{2} (c+d \sin (e+f x))^{3/2} \left (3 d \left (231 c d a^3+495 b d^2 a^2+429 b^2 c d a-5 b^3 \left (2 c^2-27 d^2\right )\right )+\left (5 \left (8 c^3+67 d^2 c\right ) b^3-33 a d \left (10 c^2-49 d^2\right ) b^2+1485 a^2 c d^2 b+693 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {-\frac {1}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 d \left (231 c d a^3+495 b d^2 a^2+429 b^2 c d a-5 b^3 \left (2 c^2-27 d^2\right )\right )+\left (5 \left (8 c^3+67 d^2 c\right ) b^3-33 a d \left (10 c^2-49 d^2\right ) b^2+1485 a^2 c d^2 b+693 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {-\frac {1}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 d \left (231 c d a^3+495 b d^2 a^2+429 b^2 c d a-5 b^3 \left (2 c^2-27 d^2\right )\right )+\left (5 \left (8 c^3+67 d^2 c\right ) b^3-33 a d \left (10 c^2-49 d^2\right ) b^2+1485 a^2 c d^2 b+693 a^3 d^3\right ) \sin (e+f x)\right )dx-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {2}{5} \int \frac {3}{2} \sqrt {c+d \sin (e+f x)} \left (d \left (231 d \left (5 c^2+3 d^2\right ) a^3+3960 b c d^2 a^2+33 b^2 d \left (55 c^2+49 d^2\right ) a-10 b^3 \left (c^3-101 c d^2\right )\right )+\left (5 \left (8 c^4+57 d^2 c^2+135 d^4\right ) b^3-66 a c d \left (5 c^2-57 d^2\right ) b^2+495 a^2 d^2 \left (3 c^2+5 d^2\right ) b+1848 a^3 c d^3\right ) \sin (e+f x)\right )dx\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (231 d \left (5 c^2+3 d^2\right ) a^3+3960 b c d^2 a^2+33 b^2 d \left (55 c^2+49 d^2\right ) a-10 b^3 \left (c^3-101 c d^2\right )\right )+\left (5 \left (8 c^4+57 d^2 c^2+135 d^4\right ) b^3-66 a c d \left (5 c^2-57 d^2\right ) b^2+495 a^2 d^2 \left (3 c^2+5 d^2\right ) b+1848 a^3 c d^3\right ) \sin (e+f x)\right )dx\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (231 d \left (5 c^2+3 d^2\right ) a^3+3960 b c d^2 a^2+33 b^2 d \left (55 c^2+49 d^2\right ) a-10 b^3 \left (c^3-101 c d^2\right )\right )+\left (5 \left (8 c^4+57 d^2 c^2+135 d^4\right ) b^3-66 a c d \left (5 c^2-57 d^2\right ) b^2+495 a^2 d^2 \left (3 c^2+5 d^2\right ) b+1848 a^3 c d^3\right ) \sin (e+f x)\right )dx\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {2}{3} \int \frac {d \left (231 c d \left (15 c^2+17 d^2\right ) a^3+495 b d^2 \left (27 c^2+5 d^2\right ) a^2+33 b^2 c d \left (155 c^2+261 d^2\right ) a+5 b^3 \left (2 c^4+663 d^2 c^2+135 d^4\right )\right )+\left (5 \left (8 c^5+51 d^2 c^3+741 d^4 c\right ) b^3-33 a d \left (10 c^4-279 d^2 c^2-147 d^4\right ) b^2+495 a^2 c d^2 \left (3 c^2+29 d^2\right ) b+231 a^3 d^3 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \int \frac {d \left (231 c d \left (15 c^2+17 d^2\right ) a^3+495 b d^2 \left (27 c^2+5 d^2\right ) a^2+33 b^2 c d \left (155 c^2+261 d^2\right ) a+5 b^3 \left (2 c^4+663 d^2 c^2+135 d^4\right )\right )+\left (5 \left (8 c^5+51 d^2 c^3+741 d^4 c\right ) b^3-33 a d \left (10 c^4-279 d^2 c^2-147 d^4\right ) b^2+495 a^2 c d^2 \left (3 c^2+29 d^2\right ) b+231 a^3 d^3 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \int \frac {d \left (231 c d \left (15 c^2+17 d^2\right ) a^3+495 b d^2 \left (27 c^2+5 d^2\right ) a^2+33 b^2 c d \left (155 c^2+261 d^2\right ) a+5 b^3 \left (2 c^4+663 d^2 c^2+135 d^4\right )\right )+\left (5 \left (8 c^5+51 d^2 c^3+741 d^4 c\right ) b^3-33 a d \left (10 c^4-279 d^2 c^2-147 d^4\right ) b^2+495 a^2 c d^2 \left (3 c^2+29 d^2\right ) b+231 a^3 d^3 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\frac {1}{7} \left (\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 c d \left (5 c^2-57 d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )\right )-\frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}}{11 d}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}\) |
Input:
Int[(a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(-2*b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^(7/2))/(11* d*f) + ((8*b^2*(b*c - 6*a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d *f) - ((-2*b*(66*a*b*c*d - 297*a^2*d^2 - b^2*(8*c^2 + 81*d^2))*Cos[e + f*x ]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + ((2*(1485*a^2*b*c*d^2 + 693*a^3*d^3 - 33*a*b^2*d*(10*c^2 - 49*d^2) + 5*b^3*(8*c^3 + 67*c*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) - (3*((-2*(1848*a^3*c*d^3 - 66*a*b^2*c*d*( 5*c^2 - 57*d^2) + 495*a^2*b*d^2*(3*c^2 + 5*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^ 2 + 135*d^4))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((2*(231*a^3* d^3*(23*c^2 + 9*d^2) + 495*a^2*b*c*d^2*(3*c^2 + 29*d^2) - 33*a*b^2*d*(10*c ^4 - 279*c^2*d^2 - 147*d^4) + 5*b^3*(8*c^5 + 51*c^3*d^2 + 741*c*d^4))*Elli pticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sq rt[(c + d*Sin[e + f*x])/(c + d)]) - (2*(c^2 - d^2)*(1848*a^3*c*d^3 - 66*a* b^2*c*d*(5*c^2 - 57*d^2) + 495*a^2*b*d^2*(3*c^2 + 5*d^2) + 5*b^3*(8*c^4 + 57*c^2*d^2 + 135*d^4))*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))/5)/7)/(9 *d))/(11*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. \(2694\) vs. \(2(615)=1230\).
Time = 83.65 (sec) , antiderivative size = 2695, normalized size of antiderivative = 4.20
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2695\) |
| parts | \(\text {Expression too large to display}\) | \(4390\) |
Input:
int((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(2*a^3*c^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)*EllipticF(((c+d*sin(f*x+e))/( c-d))^(1/2),((c-d)/(c+d))^(1/2))+3*d^2*b^2*(a*d+b*c)*(-2/9/d*sin(f*x+e)^3* (-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+16/63*c/d^2*sin(f*x+e)^2*(-(-c-d*s in(f*x+e))*cos(f*x+e)^2)^(1/2)-2/5*(7/9+16/21*c^2/d^2)/d*sin(f*x+e)*(-(-c- d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)-2/315*(-64*c^3-62*c*d^2)/d^4*(-(-c-d*sin (f*x+e))*cos(f*x+e)^2)^(1/2)+2/315*(32*c^3+36*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)*EllipticF(((c+d*sin(f*x +e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/315*(128*c^4+108*c^2*d^2+147*d^4) /d^4*(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))+Ell ipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+6*a^2*c^2*(a* d+b*c)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/ 2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2) *((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+E llipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))+3*b*d*(a^2*d ^2+3*a*b*c*d+b^2*c^2)*(-2/7/d*sin(f*x+e)^2*(-(-c-d*sin(f*x+e))*cos(f*x+...
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1039, normalized size of antiderivative = 1.62 \[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
-2/10395*((80*b^3*c^6 - 660*a*b^2*c^5*d + 30*(99*a^2*b + 16*b^3)*c^4*d^2 + 33*(7*a^3 + 93*a*b^2)*c^3*d^3 - 15*(759*a^2*b + 169*b^3)*c^2*d^4 - 99*(77 *a^3 + 163*a*b^2)*c*d^5 - 675*(11*a^2*b + 3*b^3)*d^6)*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) + (80*b^3*c^6 - 660* a*b^2*c^5*d + 30*(99*a^2*b + 16*b^3)*c^4*d^2 + 33*(7*a^3 + 93*a*b^2)*c^3*d ^3 - 15*(759*a^2*b + 169*b^3)*c^2*d^4 - 99*(77*a^3 + 163*a*b^2)*c*d^5 - 67 5*(11*a^2*b + 3*b^3)*d^6)*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) + 3*(40*I*b^3*c^5*d - 330*I*a*b^2*c^4*d^2 + 15 *I*(99*a^2*b + 17*b^3)*c^3*d^3 + 33*I*(161*a^3 + 279*a*b^2)*c^2*d^4 + 15*I *(957*a^2*b + 247*b^3)*c*d^5 + 693*I*(3*a^3 + 7*a*b^2)*d^6)*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*(-40*I*b ^3*c^5*d + 330*I*a*b^2*c^4*d^2 - 15*I*(99*a^2*b + 17*b^3)*c^3*d^3 - 33*I*( 161*a^3 + 279*a*b^2)*c^2*d^4 - 15*I*(957*a^2*b + 247*b^3)*c*d^5 - 693*I*(3 *a^3 + 7*a*b^2)*d^6)*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...
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((a+b*sin(f*x+e))**3*(c+d*sin(f*x+e))**(5/2),x)
Output:
Integral((a + b*sin(e + f*x))**3*(c + d*sin(e + f*x))**(5/2), x)
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(5/2), x)
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
integrate((b*sin(f*x + e) + a)^3*(d*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(5/2),x)
Output:
int((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^(5/2), x)
\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx=\left (\int \sqrt {\sin \left (f x +e \right ) d +c}d x \right ) a^{3} c^{2}+\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{5}d x \right ) b^{3} d^{2}+3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}d x \right ) a \,b^{2} d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}d x \right ) b^{3} c d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) a^{2} b \,d^{2}+6 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) a \,b^{2} c d +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) b^{3} c^{2}+\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) a^{3} d^{2}+6 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) a^{2} b c d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) a \,b^{2} c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) a^{3} c d +3 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) a^{2} b \,c^{2} \] Input:
int((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c),x)*a**3*c**2 + int(sqrt(sin(e + f*x)*d + c)*s in(e + f*x)**5,x)*b**3*d**2 + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)* *4,x)*a*b**2*d**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**4,x)*b**3 *c*d + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3,x)*a**2*b*d**2 + 6*i nt(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3,x)*a*b**2*c*d + int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3,x)*b**3*c**2 + int(sqrt(sin(e + f*x)*d + c) *sin(e + f*x)**2,x)*a**3*d**2 + 6*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x )**2,x)*a**2*b*c*d + 3*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*a*b **2*c**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x),x)*a**3*c*d + 3*int (sqrt(sin(e + f*x)*d + c)*sin(e + f*x),x)*a**2*b*c**2