Integrand size = 42, antiderivative size = 471 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {22 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5525 c^7 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
4/25*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 5/2)-4/35*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))^(13/2)+44/595*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))^(11/2)-44/1105*a^4*(g*cos(f*x+e))^(5/2)/c^3/f/g/(a+a*s in(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2)+22/3315*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))^(7/2)+22/5525*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))^(5/2)+22/5525* a^4*(g*cos(f*x+e))^(5/2)/c^6/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^( 3/2)-22/5525*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^7/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Time = 13.89 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.42 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx =\text {Too large to display} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(15/2),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])^15*(a*(1 + Sin[e + f*x]))^(7/2))/(5525*f*Cos[e + f*x]^(3 /2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2] )^15*(22/5525 + 32/(25*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^12) - 416/(17 5*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10) + 4656/(2975*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) - 2144/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) ^6) + 22/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 22/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(25*(Cos[(e + f* x)/2] - Sin[(e + f*x)/2])^13) - (832*Sin[(e + f*x)/2])/(175*(Cos[(e + f*x) /2] - Sin[(e + f*x)/2])^11) + (9312*Sin[(e + f*x)/2])/(2975*(Cos[(e + f*x) /2] - Sin[(e + f*x)/2])^9) - (4288*Sin[(e + f*x)/2])/(5525*(Cos[(e + f*x)/ 2] - Sin[(e + f*x)/2])^7) + (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (44*Sin[(e + f*x)/2])/(5525*(Cos[(e + f*x)/2] - S in[(e + f*x)/2])^3) + (44*Sin[(e + f*x)/2])/(5525*(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])^(15/2))
Time = 3.96 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3329, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{15/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{15/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{13/2}}dx}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{13/2}}dx}{5 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {3 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {3 a \left (\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}}{3 c}\right )}{13 c}\right )}{17 c}\right )}{21 c}\right )}{5 c}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x ])^(15/2),x]
Output:
(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(25*f*g*(c - c*Sin [e + f*x])^(15/2)) - (3*a*((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]]*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)))/(21*c)))/(5*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. \(1753\) vs. \(2(407)=814\).
Time = 46.12 (sec) , antiderivative size = 1754, normalized size of antiderivative = 3.72
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, method=_RETURNVERBOSE)
Output:
1/116025/f*g/(2*2^(1/2)-3)/(1+2^(1/2))*a^3/c^7*(-10958+231*(cos(1/2*f*x+1/ 2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*(4*cos(1/2*f*x+1/2*e)*(48*cos(1/2*f*x+1/2*e )^8-96*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/2*e)^4+40*cos(1/2*f*x+1/2*e)^2 +3)*sin(1/2*f*x+1/2*e)+64*cos(1/2*f*x+1/2*e)^12-192*cos(1/2*f*x+1/2*e)^10- 48*cos(1/2*f*x+1/2*e)^8+416*cos(1/2*f*x+1/2*e)^6-180*cos(1/2*f*x+1/2*e)^4- 60*cos(1/2*f*x+1/2*e)^2-1)*2^(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)*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^(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)*((4*cos(1/2*f*x+1/2*e)*( 48*cos(1/2*f*x+1/2*e)^8-96*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/2*e)^4+40* cos(1/2*f*x+1/2*e)^2+3)*sin(1/2*f*x+1/2*e)+64*cos(1/2*f*x+1/2*e)^12-192*co s(1/2*f*x+1/2*e)^10-48*cos(1/2*f*x+1/2*e)^8+416*cos(1/2*f*x+1/2*e)^6-180*c os(1/2*f*x+1/2*e)^4-60*cos(1/2*f*x+1/2*e)^2-1)*2^(1/2)-8*cos(1/2*f*x+1/2*e )*(48*cos(1/2*f*x+1/2*e)^8-96*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/2*e)^4+ 40*cos(1/2*f*x+1/2*e)^2+3)*sin(1/2*f*x+1/2*e)-128*cos(1/2*f*x+1/2*e)^12+38 4*cos(1/2*f*x+1/2*e)^10+96*cos(1/2*f*x+1/2*e)^8-832*cos(1/2*f*x+1/2*e)^6+3 60*cos(1/2*f*x+1/2*e)^4+120*cos(1/2*f*x+1/2*e)^2+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)*El lipticF((1+2^(1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.09 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {2 \, {\left (231 \, \sqrt {\frac {1}{2}} {\left (7 i \, a^{3} g \cos \left (f x + e\right )^{6} - 56 i \, a^{3} g \cos \left (f x + e\right )^{4} + 112 i \, a^{3} g \cos \left (f x + e\right )^{2} - 64 i \, a^{3} g + {\left (-i \, a^{3} g \cos \left (f x + e\right )^{6} + 24 i \, a^{3} g \cos \left (f x + e\right )^{4} - 80 i \, a^{3} g \cos \left (f x + e\right )^{2} + 64 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 (-7 i \, a^{3} g \cos \left (f x + e\right )^{6} + 56 i \, a^{3} g \cos \left (f x + e\right )^{4} - 112 i \, a^{3} g \cos \left (f x + e\right )^{2} + 64 i \, a^{3} g + {\left (i \, a^{3} g \cos \left (f x + e\right )^{6} - 24 i \, a^{3} g \cos \left (f x + e\right )^{4} + 80 i \, a^{3} g \cos \left (f x + e\right )^{2} - 64 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 (231 \, a^{3} g \cos \left (f x + e\right )^{6} - 5698 \, a^{3} g \cos \left (f x + e\right )^{4} + 42044 \, a^{3} g \cos \left (f x + e\right )^{2} - 42056 \, a^{3} g + 7 \, {\left (231 \, a^{3} g \cos \left (f x + e\right )^{4} + 1544 \, a^{3} g \cos \left (f x + e\right )^{2} - 4600 \, 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 )}}{116025 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{6} - 56 \, c^{8} f \cos \left (f x + e\right )^{4} + 112 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f - {\left (c^{8} f \cos \left (f x + e\right )^{6} - 24 \, c^{8} f \cos \left (f x + e\right )^{4} + 80 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} 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))^(15 /2),x, algorithm="fricas")
Output:
2/116025*(231*sqrt(1/2)*(7*I*a^3*g*cos(f*x + e)^6 - 56*I*a^3*g*cos(f*x + e )^4 + 112*I*a^3*g*cos(f*x + e)^2 - 64*I*a^3*g + (-I*a^3*g*cos(f*x + e)^6 + 24*I*a^3*g*cos(f*x + e)^4 - 80*I*a^3*g*cos(f*x + e)^2 + 64*I*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*sqrt(1/2)*(-7*I*a^3*g*cos(f*x + e)^6 + 56*I*a^3*g*cos(f*x + e)^4 - 112*I*a^3*g*cos(f*x + e)^2 + 64*I*a^3*g + (I*a ^3*g*cos(f*x + e)^6 - 24*I*a^3*g*cos(f*x + e)^4 + 80*I*a^3*g*cos(f*x + e)^ 2 - 64*I*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstra ssPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + (231*a^3*g*cos(f*x + e )^6 - 5698*a^3*g*cos(f*x + e)^4 + 42044*a^3*g*cos(f*x + e)^2 - 42056*a^3*g + 7*(231*a^3*g*cos(f*x + e)^4 + 1544*a^3*g*cos(f*x + e)^2 - 4600*a^3*g)*s in(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(7*c^8*f*cos(f*x + e)^6 - 56*c^8*f*cos(f*x + e)^4 + 112*c^8*f* cos(f*x + e)^2 - 64*c^8*f - (c^8*f*cos(f*x + e)^6 - 24*c^8*f*cos(f*x + e)^ 4 + 80*c^8*f*cos(f*x + e)^2 - 64*c^8*f)*sin(f*x + e))
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))** (15/2),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15 /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)^(15/2), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15 /2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{15/2}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x ))^(15/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x ))^(15/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {too large to display} \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*a**3*g*( - 1144*sqrt(sin(e + f*x) + 1)*sqrt( - si n(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3 - 1430*sqrt(sin(e + f*x ) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**2 - 1388 *sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x) - 262*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x)) + 210*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(c os(e + f*x))*cos(e + f*x))/(sin(e + f*x)**7 - 5*sin(e + f*x)**6 + 9*sin(e + f*x)**5 - 5*sin(e + f*x)**4 - 5*sin(e + f*x)**3 + 9*sin(e + f*x)**2 - 5* sin(e + f*x) + 1),x)*sin(e + f*x)**7*f - 1470*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)** 7 - 5*sin(e + f*x)**6 + 9*sin(e + f*x)**5 - 5*sin(e + f*x)**4 - 5*sin(e + f*x)**3 + 9*sin(e + f*x)**2 - 5*sin(e + f*x) + 1),x)*sin(e + f*x)**6*f + 4 410*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)**7 - 5*sin(e + f*x)**6 + 9*sin(e + f*x)**5 - 5*sin(e + f*x)**4 - 5*sin(e + f*x)**3 + 9*sin(e + f*x)**2 - 5*sin(e + f* x) + 1),x)*sin(e + f*x)**5*f - 7350*int((sqrt(sin(e + f*x) + 1)*sqrt( - si n(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**7 - 5*sin( e + f*x)**6 + 9*sin(e + f*x)**5 - 5*sin(e + f*x)**4 - 5*sin(e + f*x)**3 + 9*sin(e + f*x)**2 - 5*sin(e + f*x) + 1),x)*sin(e + f*x)**4*f + 7350*int((s qrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(...