Integrand size = 27, antiderivative size = 532 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}+\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (e+f x)}{15 d^2 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b d \left (7 c^3-39 c d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d^3 \left (c^2-d^2\right )^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (8 a^3 c d^3-6 a b^2 c d \left (c^2-5 d^2\right )-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d^3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
2/5*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e) )^(5/2)+8/15*(-a*d+b*c)^2*(2*a*c*d+b*(c^2-3*d^2))*cos(f*x+e)/d^2/(c^2-d^2) ^2/f/(c+d*sin(f*x+e))^(3/2)-2/15*(-a*d+b*c)*(a^2*d^2*(23*c^2+9*d^2)+2*a*b* d*(7*c^3-39*c*d^2)+b^2*(8*c^4-21*c^2*d^2+45*d^4))*cos(f*x+e)/d^2/(c^2-d^2) ^3/f/(c+d*sin(f*x+e))^(1/2)+2/15*(-a*d+b*c)*(a^2*d^2*(23*c^2+9*d^2)+2*a*b* d*(7*c^3-39*c*d^2)+b^2*(8*c^4-21*c^2*d^2+45*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)^ 3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-2/15*(8*a^3*c*d^3-6*a*b^2*c*d*(c^2-5*d^ 2)-3*a^2*b*d^2*(3*c^2+5*d^2)-b^3*(8*c^4-15*c^2*d^2+15*d^4))*InverseJacobiA M(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)^2/f/(c+d*sin(f*x+e))^(1/2)
Time = 6.32 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 \left (\frac {\left (d^2 \left (3 a^2 b d^2 \left (27 c^2+5 d^2\right )-a^3 c d \left (15 c^2+17 d^2\right )-3 a b^2 c d \left (7 c^2+25 d^2\right )+b^3 \left (2 c^4+15 c^2 d^2+15 d^4\right )\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )+\left (-a^3 d^3 \left (23 c^2+9 d^2\right )+3 a^2 b c d^2 \left (3 c^2+29 d^2\right )-3 a b^2 d \left (-2 c^4+19 c^2 d^2+15 d^4\right )+b^3 \left (8 c^5-21 c^3 d^2+45 c d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right )\right ) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{5/2}}{(c-d)^3 (c+d)}+\frac {d (b c-a d) \cos (e+f x) \left (8 b^2 c^6+14 a b c^5 d+68 a^2 c^4 d^2-2 b^2 c^4 d^2-146 a b c^3 d^3+13 a^2 c^2 d^4+45 b^2 c^2 d^4-60 a b c d^5+15 a^2 d^6+45 b^2 d^6-d^2 \left (2 a b c d \left (7 c^2-39 d^2\right )+a^2 d^2 \left (23 c^2+9 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \cos (2 (e+f x))+2 d \left (2 a^2 c d^2 \left (27 c^2+5 d^2\right )+a b d \left (27 c^4-170 c^2 d^2+15 d^4\right )+b^2 \left (9 c^5-20 c^3 d^2+75 c d^4\right )\right ) \sin (e+f x)\right )}{2 \left (-c^2+d^2\right )^3}\right )}{15 d^3 f (c+d \sin (e+f x))^{5/2}} \] Input:
Integrate[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]
Output:
(2*(((d^2*(3*a^2*b*d^2*(27*c^2 + 5*d^2) - a^3*c*d*(15*c^2 + 17*d^2) - 3*a* b^2*c*d*(7*c^2 + 25*d^2) + b^3*(2*c^4 + 15*c^2*d^2 + 15*d^4))*EllipticF[(- 2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] + (-(a^3*d^3*(23*c^2 + 9*d^2)) + 3*a^2 *b*c*d^2*(3*c^2 + 29*d^2) - 3*a*b^2*d*(-2*c^4 + 19*c^2*d^2 + 15*d^4) + b^3 *(8*c^5 - 21*c^3*d^2 + 45*c*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])/(c + d))^(5/2))/((c - d)^3*(c + d)) + (d*(b*c - a*d)*Co s[e + f*x]*(8*b^2*c^6 + 14*a*b*c^5*d + 68*a^2*c^4*d^2 - 2*b^2*c^4*d^2 - 14 6*a*b*c^3*d^3 + 13*a^2*c^2*d^4 + 45*b^2*c^2*d^4 - 60*a*b*c*d^5 + 15*a^2*d^ 6 + 45*b^2*d^6 - d^2*(2*a*b*c*d*(7*c^2 - 39*d^2) + a^2*d^2*(23*c^2 + 9*d^2 ) + b^2*(8*c^4 - 21*c^2*d^2 + 45*d^4))*Cos[2*(e + f*x)] + 2*d*(2*a^2*c*d^2 *(27*c^2 + 5*d^2) + a*b*d*(27*c^4 - 170*c^2*d^2 + 15*d^4) + b^2*(9*c^5 - 2 0*c^3*d^2 + 75*c*d^4))*Sin[e + f*x]))/(2*(-c^2 + d^2)^3)))/(15*d^3*f*(c + d*Sin[e + f*x])^(5/2))
Time = 2.94 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.06, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3271, 27, 3042, 3500, 27, 3042, 3233, 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.
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {2 \int \frac {-5 c d a^3+12 b d^2 a^2-9 b^2 c d a+2 b^3 c^2-b \left (\left (4 c^2-5 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \sin ^2(e+f x)+\left (3 a (b c-a d)^2-5 b \left (a b c^2+\left (a^2+b^2\right ) d c-3 a b d^2\right )\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\int \frac {-5 c d a^3+12 b d^2 a^2-9 b^2 c d a+2 b^3 c^2-b \left (\left (4 c^2-5 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \sin ^2(e+f x)+\left (3 a (b c-a d)^2-5 b \left (a b c^2+\left (a^2+b^2\right ) d c-3 a b d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\int \frac {-5 c d a^3+12 b d^2 a^2-9 b^2 c d a+2 b^3 c^2-b \left (\left (4 c^2-5 d^2\right ) b^2+2 a c d b-a^2 d^2\right ) \sin (e+f x)^2+\left (3 a (b c-a d)^2-5 b \left (a b c^2+\left (a^2+b^2\right ) d c-3 a b d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {-\frac {2 \int -\frac {3 d \left (-d \left (5 c^2+3 d^2\right ) a^3+24 b c d^2 a^2-3 b^2 d \left (3 c^2+5 d^2\right ) a-2 b^3 \left (c^3-5 c d^2\right )\right )+\left (-\left (\left (8 c^4-15 d^2 c^2+15 d^4\right ) b^3\right )-6 a c d \left (c^2-5 d^2\right ) b^2-3 a^2 d^2 \left (3 c^2+5 d^2\right ) b+8 a^3 c d^3\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}-\frac {8 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\int \frac {3 d \left (-d \left (5 c^2+3 d^2\right ) a^3+24 b c d^2 a^2-3 b^2 d \left (3 c^2+5 d^2\right ) a-2 b^3 \left (c^3-5 c d^2\right )\right )+\left (-\left (\left (8 c^4-15 d^2 c^2+15 d^4\right ) b^3\right )-6 a c d \left (c^2-5 d^2\right ) b^2-3 a^2 d^2 \left (3 c^2+5 d^2\right ) b+8 a^3 c d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\int \frac {3 d \left (-d \left (5 c^2+3 d^2\right ) a^3+24 b c d^2 a^2-3 b^2 d \left (3 c^2+5 d^2\right ) a-2 b^3 \left (c^3-5 c d^2\right )\right )+\left (-\left (\left (8 c^4-15 d^2 c^2+15 d^4\right ) b^3\right )-6 a c d \left (c^2-5 d^2\right ) b^2-3 a^2 d^2 \left (3 c^2+5 d^2\right ) b+8 a^3 c d^3\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {2 \int -\frac {d \left (-c d \left (15 c^2+17 d^2\right ) a^3+3 b d^2 \left (27 c^2+5 d^2\right ) a^2-3 b^2 c d \left (7 c^2+25 d^2\right ) a+b^3 \left (2 c^4+15 d^2 c^2+15 d^4\right )\right )+(b c-a d) \left (\left (8 c^4-21 d^2 c^2+45 d^4\right ) b^2+2 a c d \left (7 c^2-39 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\int \frac {d \left (-c d \left (15 c^2+17 d^2\right ) a^3+3 b d^2 \left (27 c^2+5 d^2\right ) a^2-3 b^2 c d \left (7 c^2+25 d^2\right ) a+b^3 \left (2 c^4+15 d^2 c^2+15 d^4\right )\right )+(b c-a d) \left (\left (8 c^4-21 d^2 c^2+45 d^4\right ) b^2+2 a c d \left (7 c^2-39 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\int \frac {d \left (-c d \left (15 c^2+17 d^2\right ) a^3+3 b d^2 \left (27 c^2+5 d^2\right ) a^2-3 b^2 c d \left (7 c^2+25 d^2\right ) a+b^3 \left (2 c^4+15 d^2 c^2+15 d^4\right )\right )+(b c-a d) \left (\left (8 c^4-21 d^2 c^2+45 d^4\right ) b^2+2 a c d \left (7 c^2-39 d^2\right ) b+a^2 d^2 \left (23 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\frac {(b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}+\frac {\left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\frac {(b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}+\frac {\left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\frac {(b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\frac {(b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}+\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}+\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{5 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}-\frac {\frac {\frac {2 (b c-a d) \left (23 a^2 c^2 d^2+9 a^2 d^4+14 a b c^3 d-78 a b c d^3+8 b^2 c^4-21 b^2 c^2 d^2+45 b^2 d^4\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\frac {2 (b c-a d) \left (a^2 d^2 \left (23 c^2+9 d^2\right )+2 a b c d \left (7 c^2-39 d^2\right )+b^2 \left (8 c^4-21 c^2 d^2+45 d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \left (c^2-d^2\right ) \left (8 a^3 c d^3-3 a^2 b d^2 \left (3 c^2+5 d^2\right )-6 a b^2 c d \left (c^2-5 d^2\right )-\left (b^3 \left (8 c^4-15 c^2 d^2+15 d^4\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}}{c^2-d^2}}{3 d \left (c^2-d^2\right )}-\frac {8 (b c-a d)^2 \left (2 a c d+b \left (c^2-3 d^2\right )\right ) \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 d \left (c^2-d^2\right )}\) |
Input:
Int[(a + b*Sin[e + f*x])^3/(c + d*Sin[e + f*x])^(7/2),x]
Output:
(2*(b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(5*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) - ((-8*(b*c - a*d)^2*(2*a*c*d + b*(c^2 - 3*d^2))* Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((2*(b*c - a*d)*(8*b^2*c^4 + 14*a*b*c^3*d + 23*a^2*c^2*d^2 - 21*b^2*c^2*d^2 - 78*a*b* c*d^3 + 9*a^2*d^4 + 45*b^2*d^4)*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Si n[e + f*x]]) + ((2*(b*c - a*d)*(2*a*b*c*d*(7*c^2 - 39*d^2) + a^2*d^2*(23*c ^2 + 9*d^2) + b^2*(8*c^4 - 21*c^2*d^2 + 45*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*(c^2 - d^2)*(8*a^3*c*d^3 - 6*a*b^2*c*d*(c^2 - 5*d^2) - 3 *a^2*b*d^2*(3*c^2 + 5*d^2) - b^3*(8*c^4 - 15*c^2*d^2 + 15*d^4))*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]]))/(c^2 - d^2))/(3*d*(c^2 - d^2)))/(5*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_)]), 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_)])^(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. \(1613\) vs. \(2(513)=1026\).
Time = 9.68 (sec) , antiderivative size = 1614, normalized size of antiderivative = 3.03
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1614\) |
| parts | \(\text {Expression too large to display}\) | \(4599\) |
Input:
int((a+b*sin(f*x+e))^3/(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)*(2*b^3/d^3*(c/d-1)*((c+d*sin(f*x+e ))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^( 1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*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)*(2/5/(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)*EllipticF(((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))))+3*b/d^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)*(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...
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 2984, normalized size of antiderivative = 5.61 \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")
Output:
-2/45*((16*b^3*c^9 + 12*a*b^2*c^8*d + 18*a^2*b*c^7*d^2 - (a^3 + 15*a*b^2)* c^6*d^3 - 3*(5*a^2*b + 33*b^3)*c^5*d^4 + 6*(5*a^3 - 3*a*b^2)*c^4*d^5 - 18* (14*a^2*b - 5*b^3)*c^3*d^6 + 9*(11*a^3 + 45*a*b^2)*c^2*d^7 - 135*(a^2*b + b^3)*c*d^8 - 3*(16*b^3*c^7*d^2 + 12*a*b^2*c^6*d^3 + 6*(3*a^2*b - 8*b^3)*c^ 5*d^4 - (a^3 + 51*a*b^2)*c^4*d^5 - 3*(23*a^2*b - 15*b^3)*c^3*d^6 + 3*(11*a ^3 + 45*a*b^2)*c^2*d^7 - 45*(a^2*b + b^3)*c*d^8)*cos(f*x + e)^2 + (48*b^3* c^8*d + 36*a*b^2*c^7*d^2 + 2*(27*a^2*b - 64*b^3)*c^6*d^3 - 3*(a^3 + 47*a*b ^2)*c^5*d^4 - 3*(63*a^2*b - 29*b^3)*c^4*d^5 + 2*(49*a^3 + 177*a*b^2)*c^3*d ^6 - 6*(34*a^2*b + 15*b^3)*c^2*d^7 + 3*(11*a^3 + 45*a*b^2)*c*d^8 - 45*(a^2 *b + b^3)*d^9 - (16*b^3*c^6*d^3 + 12*a*b^2*c^5*d^4 + 6*(3*a^2*b - 8*b^3)*c ^4*d^5 - (a^3 + 51*a*b^2)*c^3*d^6 - 3*(23*a^2*b - 15*b^3)*c^2*d^7 + 3*(11* a^3 + 45*a*b^2)*c*d^8 - 45*(a^2*b + b^3)*d^9)*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 ) + (16*b^3*c^9 + 12*a*b^2*c^8*d + 18*a^2*b*c^7*d^2 - (a^3 + 15*a*b^2)*c^6 *d^3 - 3*(5*a^2*b + 33*b^3)*c^5*d^4 + 6*(5*a^3 - 3*a*b^2)*c^4*d^5 - 18*(14 *a^2*b - 5*b^3)*c^3*d^6 + 9*(11*a^3 + 45*a*b^2)*c^2*d^7 - 135*(a^2*b + b^3 )*c*d^8 - 3*(16*b^3*c^7*d^2 + 12*a*b^2*c^6*d^3 + 6*(3*a^2*b - 8*b^3)*c^5*d ^4 - (a^3 + 51*a*b^2)*c^4*d^5 - 3*(23*a^2*b - 15*b^3)*c^3*d^6 + 3*(11*a^3 + 45*a*b^2)*c^2*d^7 - 45*(a^2*b + b^3)*c*d^8)*cos(f*x + e)^2 + (48*b^3*...
Timed out. \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(7/2), x)
\[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")
Output:
integrate((b*sin(f*x + e) + a)^3/(d*sin(f*x + e) + c)^(7/2), x)
Timed out. \[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2),x)
Output:
int((a + b*sin(e + f*x))^3/(c + d*sin(e + f*x))^(7/2), x)
\[ \int \frac {(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{7/2}} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) a^{3}+\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) b^{3}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) a \,b^{2}+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) a^{2} b \] Input:
int((a+b*sin(f*x+e))^3/(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**3 + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**3)/(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**3 + 3*int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x)**2)/(si n(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**2 + 3*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**2*b