Integrand size = 27, antiderivative size = 302 \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {36 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{35 d^2 f}+\frac {216 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac {2 \cos (e+f x) (27+27 \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac {36 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{35 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {36 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{35 d^3 f \sqrt {c+d \sin (e+f x)}} \]
8/35*a^3*(c-4*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f-2/7*cos(f*x+e)*(a ^3+a^3*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2)/d/f-4/105*a^3*(4*c^2-21*c*d+65*d ^2)*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/d^2/f-4/105*a^3*(4*c^3-21*c^2*d+62*c *d^2+147*d^3)*(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^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+4/105*a^3*(c^2-d^2)*(4*c^ 2-21*c*d+65*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*si n(f*x+e))/(c+d))^(1/2)/d^3/f/(c+d*sin(f*x+e))^(1/2)
Time = 0.59 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75 \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\frac {9 \left (-8 \left (d^2 \left (c^2+126 c d+65 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (\left (16 c^2-84 c d-535 d^2\right ) \cos (e+f x)+3 d (5 d \cos (3 (e+f x))-2 (c+21 d) \sin (2 (e+f x)))\right )\right )}{70 d^3 f \sqrt {c+d \sin (e+f x)}} \]
(9*(-8*(d^2*(c^2 + 126*c*d + 65*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d )/(c + d)] + (4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*((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] )*((16*c^2 - 84*c*d - 535*d^2)*Cos[e + f*x] + 3*d*(5*d*Cos[3*(e + f*x)] - 2*(c + 21*d)*Sin[2*(e + f*x)]))))/(70*d^3*f*Sqrt[c + d*Sin[e + f*x]])
Time = 1.73 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.10, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3042, 3242, 3042, 3447, 3042, 3502, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^3 \sqrt {c+d \sin (e+f x)}dx\) |
| \(\Big \downarrow \) 3242 |
| \(\displaystyle \frac {2 \int (\sin (e+f x) a+a) \left (a^2 (c+5 d)-2 a^2 (c-4 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}dx}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (\sin (e+f x) a+a) \left (a^2 (c+5 d)-2 a^2 (c-4 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)}dx}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {2 \int \sqrt {c+d \sin (e+f x)} \left (-2 (c-4 d) \sin ^2(e+f x) a^3+(c+5 d) a^3+\left (a^3 (c+5 d)-2 a^3 (c-4 d)\right ) \sin (e+f x)\right )dx}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \sqrt {c+d \sin (e+f x)} \left (-2 (c-4 d) \sin (e+f x)^2 a^3+(c+5 d) a^3+\left (a^3 (c+5 d)-2 a^3 (c-4 d)\right ) \sin (e+f x)\right )dx}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 \left (\frac {2 \int -\frac {1}{2} \sqrt {c+d \sin (e+f x)} \left (a^3 (c-49 d) d-a^3 \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x)\right )dx}{5 d}+\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\int \sqrt {c+d \sin (e+f x)} \left (a^3 (c-49 d) d-a^3 \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x)\right )dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\int \sqrt {c+d \sin (e+f x)} \left (a^3 (c-49 d) d-a^3 \left (4 c^2-21 d c+65 d^2\right ) \sin (e+f x)\right )dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2}{3} \int -\frac {d \left (c^2+126 d c+65 d^2\right ) a^3+\left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{2 \sqrt {c+d \sin (e+f x)}}dx+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^2+126 d c+65 d^2\right ) a^3+\left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}-\frac {1}{3} \int \frac {d \left (c^2+126 d c+65 d^2\right ) a^3+\left (4 c^3-21 d c^2+62 d^2 c+147 d^3\right ) \sin (e+f x) a^3}{\sqrt {c+d \sin (e+f x)}}dx}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {1}{3} \left (\frac {a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )+\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 \left (\frac {4 a^3 (c-4 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{5 d f}-\frac {\frac {2 a^3 \left (4 c^2-21 c d+65 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 f}+\frac {1}{3} \left (\frac {2 a^3 \left (c^2-d^2\right ) \left (4 c^2-21 c d+65 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a^3 \left (4 c^3-21 c^2 d+62 c d^2+147 d^3\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\right )}{5 d}\right )}{7 d}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{3/2}}{7 d f}\) |
(-2*Cos[e + f*x]*(a^3 + a^3*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2))/(7*d *f) + (2*((4*a^3*(c - 4*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(5*d*f ) - ((2*a^3*(4*c^2 - 21*c*d + 65*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x] ])/(3*f) + ((-2*a^3*(4*c^3 - 21*c^2*d + 62*c*d^2 + 147*d^3)*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*a^3*(c^2 - d^2)*(4*c^2 - 21*c*d + 65*d^2)*Ell ipticF[(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*d)))/(7*d)
3.5.98.3.1 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_)])^(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 - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c* (m - 2) + b^2*d*(n + 1) + a^2*d*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !LtQ[ n, -1] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[ c, 0]))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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. \(1315\) vs. \(2(360)=720\).
Time = 6.14 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.36
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1316\) |
| parts | \(\text {Expression too large to display}\) | \(2443\) |
2/105*a^3*(sin(f*x+e)*c^2*d^3+18*sin(f*x+e)^4*c*d^4-sin(f*x+e)^3*c^2*d^3-4 *sin(f*x+e)^2*c^3*d^2-130*sin(f*x+e)*d^5+15*sin(f*x+e)^5*d^5-21*c^2*d^3-13 0*c*d^4+4*c^3*d^2+8*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d ))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d) )^(1/2),((c-d)/(c+d))^(1/2))*c^4*d-48*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(si n(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d *sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^2+416*((c+d*sin(f*x+e ))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^ (1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2*d^ 3+48*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(s in(f*x+e)+1)/(c-d))^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/ (c+d))^(1/2))*c*d^4+42*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/( c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c -d))^(1/2),((c-d)/(c+d))^(1/2))*c^4*d-116*((c+d*sin(f*x+e))/(c-d))^(1/2)*( -(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(( (c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^3*d^2-336*((c+d*sin(f *x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c- d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^ 2*d^3+124*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*( -d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.07 \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{4} - 42 \, a^{3} c^{3} d + 121 \, a^{3} c^{2} d^{2} - 84 \, a^{3} c d^{3} - 195 \, a^{3} 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 (8 \, a^{3} c^{4} - 42 \, a^{3} c^{3} d + 121 \, a^{3} c^{2} d^{2} - 84 \, a^{3} c d^{3} - 195 \, a^{3} 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 ) + 3 \, \sqrt {2} {\left (4 i \, a^{3} c^{3} d - 21 i \, a^{3} c^{2} d^{2} + 62 i \, a^{3} c d^{3} + 147 i \, a^{3} d^{4}\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 ) + 3 \, \sqrt {2} {\left (-4 i \, a^{3} c^{3} d + 21 i \, a^{3} c^{2} d^{2} - 62 i \, a^{3} c d^{3} - 147 i \, a^{3} d^{4}\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 ) - 3 \, {\left (15 \, a^{3} d^{4} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c d^{3} + 21 \, a^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (4 \, a^{3} c^{2} d^{2} - 21 \, a^{3} c d^{3} - 145 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{315 \, d^{4} f} \]
-2/315*(sqrt(2)*(8*a^3*c^4 - 42*a^3*c^3*d + 121*a^3*c^2*d^2 - 84*a^3*c*d^3 - 195*a^3*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)*(8*a^3*c^4 - 42*a^3*c^3*d + 121*a^3*c^2*d^2 - 84*a^3 *c*d^3 - 195*a^3*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) + 3*sqrt(2)*(4*I*a^3*c^3*d - 21*I*a^3*c^2*d^2 + 62*I*a ^3*c*d^3 + 147*I*a^3*d^4)*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*si n(f*x + e) - 2*I*c)/d)) + 3*sqrt(2)*(-4*I*a^3*c^3*d + 21*I*a^3*c^2*d^2 - 6 2*I*a^3*c*d^3 - 147*I*a^3*d^4)*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)) - 3*(15*a^3*d^4*cos(f*x + e)^3 - 3*(a^3*c* d^3 + 21*a^3*d^4)*cos(f*x + e)*sin(f*x + e) + (4*a^3*c^2*d^2 - 21*a^3*c*d^ 3 - 145*a^3*d^4)*cos(f*x + e))*sqrt(d*sin(f*x + e) + c))/(d^4*f)
\[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=a^{3} \left (\int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int \sqrt {c + d \sin {\left (e + f x \right )}}\, dx\right ) \]
a**3*(Integral(3*sqrt(c + d*sin(e + f*x))*sin(e + f*x), x) + Integral(3*sq rt(c + d*sin(e + f*x))*sin(e + f*x)**2, x) + Integral(sqrt(c + d*sin(e + f *x))*sin(e + f*x)**3, x) + Integral(sqrt(c + d*sin(e + f*x)), x))
\[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
\[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {d \sin \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int (3+3 \sin (e+f x))^3 \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]