Integrand size = 25, antiderivative size = 362 \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 (b c-3 d) \cos (e+f x)}{5 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac {2 \left (3 b c^2-24 c d+5 b d^2\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 \left (3 b c^3-69 c^2 d+29 b c d^2-27 d^3\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (3 b c^3-69 c^2 d+29 b c d^2-27 d^3\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 d \left (c^2-d^2\right )^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 \left (3 b c^2-24 c d+5 b d^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 d \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}} \]
-2/5*(-a*d+b*c)*cos(f*x+e)/(c^2-d^2)/f/(c+d*sin(f*x+e))^(5/2)-2/15*(-8*a*c *d+3*b*c^2+5*b*d^2)*cos(f*x+e)/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^(3/2)-2/15*( -23*a*c^2*d-9*a*d^3+3*b*c^3+29*b*c*d^2)*cos(f*x+e)/(c^2-d^2)^3/f/(c+d*sin( f*x+e))^(1/2)+2/15*(-23*a*c^2*d-9*a*d^3+3*b*c^3+29*b*c*d^2)*(sin(1/2*e+1/4 *Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi +1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d/(c^2-d^2)^3/f/ ((c+d*sin(f*x+e))/(c+d))^(1/2)-2/15*(-8*a*c*d+3*b*c^2+5*b*d^2)*(sin(1/2*e+ 1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4 *Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d/(c^ 2-d^2)^2/f/(c+d*sin(f*x+e))^(1/2)
Time = 1.65 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.79 \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\frac {2 \left (\frac {\left (\left (3 b c^3-69 c^2 d+29 b c d^2-27 d^3\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-(c-d) \left (3 b c^2-24 c d+5 b d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{5/2}}{(c-d)^3 d}+\frac {\cos (e+f x) \left (-9 b c^5+102 c^4 d-25 b c^3 d^2-15 c^2 d^3+2 b c d^4+9 d^5+d \left (-9 b c^4+162 c^3 d-60 b c^2 d^2+30 c d^3+5 b d^4\right ) \sin (e+f x)+d^2 \left (-3 b c^3+69 c^2 d-29 b c d^2+27 d^3\right ) \sin ^2(e+f x)\right )}{\left (c^2-d^2\right )^3}\right )}{15 f (c+d \sin (e+f x))^{5/2}} \]
(2*((((3*b*c^3 - 69*c^2*d + 29*b*c*d^2 - 27*d^3)*EllipticE[(-2*e + Pi - 2* f*x)/4, (2*d)/(c + d)] - (c - d)*(3*b*c^2 - 24*c*d + 5*b*d^2)*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*d) + (Cos[e + f*x]*(-9*b*c^5 + 102*c^4*d - 25*b*c^3*d^2 - 15*c ^2*d^3 + 2*b*c*d^4 + 9*d^5 + d*(-9*b*c^4 + 162*c^3*d - 60*b*c^2*d^2 + 30*c *d^3 + 5*b*d^4)*Sin[e + f*x] + d^2*(-3*b*c^3 + 69*c^2*d - 29*b*c*d^2 + 27* d^3)*Sin[e + f*x]^2))/(c^2 - d^2)^3))/(15*f*(c + d*Sin[e + f*x])^(5/2))
Time = 1.74 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 3233, 27, 3042, 3233, 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.
| \(\displaystyle \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {2 \int -\frac {5 (a c-b d)+3 (b c-a d) \sin (e+f x)}{2 (c+d \sin (e+f x))^{5/2}}dx}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 (a c-b d)+3 (b c-a d) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 (a c-b d)+3 (b c-a d) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {3 \left (5 a c^2-8 b d c+3 a d^2\right )+\left (3 b c^2-8 a d c+5 b d^2\right ) \sin (e+f x)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (5 a c^2-8 b d c+3 a d^2\right )+\left (3 b c^2-8 a d c+5 b d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 \left (5 a c^2-8 b d c+3 a d^2\right )+\left (3 b c^2-8 a d c+5 b d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {15 a c^3-27 b d c^2+17 a d^2 c-5 b d^3-\left (3 b c^3-23 a d c^2+29 b d^2 c-9 a d^3\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {15 a c^3-27 b d c^2+17 a d^2 c-5 b d^3-\left (3 b c^3-23 a d c^2+29 b d^2 c-9 a d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {15 a c^3-27 b d c^2+17 a d^2 c-5 b d^3-\left (3 b c^3-23 a d c^2+29 b d^2 c-9 a d^3\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \left (c^2-d^2\right ) \left (-8 a c d+3 b c^2+5 b d^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}\) |
(-2*(b*c - a*d)*Cos[e + f*x])/(5*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(5/2)) + ((-2*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((-2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*d^3 )*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*(3*b*c^3 - 23*a*c^2*d + 29*b*c*d^2 - 9*a*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*(c^2 - d^2)*(3*b*c^2 - 8*a*c*d + 5*b*d^2)*EllipticF[(e - Pi/2 + f*x)/ 2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[ e + f*x]]))/(c^2 - d^2))/(3*(c^2 - d^2)))/(5*(c^2 - d^2))
3.8.29.3.1 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[(-(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]
Leaf count of result is larger than twice the leaf count of optimal. \(1048\) vs. \(2(411)=822\).
Time = 24.20 (sec) , antiderivative size = 1049, normalized size of antiderivative = 2.90
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1049\) |
| parts | \(\text {Expression too large to display}\) | \(1628\) |
(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*(b/d*(2/3/(c^2-d^2)/d*(-(-d*sin(f* x+e)-c)*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/(-(-d*sin(f*x+e)-c)*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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^ (1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e) ^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d)) ^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))+(a *d-b*c)/d*(2/5/(c^2-d^2)/d^2*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin( f*x+e)+c/d)^3+16/15*c/(c^2-d^2)^2/d*(-(-d*sin(f*x+e)-c)*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)/(-(-d* sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)+2*(15*c^3+17*c*d^2)/(15*c^6-45*c^4*d^2+4 5*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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f* x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2) )+2/15*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)*(1/(c-d)*(-sin(f*x+e)-1)*d)^(1/2)/(-(-d*si n(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 1573, normalized size of antiderivative = 4.35 \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \]
1/45*((3*sqrt(2)*(6*b*c^5*d^2 - a*c^4*d^3 - 23*b*c^3*d^4 + 33*a*c^2*d^5 - 15*b*c*d^6)*cos(f*x + e)^2 + (sqrt(2)*(6*b*c^4*d^3 - a*c^3*d^4 - 23*b*c^2* d^5 + 33*a*c*d^6 - 15*b*d^7)*cos(f*x + e)^2 - sqrt(2)*(18*b*c^6*d - 3*a*c^ 5*d^2 - 63*b*c^4*d^3 + 98*a*c^3*d^4 - 68*b*c^2*d^5 + 33*a*c*d^6 - 15*b*d^7 ))*sin(f*x + e) - sqrt(2)*(6*b*c^7 - a*c^6*d - 5*b*c^5*d^2 + 30*a*c^4*d^3 - 84*b*c^3*d^4 + 99*a*c^2*d^5 - 45*b*c*d^6))*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)*(6*b*c^5*d^2 - a*c^4 *d^3 - 23*b*c^3*d^4 + 33*a*c^2*d^5 - 15*b*c*d^6)*cos(f*x + e)^2 + (sqrt(2) *(6*b*c^4*d^3 - a*c^3*d^4 - 23*b*c^2*d^5 + 33*a*c*d^6 - 15*b*d^7)*cos(f*x + e)^2 - sqrt(2)*(18*b*c^6*d - 3*a*c^5*d^2 - 63*b*c^4*d^3 + 98*a*c^3*d^4 - 68*b*c^2*d^5 + 33*a*c*d^6 - 15*b*d^7))*sin(f*x + e) - sqrt(2)*(6*b*c^7 - a*c^6*d - 5*b*c^5*d^2 + 30*a*c^4*d^3 - 84*b*c^3*d^4 + 99*a*c^2*d^5 - 45*b* c*d^6))*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*(3*sqrt(2)*(3*I*b*c^4*d^3 - 23*I*a*c^3*d^4 + 29*I*b*c^2*d^5 - 9* I*a*c*d^6)*cos(f*x + e)^2 + (sqrt(2)*(3*I*b*c^3*d^4 - 23*I*a*c^2*d^5 + 29* I*b*c*d^6 - 9*I*a*d^7)*cos(f*x + e)^2 + sqrt(2)*(-9*I*b*c^5*d^2 + 69*I*a*c ^4*d^3 - 90*I*b*c^3*d^4 + 50*I*a*c^2*d^5 - 29*I*b*c*d^6 + 9*I*a*d^7))*sin( f*x + e) + sqrt(2)*(-3*I*b*c^6*d + 23*I*a*c^5*d^2 - 38*I*b*c^4*d^3 + 78...
Timed out. \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {3+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\int \frac {a+b\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]