Integrand size = 42, antiderivative size = 414 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {22 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3315 c^6 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
4/21*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/f/g/(c-c*sin(f*x+e))^(1 3/2)+44/663*a^3*(g*cos(f*x+e))^(5/2)/c^2/f/g/(c-c*sin(f*x+e))^(9/2)/(a+a*s in(f*x+e))^(1/2)-22/1989*a^3*(g*cos(f*x+e))^(5/2)/c^3/f/g/(c-c*sin(f*x+e)) ^(7/2)/(a+a*sin(f*x+e))^(1/2)-22/3315*a^3*(g*cos(f*x+e))^(5/2)/c^4/f/g/(c- c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-22/3315*a^3*(g*cos(f*x+e))^(5/2 )/c^5/f/g/(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-44/357*a^2*(g*cos( f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/c/f/g/(c-c*sin(f*x+e))^(11/2)+22/3315 *a^3*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^6/f/(a+a*sin(f* x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Time = 11.78 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.45 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {22 (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 )^{13} (a (1+\sin (e+f x)))^{5/2}}{3315 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 )^5 (c-c \sin (e+f x))^{13/2}}+\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 )^{13} \left (-\frac {22}{3315}+\frac {16}{21 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}}-\frac {120}{119 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}+\frac {84}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {22}{1989 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {22}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {32 \sin \left (\frac {1}{2} (e+f x)\right )}{21 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}}-\frac {240 \sin \left (\frac {1}{2} (e+f x)\right )}{119 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}+\frac {168 \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 {44 \sin \left (\frac {1}{2} (e+f x)\right )}{1989 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{3315 \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)))^{5/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{13/2}} \]
(22*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - S in[(e + f*x)/2])^13*(a*(1 + Sin[e + f*x]))^(5/2))/(3315*f*Cos[e + f*x]^(3/ 2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(13/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) ^13*(-22/3315 + 16/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10) - 120/(11 9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) + 84/(221*(Cos[(e + f*x)/2] - S in[(e + f*x)/2])^6) - 22/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 22/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (32*Sin[(e + f*x)/2])/ (21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) - (240*Sin[(e + f*x)/2])/(11 9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) + (168*Sin[(e + f*x)/2])/(221*( Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (44*Sin[(e + f*x)/2])/(1989*(Cos [(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f *x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(13/2))
Time = 3.30 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3329, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{13/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{13/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx}{21 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}}{3 c}\right )}{13 c}\right )}{17 c}\right )}{21 c}\) |
(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*(c - c*Sin [e + f*x])^(13/2)) - (11*a*((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*Sin[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]]*Sq rt[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)))/(21*c)
3.2.15.3.1 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 = 5.08 (sec) , antiderivative size = 1991, normalized size of antiderivative = 4.81
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, method=_RETURNVERBOSE)
2/69615*I/f*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g*a^2/(1+cos(f*x +e))/(cos(f*x+e)^4+4*cos(f*x+e)^2*sin(f*x+e)-8*cos(f*x+e)^2-8*sin(f*x+e)+8 )/(-c*(sin(f*x+e)-1))^(1/2)/c^6*(-924*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e) /(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)+924*(1/(1+co s(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e) -cot(f*x+e)),I)-26520*I*sec(f*x+e)^2+9688*I*sin(f*x+e)-30216*I*tan(f*x+e)- 22824*I*sec(f*x+e)+231*I*cos(f*x+e)^4+77*I*cos(f*x+e)^3-2926*I*cos(f*x+e)^ 2+18944*I*cos(f*x+e)-231*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Elli pticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^4+2 31*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-co t(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^4-462*(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*sin(f*x+e)+462*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellipti cF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^3*sin( f*x+e)+2541*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)^2-2541*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+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+1155*(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)^4-1155*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-c...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.20 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {2 \, {\left (1386 \, a^{2} g \cos \left (f x + e\right )^{4} + 5607 \, a^{2} g \cos \left (f x + e\right )^{2} - 10796 \, a^{2} g - {\left (231 \, a^{2} g \cos \left (f x + e\right )^{4} - 4081 \, a^{2} g \cos \left (f x + e\right )^{2} + 15724 \, a^{2} 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 (i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{6} - 18 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} + 48 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} - 32 i \, \sqrt {2} a^{2} g + 2 \, {\left (3 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} - 16 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a^{2} 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 (-i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{6} + 18 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} - 48 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} + 32 i \, \sqrt {2} a^{2} g + 2 \, {\left (-3 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} + 16 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a^{2} 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 )}{69615 \, {\left (c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{4} - 16 \, c^{7} f \cos \left (f x + e\right )^{2} + 16 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13 /2),x, algorithm="fricas")
1/69615*(2*(1386*a^2*g*cos(f*x + e)^4 + 5607*a^2*g*cos(f*x + e)^2 - 10796* a^2*g - (231*a^2*g*cos(f*x + e)^4 - 4081*a^2*g*cos(f*x + e)^2 + 15724*a^2* 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*(I*sqrt(2)*a^2*g*cos(f*x + e)^6 - 18*I*sqrt(2)*a^2*g* cos(f*x + e)^4 + 48*I*sqrt(2)*a^2*g*cos(f*x + e)^2 - 32*I*sqrt(2)*a^2*g + 2*(3*I*sqrt(2)*a^2*g*cos(f*x + e)^4 - 16*I*sqrt(2)*a^2*g*cos(f*x + e)^2 + 16*I*sqrt(2)*a^2*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*(-I*sqrt(2)*a^ 2*g*cos(f*x + e)^6 + 18*I*sqrt(2)*a^2*g*cos(f*x + e)^4 - 48*I*sqrt(2)*a^2* g*cos(f*x + e)^2 + 32*I*sqrt(2)*a^2*g + 2*(-3*I*sqrt(2)*a^2*g*cos(f*x + e) ^4 + 16*I*sqrt(2)*a^2*g*cos(f*x + e)^2 - 16*I*sqrt(2)*a^2*g)*sin(f*x + e)) *sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e ) - I*sin(f*x + e))))/(c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x + e)^4 + 48 *c^7*f*cos(f*x + e)^2 - 32*c^7*f + 2*(3*c^7*f*cos(f*x + e)^4 - 16*c^7*f*co s(f*x + e)^2 + 16*c^7*f)*sin(f*x + e))
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13 /2),x, algorithm="maxima")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13 /2),x, algorithm="giac")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \]