Integrand size = 27, antiderivative size = 333 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=-\frac {2 \left (-6 b c (b c-21 d)+5 \left (63+5 b^2\right ) d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{105 d f}+\frac {4 b (b c-21 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d f}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}+\frac {4 \left (630 c d^2+63 b d \left (c^2+3 d^2\right )-b^2 \left (3 c^3-41 c d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{105 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (126 b c d+315 d^2-b^2 \left (6 c^2-25 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{105 d^2 f \sqrt {c+d \sin (e+f x)}} \]
4/35*b*(-7*a*d+b*c)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d/f-2/7*b^2*cos(f*x+ e)*(c+d*sin(f*x+e))^(5/2)/d/f-2/105*(5*(7*a^2+5*b^2)*d^2-6*b*c*(-7*a*d+b*c ))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d/f-4/105*(70*a^2*c*d^2+21*a*b*d*(c^2 +3*d^2)-b^2*(3*c^3-41*c*d^2))*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2* e+1/4*Pi+1/2*f*x)*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)+2/105*(c^ 2-d^2)*(42*a*b*c*d+35*a^2*d^2-b^2*(6*c^2-25*d^2))*(sin(1/2*e+1/4*Pi+1/2*f* x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(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))/(c+d))^(1/2)/d^2/f/(c+d*sin(f*x +e))^(1/2)
Time = 2.36 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.83 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\frac {4 \left (-d^2 \left (\left (945+51 b^2\right ) c^2+504 b c d+5 \left (63+5 b^2\right ) d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+2 \left (-630 c d^2-63 b d \left (c^2+3 d^2\right )+b^2 \left (3 c^3-41 c d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-d (c+d \sin (e+f x)) \left (\left (1008 b c d+1260 d^2+b^2 \left (12 c^2+115 d^2\right )\right ) \cos (e+f x)+3 b d (-5 b d \cos (3 (e+f x))+4 (4 b c+21 d) \sin (2 (e+f x)))\right )}{210 d^2 f \sqrt {c+d \sin (e+f x)}} \]
(4*(-(d^2*((945 + 51*b^2)*c^2 + 504*b*c*d + 5*(63 + 5*b^2)*d^2)*EllipticF[ (-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]) + 2*(-630*c*d^2 - 63*b*d*(c^2 + 3*d ^2) + b^2*(3*c^3 - 41*c*d^2))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2 *d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*Sqrt[(c + d*Sin[e + f*x])/(c + d)] - d*(c + d*Sin[e + f*x])*((1008*b*c*d + 1260*d ^2 + b^2*(12*c^2 + 115*d^2))*Cos[e + f*x] + 3*b*d*(-5*b*d*Cos[3*(e + f*x)] + 4*(4*b*c + 21*d)*Sin[2*(e + f*x)])))/(210*d^2*f*Sqrt[c + d*Sin[e + f*x] ])
Time = 1.70 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.07, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3270, 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.
| \(\displaystyle \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3270 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (c+d \sin (e+f x))^{3/2} \left (\left (7 a^2+5 b^2\right ) d-2 b (b c-7 a d) \sin (e+f x)\right )dx}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^{3/2} \left (\left (7 a^2+5 b^2\right ) d-2 b (b c-7 a d) \sin (e+f x)\right )dx}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (c+d \sin (e+f x))^{3/2} \left (\left (7 a^2+5 b^2\right ) d-2 b (b c-7 a d) \sin (e+f x)\right )dx}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {1}{2} \sqrt {c+d \sin (e+f x)} \left (d \left (35 c a^2+42 b d a+19 b^2 c\right )+\left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \sin (e+f x)\right )dx+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (35 c a^2+42 b d a+19 b^2 c\right )+\left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \sin (e+f x)\right )dx+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \sqrt {c+d \sin (e+f x)} \left (d \left (35 c a^2+42 b d a+19 b^2 c\right )+\left (5 \left (7 a^2+5 b^2\right ) d^2-6 b c (b c-7 a d)\right ) \sin (e+f x)\right )dx+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {d \left (35 \left (3 c^2+d^2\right ) a^2+168 b c d a+b^2 \left (51 c^2+25 d^2\right )\right )+2 \left (-\left (\left (3 c^3-41 c d^2\right ) b^2\right )+21 a d \left (c^2+3 d^2\right ) b+70 a^2 c d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {d \left (35 \left (3 c^2+d^2\right ) a^2+168 b c d a+b^2 \left (51 c^2+25 d^2\right )\right )+2 \left (-\left (\left (3 c^3-41 c d^2\right ) b^2\right )+21 a d \left (c^2+3 d^2\right ) b+70 a^2 c d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {d \left (35 \left (3 c^2+d^2\right ) a^2+168 b c d a+b^2 \left (51 c^2+25 d^2\right )\right )+2 \left (-\left (\left (3 c^3-41 c d^2\right ) b^2\right )+21 a d \left (c^2+3 d^2\right ) b+70 a^2 c d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}-\frac {\left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {4 \left (70 a^2 c d^2+21 a b d \left (c^2+3 d^2\right )-\left (b^2 \left (3 c^3-41 c d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (35 a^2 d^2+42 a b c d-\left (b^2 \left (6 c^2-25 d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}\right )-\frac {2 \left (5 d^2 \left (7 a^2+5 b^2\right )-6 b c (b c-7 a d)\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}\right )+\frac {4 b (b c-7 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 f}}{7 d}-\frac {2 b^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{7 d f}\) |
(-2*b^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(5/2))/(7*d*f) + ((4*b*(b*c - 7* a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*f) + ((-2*(5*(7*a^2 + 5*b ^2)*d^2 - 6*b*c*(b*c - 7*a*d))*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*f ) + ((4*(70*a^2*c*d^2 + 21*a*b*d*(c^2 + 3*d^2) - b^2*(3*c^3 - 41*c*d^2))*E llipticE[(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)*(42*a*b*c*d + 35*a^2 *d^2 - b^2*(6*c^2 - 25*d^2))*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]* Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/3)/5)/ (7*d)
3.8.31.3.1 Defintions of rubi rules used
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. \(1574\) vs. \(2(389)=778\).
Time = 13.22 (sec) , antiderivative size = 1575, normalized size of antiderivative = 4.73
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1575\) |
| parts | \(\text {Expression too large to display}\) | \(2397\) |
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(2*a^2*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)*(1/(c-d)*(-sin(f*x+e)-1)*d) ^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e)) /(c-d))^(1/2),((c-d)/(c+d))^(1/2))+d^2*b^2*(-2/7/d*sin(f*x+e)^2*(-(-d*sin( f*x+e)-c)*cos(f*x+e)^2)^(1/2)+12/35*c/d^2*sin(f*x+e)*(-(-d*sin(f*x+e)-c)*c os(f*x+e)^2)^(1/2)-2/3*(5/7+24/35*c^2/d^2)/d*(-(-d*sin(f*x+e)-c)*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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin( f*x+e)-c)*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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/ 2)/(-(-d*sin(f*x+e)-c)*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*b*d^2+2*b^2*c*d)*(-2/5/d*sin(f*x+e)*(- (-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+8/15*c/d^2*(-(-d*sin(f*x+e)-c)*cos(f *x+e)^2)^(1/2)+4/15*c/d*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f *x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)* 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*(3/5+8/15*c^2/d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1- sin(f*x+e))/(c+d))^(1/2)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.02 \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {2} {\left (12 \, b^{2} c^{4} - 84 \, a b c^{3} d + 252 \, a b c d^{3} + {\left (35 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} + 15 \, {\left (7 \, a^{2} + 5 \, b^{2}\right )} d^{4}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (12 \, b^{2} c^{4} - 84 \, a b c^{3} d + 252 \, a b c d^{3} + {\left (35 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} + 15 \, {\left (7 \, a^{2} + 5 \, b^{2}\right )} d^{4}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) - 6 \, \sqrt {2} {\left (-3 i \, b^{2} c^{3} d + 21 i \, a b c^{2} d^{2} + 63 i \, a b d^{4} + i \, {\left (70 \, a^{2} + 41 \, b^{2}\right )} c d^{3}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 6 \, \sqrt {2} {\left (3 i \, b^{2} c^{3} d - 21 i \, a b c^{2} d^{2} - 63 i \, a b d^{4} - i \, {\left (70 \, a^{2} + 41 \, b^{2}\right )} c d^{3}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 6 \, {\left (15 \, b^{2} d^{4} \cos \left (f x + e\right )^{3} - 6 \, {\left (4 \, b^{2} c d^{3} + 7 \, a b d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (3 \, b^{2} c^{2} d^{2} + 84 \, a b c d^{3} + 5 \, {\left (7 \, a^{2} + 8 \, b^{2}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, d^{3} f} \]
1/315*(sqrt(2)*(12*b^2*c^4 - 84*a*b*c^3*d + 252*a*b*c*d^3 + (35*a^2 - 11*b ^2)*c^2*d^2 + 15*(7*a^2 + 5*b^2)*d^4)*sqrt(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) + sqrt(2)*(12*b^2*c^4 - 84*a*b*c^3*d + 2 52*a*b*c*d^3 + (35*a^2 - 11*b^2)*c^2*d^2 + 15*(7*a^2 + 5*b^2)*d^4)*sqrt(-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) - 6*sqrt( 2)*(-3*I*b^2*c^3*d + 21*I*a*b*c^2*d^2 + 63*I*a*b*d^4 + I*(70*a^2 + 41*b^2) *c*d^3)*sqrt(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)) - 6*sqrt(2)*(3*I*b^2*c^3*d - 21*I*a*b*c^2*d^2 - 63*I*a*b*d^4 - I*(70 *a^2 + 41*b^2)*c*d^3)*sqrt(-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)) + 6*(15*b^2*d^4*cos(f*x + e)^3 - 6*(4*b^2*c*d^3 + 7 *a*b*d^4)*cos(f*x + e)*sin(f*x + e) - (3*b^2*c^2*d^2 + 84*a*b*c*d^3 + 5*(7 *a^2 + 8*b^2)*d^4)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^3*f)
\[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{2} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (3+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]