Integrand size = 27, antiderivative size = 451 \[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-b^2 \left (5 c^3-57 c d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}-\frac {2 \left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b d \left (3 c^3+29 c d^2\right )-b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 \left (c^2-d^2\right ) \left (5 b^2 c^3-45 a b c^2 d-84 a^2 c d^2-57 b^2 c d^2-75 a b d^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{315 d^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
-4/315*(84*a^2*c*d^2+15*a*b*d*(3*c^2+5*d^2)-b^2*(5*c^3-57*c*d^2))*cos(f*x+ e)*(c+d*sin(f*x+e))^(1/2)/d/f-2/315*(7*(9*a^2+7*b^2)*d^2-10*b*c*(-9*a*d+b* c))*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d/f+4/63*b*(-9*a*d+b*c)*cos(f*x+e)*( c+d*sin(f*x+e))^(5/2)/d/f-2/9*b^2*cos(f*x+e)*(c+d*sin(f*x+e))^(7/2)/d/f-2/ 315*(21*a^2*d^2*(23*c^2+9*d^2)+30*a*b*d*(3*c^3+29*c*d^2)-b^2*(10*c^4-279*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^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+4/315*(c^2 -d^2)*(-84*a^2*c*d^2-45*a*b*c^2*d-75*a*b*d^3+5*b^2*c^3-57*b^2*c*d^2)*Inver seJacobiAM(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^2/f/(c+d*sin(f*x+e))^(1/2)
Time = 3.18 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.85 \[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\frac {8 \left (-d^2 \left (30 a b d \left (27 c^2+5 d^2\right )+b^2 c \left (155 c^2+261 d^2\right )+21 a^2 \left (15 c^3+17 c d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (-21 a^2 d^2 \left (23 c^2+9 d^2\right )-30 a b c d \left (3 c^2+29 d^2\right )+b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d (c+d \sin (e+f x)) \left (2 \left (924 a^2 c d^2+30 a b d \left (36 c^2+23 d^2\right )+b^2 \left (20 c^3+747 c d^2\right )\right ) \cos (e+f x)-10 b d^2 (19 b c+18 a d) \cos (3 (e+f x))+2 d \left (540 a b c d+126 a^2 d^2+b^2 \left (150 c^2+133 d^2\right )\right ) \sin (2 (e+f x))-35 b^2 d^3 \sin (4 (e+f x))\right )}{1260 d^2 f \sqrt {c+d \sin (e+f x)}} \] Input:
Integrate[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(8*(-(d^2*(30*a*b*d*(27*c^2 + 5*d^2) + b^2*c*(155*c^2 + 261*d^2) + 21*a^2* (15*c^3 + 17*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + (- 21*a^2*d^2*(23*c^2 + 9*d^2) - 30*a*b*c*d*(3*c^2 + 29*d^2) + b^2*(10*c^4 - 279*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*S in[e + f*x])/(c + d)] - d*(c + d*Sin[e + f*x])*(2*(924*a^2*c*d^2 + 30*a*b* d*(36*c^2 + 23*d^2) + b^2*(20*c^3 + 747*c*d^2))*Cos[e + f*x] - 10*b*d^2*(1 9*b*c + 18*a*d)*Cos[3*(e + f*x)] + 2*d*(540*a*b*c*d + 126*a^2*d^2 + b^2*(1 50*c^2 + 133*d^2))*Sin[2*(e + f*x)] - 35*b^2*d^3*Sin[4*(e + f*x)]))/(1260* d^2*f*Sqrt[c + d*Sin[e + f*x]])
Time = 2.50 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.02, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3270, 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.
| \(\displaystyle \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 3270 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (c+d \sin (e+f x))^{5/2} \left (\left (9 a^2+7 b^2\right ) d-2 b (b c-9 a d) \sin (e+f x)\right )dx}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^{5/2} \left (\left (9 a^2+7 b^2\right ) d-2 b (b c-9 a d) \sin (e+f x)\right )dx}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^{5/2} \left (\left (9 a^2+7 b^2\right ) d-2 b (b c-9 a d) \sin (e+f x)\right )dx}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {1}{2} (c+d \sin (e+f x))^{3/2} \left (3 d \left (21 c a^2+30 b d a+13 b^2 c\right )+\left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \sin (e+f x)\right )dx+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 d \left (21 c a^2+30 b d a+13 b^2 c\right )+\left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \sin (e+f x)\right )dx+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int (c+d \sin (e+f x))^{3/2} \left (3 d \left (21 c a^2+30 b d a+13 b^2 c\right )+\left (7 \left (9 a^2+7 b^2\right ) d^2-10 b c (b c-9 a d)\right ) \sin (e+f x)\right )dx+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {c+d \sin (e+f x)} \left (d \left (21 \left (5 c^2+3 d^2\right ) a^2+240 b c d a+b^2 \left (55 c^2+49 d^2\right )\right )+2 \left (-\left (\left (5 c^3-57 c d^2\right ) b^2\right )+15 a d \left (3 c^2+5 d^2\right ) b+84 a^2 c d^2\right ) \sin (e+f x)\right )dx-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (21 \left (5 c^2+3 d^2\right ) a^2+240 b c d a+b^2 \left (55 c^2+49 d^2\right )\right )+2 \left (-\left (\left (5 c^3-57 c d^2\right ) b^2\right )+15 a d \left (3 c^2+5 d^2\right ) b+84 a^2 c d^2\right ) \sin (e+f x)\right )dx-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (21 \left (5 c^2+3 d^2\right ) a^2+240 b c d a+b^2 \left (55 c^2+49 d^2\right )\right )+2 \left (-\left (\left (5 c^3-57 c d^2\right ) b^2\right )+15 a d \left (3 c^2+5 d^2\right ) b+84 a^2 c d^2\right ) \sin (e+f x)\right )dx-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {d \left (21 \left (15 c^3+17 d^2 c\right ) a^2+30 b d \left (27 c^2+5 d^2\right ) a+b^2 c \left (155 c^2+261 d^2\right )\right )+\left (-\left (\left (10 c^4-279 d^2 c^2-147 d^4\right ) b^2\right )+30 a c d \left (3 c^2+29 d^2\right ) b+21 a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {d \left (21 \left (15 c^3+17 d^2 c\right ) a^2+30 b d \left (27 c^2+5 d^2\right ) a+b^2 c \left (155 c^2+261 d^2\right )\right )+\left (-\left (\left (10 c^4-279 d^2 c^2-147 d^4\right ) b^2\right )+30 a c d \left (3 c^2+29 d^2\right ) b+21 a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \int \frac {d \left (21 \left (15 c^3+17 d^2 c\right ) a^2+30 b d \left (27 c^2+5 d^2\right ) a+b^2 c \left (155 c^2+261 d^2\right )\right )+\left (-\left (\left (10 c^4-279 d^2 c^2-147 d^4\right ) b^2\right )+30 a c d \left (3 c^2+29 d^2\right ) b+21 a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}+\frac {2 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}+\frac {2 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}+\frac {2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (21 a^2 d^2 \left (23 c^2+9 d^2\right )+30 a b c d \left (3 c^2+29 d^2\right )-\left (b^2 \left (10 c^4-279 c^2 d^2-147 d^4\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 \left (c^2-d^2\right ) \left (-84 a^2 c d^2-45 a b c^2 d-75 a b d^3+5 b^2 c^3-57 b^2 c d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\right )-\frac {4 \left (84 a^2 c d^2+15 a b d \left (3 c^2+5 d^2\right )-\left (b^2 \left (5 c^3-57 c d^2\right )\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )-\frac {2 \left (7 d^2 \left (9 a^2+7 b^2\right )-10 b c (b c-9 a d)\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}\right )+\frac {4 b (b c-9 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 f}}{9 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}\) |
Input:
Int[(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^(5/2),x]
Output:
(-2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(7/2))/(9*d*f) + ((4*b*(b*c - 9* a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*f) + ((-2*(7*(9*a^2 + 7*b ^2)*d^2 - 10*b*c*(b*c - 9*a*d))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/( 5*f) + (3*((-4*(84*a^2*c*d^2 + 15*a*b*d*(3*c^2 + 5*d^2) - b^2*(5*c^3 - 57* c*d^2))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f) + ((2*(21*a^2*d^2*(23 *c^2 + 9*d^2) + 30*a*b*c*d*(3*c^2 + 29*d^2) - b^2*(10*c^4 - 279*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)]) + (4*(c^2 - d^2)*(5*b^2*c^ 3 - 45*a*b*c^2*d - 84*a^2*c*d^2 - 57*b^2*c*d^2 - 75*a*b*d^3)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*S qrt[c + d*Sin[e + f*x]]))/3))/5)/7)/(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_)])^2, x_Symbol] :> Simp[(-d^2)*Cos[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[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x ], x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && Ne Q[a^2 - b^2, 0] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2089\) vs. \(2(428)=856\).
Time = 20.03 (sec) , antiderivative size = 2090, normalized size of antiderivative = 4.63
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2090\) |
| parts | \(\text {Expression too large to display}\) | \(3030\) |
Input:
int((a+b*sin(f*x+e))^2*(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^2*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))+b*d^2*(2*a*d+3*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*c^2*(3*a*d+2*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)))+d*(a^2*d^2+6*a*b*c*d+3*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 = 0.15 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.72 \[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx =\text {Too large to display} \] Input:
integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
2/945*((20*b^2*c^5 - 180*a*b*c^4*d + 690*a*b*c^2*d^3 + 450*a*b*d^5 - 3*(7* a^2 + 31*b^2)*c^3*d^2 + 3*(231*a^2 + 163*b^2)*c*d^4)*sqrt(1/2*I*d)*weierst rassPInverse(-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) + (20*b^2*c^5 - 180*a *b*c^4*d + 690*a*b*c^2*d^3 + 450*a*b*d^5 - 3*(7*a^2 + 31*b^2)*c^3*d^2 + 3* (231*a^2 + 163*b^2)*c*d^4)*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*(-10*I*b^2*c^4*d + 90*I*a*b*c^3*d^2 + 870 *I*a*b*c*d^4 + 3*I*(161*a^2 + 93*b^2)*c^2*d^3 + 21*I*(9*a^2 + 7*b^2)*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, -8/27*(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*(10*I*b^2*c^4*d - 90*I*a*b*c^3*d^2 - 870*I*a*b*c*d^4 - 3*I*(161*a^2 + 93*b^2)*c^2*d^3 - 21*I*(9*a^2 + 7*b^2)*d^5)*sqrt(-1/2*I*d)*weierstrassZe ta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrass PInverse(-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*(5*(19*b^2*c*d^4 + 18*a*b*d^5)*cos(f*x + e)^3 - (5*b^2*c^3*d^2 + 270*a*b*c^2*d^3 + 240*a*b*d^ 5 + 3*(77*a^2 + 86*b^2)*c*d^4)*cos(f*x + e) + (35*b^2*d^5*cos(f*x + e)^3 - 3*(25*b^2*c^2*d^3 + 90*a*b*c*d^4 + 7*(3*a^2 + 4*b^2)*d^5)*cos(f*x + e)...
\[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{2} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((a+b*sin(f*x+e))**2*(c+d*sin(f*x+e))**(5/2),x)
Output:
Integral((a + b*sin(e + f*x))**2*(c + d*sin(e + f*x))**(5/2), x)
\[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(5/2), x)
\[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
integrate((b*sin(f*x + e) + a)^2*(d*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2),x)
Output:
int((a + b*sin(e + f*x))^2*(c + d*sin(e + f*x))^(5/2), x)
\[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx=\left (\int \sqrt {\sin \left (f x +e \right ) d +c}d x \right ) a^{2} c^{2}+\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{4}d x \right ) b^{2} d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) a b \,d^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}d x \right ) b^{2} c d +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) a^{2} d^{2}+4 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) a b c d +\left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}d x \right ) b^{2} c^{2}+2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) a^{2} c d +2 \left (\int \sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )d x \right ) a b \,c^{2} \] Input:
int((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c),x)*a**2*c**2 + int(sqrt(sin(e + f*x)*d + c)*s in(e + f*x)**4,x)*b**2*d**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)* *3,x)*a*b*d**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3,x)*b**2*c* d + int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*a**2*d**2 + 4*int(sqrt (sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*a*b*c*d + int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2,x)*b**2*c**2 + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x),x)*a**2*c*d + 2*int(sqrt(sin(e + f*x)*d + c)*sin(e + f*x),x)*a*b*c** 2