Integrand size = 42, antiderivative size = 414 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {22 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{663 c^6 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
4/21*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 3/2)-20/119*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))^(11/2)+220/1547*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))^(9/2)-220/1989*a^4*(g*cos(f*x+e))^(5/2)/c^3/f/g/(a +a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2)+22/663*a^4*(g*cos(f*x+e))^(5/2 )/c^4/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(5/2)+22/663*a^4*(g*cos( f*x+e))^(5/2)/c^5/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(3/2)-22/663 *a^4*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*EllipticE(sin(1/2*f*x+1/2*e), 2^(1/2))/c^6/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Time = 13.14 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.45 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx =\text {Too large to display} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(13/2),x]
Output:
(-22*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13*(a*(1 + Sin[e + f*x]))^(7/2))/(663*f*Cos[e + f*x]^(3/ 2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(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*(22/663 + 32/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10) - 352/(119* (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) + 464/(221*(Cos[(e + f*x)/2] - Si n[(e + f*x)/2])^6) - 1216/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 22/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/ (21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) - (704*Sin[(e + f*x)/2])/(11 9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) + (928*Sin[(e + f*x)/2])/(221*( Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (2432*Sin[(e + f*x)/2])/(1989*(C os[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (44*Sin[(e + f*x)/2])/(663*(Cos[( e + f*x)/2] - Sin[(e + f*x)/2])^3) + (44*Sin[(e + f*x)/2])/(663*(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])^(13/2))
Time = 3.44 (sec) , antiderivative size = 424, 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, 3329, 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.
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{13/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{13/2}}dx\) |
\(\Big \downarrow \) 3329 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{11/2}}dx}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{11/2}}dx}{7 c}\) |
\(\Big \downarrow \) 3329 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3329 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3329 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3331 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3331 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3321 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}\right )}{3 c}\right )}{13 c}\right )}{17 c}\right )}{7 c}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x ])^(13/2),x]
Output:
(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(21*f*g*(c - c*Sin [e + f*x])^(13/2)) - (5*a*((4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x] )^(3/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (11*a*((4*a*(g*Cos[e + f*x ])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - ( 7*a*((4*a*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*S in[e + f*x])^(7/2)) - (a*((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)))/(7*c )
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1545\) vs. \(2(358)=716\).
Time = 38.45 (sec) , antiderivative size = 1546, normalized size of antiderivative = 3.73
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13/2),x, method=_RETURNVERBOSE)
Output:
-1/13923/f*g/(2*2^(1/2)-3)/(1+2^(1/2))*a^3/c^6*(-1676+231*(cos(1/2*f*x+1/2 *e)^2+2*cos(1/2*f*x+1/2*e)+1)*(2*cos(1/2*f*x+1/2*e)*(16*cos(1/2*f*x+1/2*e) ^8-32*cos(1/2*f*x+1/2*e)^6-24*cos(1/2*f*x+1/2*e)^4+40*cos(1/2*f*x+1/2*e)^2 +5)*sin(1/2*f*x+1/2*e)-80*cos(1/2*f*x+1/2*e)^8+160*cos(1/2*f*x+1/2*e)^6-40 *cos(1/2*f*x+1/2*e)^4-40*cos(1/2*f*x+1/2*e)^2-1)*2^(1/2)*EllipticE((1+2^(1 /2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3)*(-2*(2^(1/2)*co s(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1))^( 1/2)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos(1/2 *f*x+1/2*e)+1))^(1/2)+462*(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*(( 2*cos(1/2*f*x+1/2*e)*(16*cos(1/2*f*x+1/2*e)^8-32*cos(1/2*f*x+1/2*e)^6-24*c os(1/2*f*x+1/2*e)^4+40*cos(1/2*f*x+1/2*e)^2+5)*sin(1/2*f*x+1/2*e)-80*cos(1 /2*f*x+1/2*e)^8+160*cos(1/2*f*x+1/2*e)^6-40*cos(1/2*f*x+1/2*e)^4-40*cos(1/ 2*f*x+1/2*e)^2-1)*2^(1/2)-4*cos(1/2*f*x+1/2*e)*(16*cos(1/2*f*x+1/2*e)^8-32 *cos(1/2*f*x+1/2*e)^6-24*cos(1/2*f*x+1/2*e)^4+40*cos(1/2*f*x+1/2*e)^2+5)*s in(1/2*f*x+1/2*e)+160*cos(1/2*f*x+1/2*e)^8-320*cos(1/2*f*x+1/2*e)^6+80*cos (1/2*f*x+1/2*e)^4+80*cos(1/2*f*x+1/2*e)^2+2)*((2^(1/2)*cos(1/2*f*x+1/2*e)- 2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos(1/2*f*x+1/2*e)+1))^(1/2)*(-2*(2^(1/2) *cos(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1) )^(1/2)*EllipticF((1+2^(1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2 ^(1/2)+3)+2*(838+(36960*cos(1/2*f*x+1/2*e)^9+18480*cos(1/2*f*x+1/2*e)^8...
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.11 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {2 \, {\left (231 \, \sqrt {\frac {1}{2}} {\left (-i \, a^{3} g \cos \left (f x + e\right )^{6} + 18 i \, a^{3} g \cos \left (f x + e\right )^{4} - 48 i \, a^{3} g \cos \left (f x + e\right )^{2} + 32 i \, a^{3} g + 2 \, {\left (-3 i \, a^{3} g \cos \left (f x + e\right )^{4} + 16 i \, a^{3} g \cos \left (f x + e\right )^{2} - 16 i \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, \sqrt {\frac {1}{2}} {\left (i \, a^{3} g \cos \left (f x + e\right )^{6} - 18 i \, a^{3} g \cos \left (f x + e\right )^{4} + 48 i \, a^{3} g \cos \left (f x + e\right )^{2} - 32 i \, a^{3} g + 2 \, {\left (3 i \, a^{3} g \cos \left (f x + e\right )^{4} - 16 i \, a^{3} g \cos \left (f x + e\right )^{2} + 16 i \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (1386 \, a^{3} g \cos \left (f x + e\right )^{4} - 8316 \, a^{3} g \cos \left (f x + e\right )^{2} + 7768 \, a^{3} g - {\left (231 \, a^{3} g \cos \left (f x + e\right )^{4} + 560 \, a^{3} g \cos \left (f x + e\right )^{2} - 2840 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{13923 \, {\left (c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{4} - 16 \, c^{7} f \cos \left (f x + e\right )^{2} + 16 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13 /2),x, algorithm="fricas")
Output:
-2/13923*(231*sqrt(1/2)*(-I*a^3*g*cos(f*x + e)^6 + 18*I*a^3*g*cos(f*x + e) ^4 - 48*I*a^3*g*cos(f*x + e)^2 + 32*I*a^3*g + 2*(-3*I*a^3*g*cos(f*x + e)^4 + 16*I*a^3*g*cos(f*x + e)^2 - 16*I*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weier strassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e) )) + 231*sqrt(1/2)*(I*a^3*g*cos(f*x + e)^6 - 18*I*a^3*g*cos(f*x + e)^4 + 4 8*I*a^3*g*cos(f*x + e)^2 - 32*I*a^3*g + 2*(3*I*a^3*g*cos(f*x + e)^4 - 16*I *a^3*g*cos(f*x + e)^2 + 16*I*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZ eta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + (1 386*a^3*g*cos(f*x + e)^4 - 8316*a^3*g*cos(f*x + e)^2 + 7768*a^3*g - (231*a ^3*g*cos(f*x + e)^4 + 560*a^3*g*cos(f*x + e)^2 - 2840*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))/ (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*cos(f*x + e)^2 + 16*c^7* f)*sin(f*x + e))
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))** (13/2),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13 /2),x, algorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(13/2), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13 /2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x ))^(13/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x ))^(13/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {too large to display} \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13/2),x)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*a**3*g*(792*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3 + 528*sqrt(sin(e + f*x) + 1 )*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**2 + 920*sqrt( sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x ) + 64*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x)) - 280*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**6 - 4*sin(e + f*x)**5 + 5*sin(e + f*x)* *4 - 5*sin(e + f*x)**2 + 4*sin(e + f*x) - 1),x)*sin(e + f*x)**6*f + 1680*i nt((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*co s(e + f*x))/(sin(e + f*x)**6 - 4*sin(e + f*x)**5 + 5*sin(e + f*x)**4 - 5*s in(e + f*x)**2 + 4*sin(e + f*x) - 1),x)*sin(e + f*x)**5*f - 4200*int((sqrt (sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f* x))/(sin(e + f*x)**6 - 4*sin(e + f*x)**5 + 5*sin(e + f*x)**4 - 5*sin(e + f *x)**2 + 4*sin(e + f*x) - 1),x)*sin(e + f*x)**4*f + 5600*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin (e + f*x)**6 - 4*sin(e + f*x)**5 + 5*sin(e + f*x)**4 - 5*sin(e + f*x)**2 + 4*sin(e + f*x) - 1),x)*sin(e + f*x)**3*f - 4200*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x )**6 - 4*sin(e + f*x)**5 + 5*sin(e + f*x)**4 - 5*sin(e + f*x)**2 + 4*sin(e + f*x) - 1),x)*sin(e + f*x)**2*f + 1680*int((sqrt(sin(e + f*x) + 1)*sq...