Integrand size = 42, antiderivative size = 357 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{1105 c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
-28/221*a^2*(g*cos(f*x+e))^(5/2)/c/f/g/(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x +e))^(1/2)+14/663*a^2*(g*cos(f*x+e))^(5/2)/c^2/f/g/(c-c*sin(f*x+e))^(7/2)/ (a+a*sin(f*x+e))^(1/2)+14/1105*a^2*(g*cos(f*x+e))^(5/2)/c^3/f/g/(c-c*sin(f *x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)+14/1105*a^2*(g*cos(f*x+e))^(5/2)/c^4/f /g/(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)+4/17*a*(g*cos(f*x+e))^(5/ 2)*(a+a*sin(f*x+e))^(1/2)/f/g/(c-c*sin(f*x+e))^(11/2)-14/1105*a^2*g*(cos(1 /2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^( 1/2))*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)/c^5/f/(a+a*sin(f*x+e))^(1/2)/( c-c*sin(f*x+e))^(1/2)
Time = 10.55 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.49 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {14 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} (a (1+\sin (e+f x)))^{3/2}}{1105 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{11/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} \left (\frac {14}{1105}+\frac {8}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}-\frac {80}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {14}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {14}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {16 \sin \left (\frac {1}{2} (e+f x)\right )}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {160 \sin \left (\frac {1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{3/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{11/2}} \]
(-14*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(3/2))/(1105*f*Cos[e + f*x]^(3 /2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(11/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2] )^11*(14/1105 + 8/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) - 80/(221*( Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 14/(663*(Cos[(e + f*x)/2] - Sin[ (e + f*x)/2])^4) + 14/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (16 *Sin[(e + f*x)/2])/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (160*Sin [(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) + (28*Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (28*Sin[(e + f* x)/2])/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (28*Sin[(e + f*x)/ 2])/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^( 3/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(11/ 2))
Time = 2.70 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3329, 3042, 3329, 3042, 3331, 3042, 3331, 3042, 3331, 3042, 3321, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}}{3 c}\right )}{13 c}\right )}{17 c}\) |
(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (7*a*((4*a*(g*Cos[e + f*x])^(5/2))/(13*f*g*Sqrt[a + a*S in[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (3*a*((2*(g*Cos[e + f*x])^(5/2) )/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + ((2*(g*Cos [e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/ 2)) + ((2*(g*Cos[e + f*x])^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin [e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE [(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])) /(5*c))/(3*c)))/(13*c)))/(17*c)
3.2.5.3.1 Defintions of rubi rules used
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* (Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) Int[(g *Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2 *b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* x])^n/(f*g*(2*n + p + 1))), x] - Simp[b*((2*m + p - 1)/(d*(2*n + p + 1))) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^( n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] & & EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && In tegersQ[2*m, 2*n, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b* (g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(a* f*g*(2*m + p + 1))), x] + Simp[(m + n + p + 1)/(a*(2*m + p + 1)) Int[(g*C os[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && !LtQ[m, n, -1] && Integers Q[2*m, 2*n, 2*p]
Result contains complex when optimal does not.
Time = 3.76 (sec) , antiderivative size = 1578, normalized size of antiderivative = 4.42
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11/2),x, method=_RETURNVERBOSE)
-2/3315*I/f*(g*cos(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*g*a/(cos(f*x+e)^ 2*sin(f*x+e)-3*cos(f*x+e)^2-4*sin(f*x+e)+4)/(-c*(sin(f*x+e)-1))^(1/2)/c^5/ (1+cos(f*x+e))*(21*I*cos(f*x+e)^2*sin(f*x+e)+7*I*cos(f*x+e)*sin(f*x+e)-439 *I+84*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e) -cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-84*sin(f*x+e)*(cos(f *x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+c os(f*x+e)))^(1/2)*cos(f*x+e)^2+21*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellipt icE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^4-21* (cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*( 1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^4+42*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)* EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e) ^3-42*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e) ),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^3+780*I*tan(f*x+e)*sec(f*x+e)+168 *(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)* (1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)*sin(f*x+e)-168*(cos(f*x+e)/(1+cos(f*x+ e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2) *cos(f*x+e)*sin(f*x+e)-147*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*( csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+147*(cos(f *x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+c os(f*x+e)))^(1/2)*cos(f*x+e)^2-336*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*El...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.11 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {2 \, {\left (21 \, a g \cos \left (f x + e\right )^{4} - 266 \, a g \cos \left (f x + e\right )^{2} + 502 \, a g + {\left (105 \, a g \cos \left (f x + e\right )^{2} + 278 \, a g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 21 \, {\left (5 i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} - 20 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a g + {\left (-i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} + 12 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, {\left (-5 i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} + 20 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a g + {\left (i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} - 12 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{3315 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{4} - 20 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f - {\left (c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11 /2),x, algorithm="fricas")
1/3315*(2*(21*a*g*cos(f*x + e)^4 - 266*a*g*cos(f*x + e)^2 + 502*a*g + (105 *a*g*cos(f*x + e)^2 + 278*a*g)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*s in(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) + 21*(5*I*sqrt(2)*a*g*cos(f*x + e)^4 - 20*I*sqrt(2)*a*g*cos(f*x + e)^2 + 16*I*sqrt(2)*a*g + (-I*sqrt(2)*a *g*cos(f*x + e)^4 + 12*I*sqrt(2)*a*g*cos(f*x + e)^2 - 16*I*sqrt(2)*a*g)*si n(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*(-5*I*sqrt(2)*a*g*cos(f*x + e)^4 + 20 *I*sqrt(2)*a*g*cos(f*x + e)^2 - 16*I*sqrt(2)*a*g + (I*sqrt(2)*a*g*cos(f*x + e)^4 - 12*I*sqrt(2)*a*g*cos(f*x + e)^2 + 16*I*sqrt(2)*a*g)*sin(f*x + e)) *sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e ) - I*sin(f*x + e))))/(5*c^6*f*cos(f*x + e)^4 - 20*c^6*f*cos(f*x + e)^2 + 16*c^6*f - (c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^2 + 16*c^6*f)*sin (f*x + e))
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11 /2),x, algorithm="maxima")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11 /2),x, algorithm="giac")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]