Integrand size = 27, antiderivative size = 460 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \cos (e+f x)}{15 d \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
2/5*(-a*d+b*c)^2*cos(f*x+e)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(5/2)-4/15*(-a* d+b*c)*(4*a*c*d+b*(c^2-5*d^2))*cos(f*x+e)/d/(c^2-d^2)^2/f/(c+d*sin(f*x+e)) ^(3/2)+2/15*(a^2*d^2*(23*c^2+9*d^2)-a*b*(6*c^3*d+58*c*d^3)-b^2*(2*c^4-19*c ^2*d^2-15*d^4))*cos(f*x+e)/d/(c^2-d^2)^3/f/(c+d*sin(f*x+e))^(1/2)-2/15*(a^ 2*d^2*(23*c^2+9*d^2)-a*b*(6*c^3*d+58*c*d^3)-b^2*(2*c^4-19*c^2*d^2-15*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/(c^2-d^2)^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+4/15*(-a*d+b*c )*(4*a*c*d+b*(c^2-5*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^2/(c^2-d^2)^2/f/(c+d*sin(f* x+e))^(1/2)
Time = 4.63 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 \left (-\frac {\left (d^2 \left (-2 a b d \left (27 c^2+5 d^2\right )+b^2 c \left (7 c^2+25 d^2\right )+a^2 \left (15 c^3+17 c d^2\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )-\left (-a^2 d^2 \left (23 c^2+9 d^2\right )+a b \left (6 c^3 d+58 c d^3\right )+b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{5/2}}{(c-d)^3 (c+d)}+\frac {d \cos (e+f x) \left (3 (b c-a d)^2 \left (c^2-d^2\right )^2-2 \left (c^2-d^2\right ) \left (-4 a^2 c d^2+a b d \left (3 c^2+5 d^2\right )+b^2 \left (c^3-5 c d^2\right )\right ) (c+d \sin (e+f x))-\left (-a^2 d^2 \left (23 c^2+9 d^2\right )+a b \left (6 c^3 d+58 c d^3\right )+b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right ) (c+d \sin (e+f x))^2\right )}{\left (c^2-d^2\right )^3}\right )}{15 d^2 f (c+d \sin (e+f x))^{5/2}} \] Input:
Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]
Output:
(2*(-(((d^2*(-2*a*b*d*(27*c^2 + 5*d^2) + b^2*c*(7*c^2 + 25*d^2) + a^2*(15* c^3 + 17*c*d^2))*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - (-(a^2* d^2*(23*c^2 + 9*d^2)) + a*b*(6*c^3*d + 58*c*d^3) + b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*E llipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]))*((c + d*Sin[e + f*x])/(c + d))^(5/2))/((c - d)^3*(c + d))) + (d*Cos[e + f*x]*(3*(b*c - a*d)^2*(c^2 - d^2)^2 - 2*(c^2 - d^2)*(-4*a^2*c*d^2 + a*b*d*(3*c^2 + 5*d^2) + b^2*(c^3 - 5*c*d^2))*(c + d*Sin[e + f*x]) - (-(a^2*d^2*(23*c^2 + 9*d^2)) + a*b*(6*c ^3*d + 58*c*d^3) + b^2*(2*c^4 - 19*c^2*d^2 - 15*d^4))*(c + d*Sin[e + f*x]) ^2))/(c^2 - d^2)^3))/(15*d^2*f*(c + d*Sin[e + f*x])^(5/2))
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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3269 |
| \(\displaystyle \frac {2 \int \frac {5 d \left (\left (a^2+b^2\right ) c-2 a b d\right )+\left (\left (2 c^2-5 d^2\right ) b^2+6 a c d b-3 a^2 d^2\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 d \left (\left (a^2+b^2\right ) c-2 a b d\right )+\left (\left (2 c^2-5 d^2\right ) b^2+6 a c d b-3 a^2 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 d \left (\left (a^2+b^2\right ) c-2 a b d\right )+\left (\left (2 c^2-5 d^2\right ) b^2+6 a c d b-3 a^2 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {2 \int \frac {3 d \left (-\left (\left (5 c^2+3 d^2\right ) a^2\right )+16 b c d a-b^2 \left (3 c^2+5 d^2\right )\right )-2 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {3 d \left (-\left (\left (5 c^2+3 d^2\right ) a^2\right )+16 b c d a-b^2 \left (3 c^2+5 d^2\right )\right )-2 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {3 d \left (-\left (\left (5 c^2+3 d^2\right ) a^2\right )+16 b c d a-b^2 \left (3 c^2+5 d^2\right )\right )-2 (b c-a d) \left (4 a c d+b \left (c^2-5 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {-\frac {2 \int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {d \left (15 a^2 c^3+7 b^2 c^3-54 a b d c^2+17 a^2 d^2 c+25 b^2 d^2 c-10 a b d^3\right )+\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-a \left (6 d c^3+58 d^3 c\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {d \left (-\left (\left (15 c^3+17 d^2 c\right ) a^2\right )+2 b d \left (27 c^2+5 d^2\right ) a-b^2 c \left (7 c^2+25 d^2\right )\right )-\left (-\left (\left (2 c^4-19 d^2 c^2-15 d^4\right ) b^2\right )-2 a c d \left (3 c^2+29 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (a^2 d^2 \left (23 c^2+9 d^2\right )-a b \left (6 c^3 d+58 c d^3\right )-\left (b^2 \left (2 c^4-19 c^2 d^2-15 d^4\right )\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {4 (b c-a d) \left (4 a c d+b c^2-5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x)}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
Input:
Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^(7/2),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*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] && LtQ[m, -1] && IntegerQ[2*m]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*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*(m + 1) *(2*b*c*d - a*(c^2 + d^2)) + (a^2*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1 ) + c^2*(m + 2)))*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] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1443\) vs. \(2(441)=882\).
Time = 6.00 (sec) , antiderivative size = 1444, normalized size of antiderivative = 3.14
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1444\) |
| parts | \(\text {Expression too large to display}\) | \(3043\) |
Input:
int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(b^2/d^2*(2*d*cos(f*x+e)^2/(c^2-d^ 2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2/(c^2-d^2)*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))+2/(c^2-d^2)*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*s in(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))))+(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^2*(2/5/(c ^2-d^2)/d^2*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^3+16/ 15*c/(c^2-d^2)^2/d*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d )^2+2/15*cos(f*x+e)^2*d/(c^2-d^2)^3*(23*c^2+9*d^2)/(-(-c-d*sin(f*x+e))*cos (f*x+e)^2)^(1/2)+2*(15*c^3+17*c*d^2)/(15*c^6-45*c^4*d^2+45*c^2*d^4-15*d^6) *(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)*Ellip ticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/15*d*(23*c^2+9* d^2)/(c^2-d^2)^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)))...
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 2227, normalized size of antiderivative = 4.84 \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")
Output:
-2/45*((4*b^2*c^8 + 12*a*b*c^7*d - 10*a*b*c^5*d^3 - 168*a*b*c^3*d^5 - 90*a *b*c*d^7 - (a^2 + 5*b^2)*c^6*d^2 + 6*(5*a^2 - b^2)*c^4*d^4 + 9*(11*a^2 + 1 5*b^2)*c^2*d^6 - 3*(4*b^2*c^6*d^2 + 12*a*b*c^5*d^3 - 46*a*b*c^3*d^5 - 30*a *b*c*d^7 - (a^2 + 17*b^2)*c^4*d^4 + 3*(11*a^2 + 15*b^2)*c^2*d^6)*cos(f*x + e)^2 + (12*b^2*c^7*d + 36*a*b*c^6*d^2 - 126*a*b*c^4*d^4 - 136*a*b*c^2*d^6 - 30*a*b*d^8 - (3*a^2 + 47*b^2)*c^5*d^3 + 2*(49*a^2 + 59*b^2)*c^3*d^5 + 3 *(11*a^2 + 15*b^2)*c*d^7 - (4*b^2*c^5*d^3 + 12*a*b*c^4*d^4 - 46*a*b*c^2*d^ 6 - 30*a*b*d^8 - (a^2 + 17*b^2)*c^3*d^5 + 3*(11*a^2 + 15*b^2)*c*d^7)*cos(f *x + e)^2)*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) + (4*b^2*c^8 + 12*a*b*c^7*d - 10*a*b*c^5*d^3 - 16 8*a*b*c^3*d^5 - 90*a*b*c*d^7 - (a^2 + 5*b^2)*c^6*d^2 + 6*(5*a^2 - b^2)*c^4 *d^4 + 9*(11*a^2 + 15*b^2)*c^2*d^6 - 3*(4*b^2*c^6*d^2 + 12*a*b*c^5*d^3 - 4 6*a*b*c^3*d^5 - 30*a*b*c*d^7 - (a^2 + 17*b^2)*c^4*d^4 + 3*(11*a^2 + 15*b^2 )*c^2*d^6)*cos(f*x + e)^2 + (12*b^2*c^7*d + 36*a*b*c^6*d^2 - 126*a*b*c^4*d ^4 - 136*a*b*c^2*d^6 - 30*a*b*d^8 - (3*a^2 + 47*b^2)*c^5*d^3 + 2*(49*a^2 + 59*b^2)*c^3*d^5 + 3*(11*a^2 + 15*b^2)*c*d^7 - (4*b^2*c^5*d^3 + 12*a*b*c^4 *d^4 - 46*a*b*c^2*d^6 - 30*a*b*d^8 - (a^2 + 17*b^2)*c^3*d^5 + 3*(11*a^2 + 15*b^2)*c*d^7)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassPIn verse(-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*(...
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e) + a)^2/(d*sin(f*x + e) + c)^(7/2), x)
\[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")
Output:
integrate((b*sin(f*x + e) + a)^2/(d*sin(f*x + e) + c)^(7/2), x)
Timed out. \[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(7/2),x)
Output:
int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^(7/2), x)
\[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{7/2}} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) a^{2}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) b^{2}+2 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) a b \] Input:
int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**4*d**4 + 4*sin(e + f*x)**3*c*d **3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c**3*d + c**4),x)*a**2 + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2)/(sin(e + f*x)**4*d**4 + 4 *sin(e + f*x)**3*c*d**3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c** 3*d + c**4),x)*b**2 + 2*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)**4*d**4 + 4*sin(e + f*x)**3*c*d**3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c**3*d + c**4),x)*a*b