Integrand size = 42, antiderivative size = 357 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {154 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{221 c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
4/17*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(1 1/2)-60/221*a^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/c/f/g/(c-c*sin (f*x+e))^(9/2)-308/663*a^4*(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)+154/221*a^4*(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)+220/663*a^3*(g*cos(f*x+e))^(5/2)* (a+a*sin(f*x+e))^(1/2)/c^2/f/g/(c-c*sin(f*x+e))^(7/2)-154/221*a^4*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 = 11.46 (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))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {154 (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)))^{7/2}}{221 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 )^7 (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 {154}{221}+\frac {32}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}-\frac {864}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {2096}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {288}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {64 \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 {1728 \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 {4192 \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 {576 \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 )^3}+\frac {308 \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 )}\right ) (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}} \]
(-154*(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]))^(7/2))/(221*f*Cos[e + f*x]^(3 /2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(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*(154/221 + 32/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) - 864/(221 *(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 2096/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 288/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(17*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (172 8*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) + (4192* Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (576*Sin [(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (308*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f *x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x ])^(11/2))
Time = 2.72 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.02, 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, 3329, 3042, 3329, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{5/2}}dx}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{5/2}}dx}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\right )}{17 c}\) |
(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(17*f*g*(c - c*Sin [e + f*x])^(11/2)) - (15*a*((4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x ])^(3/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (11*a*((4*a*(g*Cos[e + f*x ])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (7 *a*((4*a*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Si n[e + f*x])^(5/2)) - (3*a*((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)))/(9*c)))/(13*c)))/(17*c)
3.2.24.3.1 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 = 2.90 (sec) , antiderivative size = 3093, normalized size of antiderivative = 8.66
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, method=_RETURNVERBOSE)
1/1326/f*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g*a^3/(1+cos(f*x+e) )^3/(cos(f*x+e)^2*sin(f*x+e)-3*cos(f*x+e)^2-4*sin(f*x+e)+4)/(-cos(f*x+e)/( 1+cos(f*x+e))^2)^(3/2)/(-c*(sin(f*x+e)-1))^(1/2)/c^5*(2652*cos(f*x+e)^2*si n(f*x+e)*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x +e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))-2652*cos(f*x+e)^ 2*sin(f*x+e)*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-c os(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))+1044*cos(f *x+e)^3*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-924*cos(f*x+e)^3*(-cos(f*x+e) /(1+cos(f*x+e))^2)^(1/2)*sin(f*x+e)-9888*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1 /2)-5304*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f *x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))+5304*ln((2*(-c os(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e)) ^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))+5304*ln(2*(2*(-cos(f*x+e)/(1+cos(f *x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+ e)+1)/(1+cos(f*x+e)))*sin(f*x+e)-5304*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2) ^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+ cos(f*x+e)))*sin(f*x+e)+4896*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sin(f*x+ e)-2496*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*tan(f*x+e)-2496*(-cos(f*x+e)/ (1+cos(f*x+e))^2)^(1/2)*sec(f*x+e)-14784*I*EllipticE(I*(csc(f*x+e)-cot(f*x +e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-co...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.21 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {2 \, {\left (231 \, a^{3} g \cos \left (f x + e\right )^{4} - 1600 \, a^{3} g \cos \left (f x + e\right )^{2} + 1544 \, a^{3} g + 4 \, {\left (123 \, a^{3} g \cos \left (f x + e\right )^{2} - 230 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 231 \, {\left (5 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 20 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a^{3} g + {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 12 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, {\left (-5 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 20 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a^{3} g + {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 12 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{663 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{4} - 20 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f - {\left (c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11 /2),x, algorithm="fricas")
1/663*(2*(231*a^3*g*cos(f*x + e)^4 - 1600*a^3*g*cos(f*x + e)^2 + 1544*a^3* g + 4*(123*a^3*g*cos(f*x + e)^2 - 230*a^3*g)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) + 231*(5*I*sqrt(2 )*a^3*g*cos(f*x + e)^4 - 20*I*sqrt(2)*a^3*g*cos(f*x + e)^2 + 16*I*sqrt(2)* a^3*g + (-I*sqrt(2)*a^3*g*cos(f*x + e)^4 + 12*I*sqrt(2)*a^3*g*cos(f*x + e) ^2 - 16*I*sqrt(2)*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 231*(-5*I*sqr t(2)*a^3*g*cos(f*x + e)^4 + 20*I*sqrt(2)*a^3*g*cos(f*x + e)^2 - 16*I*sqrt( 2)*a^3*g + (I*sqrt(2)*a^3*g*cos(f*x + e)^4 - 12*I*sqrt(2)*a^3*g*cos(f*x + e)^2 + 16*I*sqrt(2)*a^3*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. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11 /2),x, algorithm="maxima")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11 /2),x, algorithm="giac")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]