Integrand size = 27, antiderivative size = 391 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{3 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d^3 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (b c-a d) \left (2 a b c d-a^2 d^2+b^2 \left (8 c^2-9 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d^3 \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}} \] Output:
2/3*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e) )^(3/2)+8/3*(-a*d+b*c)^2*(a*c*d+b*(c^2-2*d^2))*cos(f*x+e)/d^2/(c^2-d^2)^2/ f/(c+d*sin(f*x+e))^(1/2)-2/3*(4*a^3*c*d^3-6*a*b^2*c*d*(c^2-3*d^2)-3*a^2*b* d^2*(c^2+3*d^2)+b^3*(8*c^4-15*c^2*d^2+3*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/(c^2-d^2)^2/f/ ((c+d*sin(f*x+e))/(c+d))^(1/2)-2/3*(-a*d+b*c)*(2*a*b*c*d-a^2*d^2+b^2*(8*c^ 2-9*d^2))*InverseJacobiAM(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/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)
Time = 5.12 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 \left (\frac {\left (d^2 \left (-12 a^2 b c d^2+a^3 d \left (3 c^2+d^2\right )+3 a b^2 d \left (c^2+3 d^2\right )+2 b^3 \left (c^3-3 c d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (4 a^3 c d^3-6 a b^2 c d \left (c^2-3 d^2\right )-3 a^2 b d^2 \left (c^2+3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) (-c-d \sin (e+f x)) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(c-d)^2 (c+d)^2}-\frac {d (b c-a d)^2 \cos (e+f x) \left (-4 b c^3-5 a c^2 d+8 b c d^2+a d^3+d \left (-5 b c^2-4 a c d+9 b d^2\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^2}\right )}{3 d^3 f (c+d \sin (e+f x))^{3/2}} \] Input:
Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]
Output:
(2*(((d^2*(-12*a^2*b*c*d^2 + a^3*d*(3*c^2 + d^2) + 3*a*b^2*d*(c^2 + 3*d^2) + 2*b^3*(c^3 - 3*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (4*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 3*d^2) - 3*a^2*b*d^2*(c^2 + 3*d^2) + b ^3*(8*c^4 - 15*c^2*d^2 + 3*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)]))*(-c - d*Sin[e + f*x])*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/((c - d)^2*(c + d)^2 ) - (d*(b*c - a*d)^2*Cos[e + f*x]*(-4*b*c^3 - 5*a*c^2*d + 8*b*c*d^2 + a*d^ 3 + d*(-5*b*c^2 - 4*a*c*d + 9*b*d^2)*Sin[e + f*x]))/(c^2 - d^2)^2))/(3*d^3 *f*(c + d*Sin[e + f*x])^(3/2))
Time = 2.08 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3271, 27, 3042, 3500, 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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {2 \int \frac {-3 c d a^3+8 b d^2 a^2-7 b^2 c d a+2 b^3 c^2+b \left (-\left (\left (4 c^2-3 d^2\right ) b^2\right )+2 a c d b-a^2 d^2\right ) \sin ^2(e+f x)-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 c d a^3+8 b d^2 a^2-7 b^2 c d a+2 b^3 c^2+b \left (-\left (\left (4 c^2-3 d^2\right ) b^2\right )+2 a c d b-a^2 d^2\right ) \sin ^2(e+f x)-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\int \frac {-3 c d a^3+8 b d^2 a^2-7 b^2 c d a+2 b^3 c^2+b \left (-\left (\left (4 c^2-3 d^2\right ) b^2\right )+2 a c d b-a^2 d^2\right ) \sin (e+f x)^2-\left (-d^2 a^3+5 b c d a^2+b^2 \left (2 c^2-9 d^2\right ) a+3 b^3 c d\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {-\frac {2 \int -\frac {d \left (-d \left (3 c^2+d^2\right ) a^3+12 b c d^2 a^2-3 b^2 d \left (c^2+3 d^2\right ) a-2 b^3 \left (c^3-3 c d^2\right )\right )-\left (\left (8 c^4-15 d^2 c^2+3 d^4\right ) b^3-6 a c d \left (c^2-3 d^2\right ) b^2-3 a^2 d^2 \left (c^2+3 d^2\right ) b+4 a^3 c d^3\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{d \left (c^2-d^2\right )}-\frac {8 \left (a c d+b \left (c^2-2 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\int \frac {d \left (-d \left (3 c^2+d^2\right ) a^3+12 b c d^2 a^2-3 b^2 d \left (c^2+3 d^2\right ) a-2 b^3 \left (c^3-3 c d^2\right )\right )-\left (\left (8 c^4-15 d^2 c^2+3 d^4\right ) b^3-6 a c d \left (c^2-3 d^2\right ) b^2-3 a^2 d^2 \left (c^2+3 d^2\right ) b+4 a^3 c d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\int \frac {d \left (-d \left (3 c^2+d^2\right ) a^3+12 b c d^2 a^2-3 b^2 d \left (c^2+3 d^2\right ) a-2 b^3 \left (c^3-3 c d^2\right )\right )-\left (\left (8 c^4-15 d^2 c^2+3 d^4\right ) b^3-6 a c d \left (c^2-3 d^2\right ) b^2-3 a^2 d^2 \left (c^2+3 d^2\right ) b+4 a^3 c d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {\left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {\left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {\left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {\left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {\left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 \left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {\left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {\left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}-\frac {\frac {\frac {2 \left (c^2-d^2\right ) (b c-a d) \left (-a^2 d^2+2 a b c d+8 b^2 c^2-9 b^2 d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (4 a^3 c d^3-3 a^2 b d^2 \left (c^2+3 d^2\right )-6 a b^2 c d \left (c^2-3 d^2\right )+b^3 \left (8 c^4-15 c^2 d^2+3 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (a c d+b \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{d f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}\) |
Input:
Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(5/2),x]
Output:
(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) - ((-8*(b*c - a*d)^2*(a*c*d + b*(c^2 - 2*d^2))*Co s[e + f*x])/(d*(c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*(4*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 3*d^2) - 3*a^2*b*d^2*(c^2 + 3*d^2) + b^3*(8*c^4 - 15 *c^2*d^2 + 3*d^4))*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c + d *Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*(b*c - a*d)* (c^2 - d^2)*(8*b^2*c^2 + 2*a*b*c*d - a^2*d^2 - 9*b^2*d^2)*EllipticF[(e - P i/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]]))/(d*(c^2 - d^2)))/(3*d*(c^2 - d^2))
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_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* (n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin [e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1371\) vs. \(2(376)=752\).
Time = 8.38 (sec) , antiderivative size = 1372, normalized size of antiderivative = 3.51
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1372\) |
| parts | \(\text {Expression too large to display}\) | \(3581\) |
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)*(b^2/d^3*(2*b*d*(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)))+6*a*d*(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))-4*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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/( c+d))^(1/2)))+1/d^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(2/3/(c^ 2-d^2)/d*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d* cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2*(3*c^ 2+d^2)/(3*c^4-6*c^2*d^2+3*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)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/( c+d))^(1/2))+8/3*c*d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d *(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f *x+e))*cos(f*x+e)^2)^(1/2)*((-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...
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 1848, normalized size of antiderivative = 4.73 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
2/9*((16*b^3*c^7 - 12*a*b^2*c^6*d - 4*a^3*c^2*d^5 - 2*(3*a^2*b + 10*b^3)*c ^5*d^2 - (a^3 - 15*a*b^2)*c^4*d^3 + 12*(a^2*b - b^3)*c^3*d^4 + 6*(3*a^2*b + 4*b^3)*c*d^6 - 3*(a^3 + 9*a*b^2)*d^7 - (16*b^3*c^5*d^2 - 12*a*b^2*c^4*d^ 3 - 6*(a^2*b + 6*b^3)*c^3*d^4 - (a^3 - 27*a*b^2)*c^2*d^5 + 6*(3*a^2*b + 4* b^3)*c*d^6 - 3*(a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^2 + 2*(16*b^3*c^6*d - 12* a*b^2*c^5*d^2 - 6*(a^2*b + 6*b^3)*c^4*d^3 - (a^3 - 27*a*b^2)*c^3*d^4 + 6*( 3*a^2*b + 4*b^3)*c^2*d^5 - 3*(a^3 + 9*a*b^2)*c*d^6)*sin(f*x + e))*sqrt(1/2 *I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c *d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (16*b^ 3*c^7 - 12*a*b^2*c^6*d - 4*a^3*c^2*d^5 - 2*(3*a^2*b + 10*b^3)*c^5*d^2 - (a ^3 - 15*a*b^2)*c^4*d^3 + 12*(a^2*b - b^3)*c^3*d^4 + 6*(3*a^2*b + 4*b^3)*c* d^6 - 3*(a^3 + 9*a*b^2)*d^7 - (16*b^3*c^5*d^2 - 12*a*b^2*c^4*d^3 - 6*(a^2* b + 6*b^3)*c^3*d^4 - (a^3 - 27*a*b^2)*c^2*d^5 + 6*(3*a^2*b + 4*b^3)*c*d^6 - 3*(a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^2 + 2*(16*b^3*c^6*d - 12*a*b^2*c^5*d ^2 - 6*(a^2*b + 6*b^3)*c^4*d^3 - (a^3 - 27*a*b^2)*c^3*d^4 + 6*(3*a^2*b + 4 *b^3)*c^2*d^5 - 3*(a^3 + 9*a*b^2)*c*d^6)*sin(f*x + e))*sqrt(-1/2*I*d)*weie rstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3 , 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(8*I*b^3*c^6* d - 6*I*a*b^2*c^5*d^2 - I*(3*a^2*b + 7*b^3)*c^4*d^3 + 4*I*(a^3 + 3*a*b^2)* c^3*d^4 - 12*I*(a^2*b + b^3)*c^2*d^5 + 2*I*(2*a^3 + 9*a*b^2)*c*d^6 - 3*...
Timed out. \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] 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)
\[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] 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. \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \] 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)
\[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) a^{3}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) b^{3}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) a \,b^{2}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) a^{2} b \] 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)/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d **2 + 3*sin(e + f*x)*c**2*d + c**3),x)*a**3 + int((sqrt(sin(e + f*x)*d + c )*sin(e + f*x)**3)/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d**2 + 3*si n(e + f*x)*c**2*d + c**3),x)*b**3 + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e + f* x)*c**2*d + c**3),x)*a*b**2 + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x) )/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e + f*x)*c**2*d + c**3),x)*a**2*b