Integrand size = 42, antiderivative size = 409 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {14 a^4 c^2 (g \cos (e+f x))^{5/2}}{39 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^4 c^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{13 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{143 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{429 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{143 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{13 f g} \] Output:
-14/39*a^4*c^2*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f* x+e))^(1/2)+14/13*a^4*c^2*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*Elliptic E(sin(1/2*f*x+1/2*e),2^(1/2))/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1 /2)-2/13*a^3*c^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/f/g/(c-c*sin( f*x+e))^(1/2)-10/143*a^2*c^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/f /g/(c-c*sin(f*x+e))^(1/2)-14/429*a*c^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e ))^(5/2)/f/g/(c-c*sin(f*x+e))^(1/2)+14/143*c^2*(g*cos(f*x+e))^(5/2)*(a+a*s in(f*x+e))^(7/2)/f/g/(c-c*sin(f*x+e))^(1/2)+2/13*c*(g*cos(f*x+e))^(5/2)*(a +a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2)/f/g
Time = 10.45 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.52 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {a^3 c (g \cos (e+f x))^{3/2} (-1+\sin (e+f x)) (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} \left (-7392 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (1560 \cos (e+f x)+780 \cos (3 (e+f x))+156 \cos (5 (e+f x))-1507 \sin (2 (e+f x))-88 \sin (4 (e+f x))+33 \sin (6 (e+f x)))\right )}{6864 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2),x]
Output:
(a^3*c*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])*(1 + Sin[e + f*x])^3*Sqr t[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-7392*EllipticE[(e + f*x )/2, 2] + Sqrt[Cos[e + f*x]]*(1560*Cos[e + f*x] + 780*Cos[3*(e + f*x)] + 1 56*Cos[5*(e + f*x)] - 1507*Sin[2*(e + f*x)] - 88*Sin[4*(e + f*x)] + 33*Sin [6*(e + f*x)])))/(6864*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f *x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)
Time = 3.34 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3330, 3042, 3330, 3042, 3330, 3042, 3330, 3042, 3330, 3042, 3330, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{13} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2} \sqrt {c-c \sin (e+f x)}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2} \sqrt {c-c \sin (e+f x)}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2}}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2}}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}+\frac {7}{13} c \left (\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )\right )\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x]) ^(3/2),x]
Output:
(2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(13*f*g) + (7*c*((2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^( 7/2))/(11*f*g*Sqrt[c - c*Sin[e + f*x]]) + (3*c*((-2*a*(g*Cos[e + f*x])^(5/ 2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[c - c*Sin[e + f*x]]) + (5*a*((- 2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[c - c*S in[e + f*x]]) + (11*a*((-2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x ]])/(5*f*g*Sqrt[c - c*Sin[e + f*x]]) + (7*a*((-2*a*(g*Cos[e + f*x])^(5/2)) /(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*g*Sqrt[C os[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a *Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])))/5))/7))/3))/11))/13
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[(- 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*(m + n + p))), x] + Simp[a*((2*m + p - 1)/(m + n + p)) Int[(g* Cos[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] && GtQ[m, 0] && NeQ[m + n + p, 0] && !LtQ[0, n, m] && IntegersQ[2 *m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(1181\) vs. \(2(353)=706\).
Time = 39.98 (sec) , antiderivative size = 1182, normalized size of antiderivative = 2.89
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2),x,m ethod=_RETURNVERBOSE)
Output:
-1/429/f*g/(2*2^(1/2)-3)/(1+2^(1/2))*a^3*c*(2*(-1+2*cos(1/2*f*x+1/2*e)^2)* (-11*(192*cos(1/2*f*x+1/2*e)^12+192*cos(1/2*f*x+1/2*e)^11-480*cos(1/2*f*x+ 1/2*e)^10-480*cos(1/2*f*x+1/2*e)^9+400*cos(1/2*f*x+1/2*e)^8+400*cos(1/2*f* x+1/2*e)^7-120*cos(1/2*f*x+1/2*e)^6-120*cos(1/2*f*x+1/2*e)^5-28*cos(1/2*f* x+1/2*e)^4-28*cos(1/2*f*x+1/2*e)^3+18*cos(1/2*f*x+1/2*e)^2+18*cos(1/2*f*x+ 1/2*e)-21)*sin(1/2*f*x+1/2*e)-2496*cos(1/2*f*x+1/2*e)^11-2496*cos(1/2*f*x+ 1/2*e)^10+6240*cos(1/2*f*x+1/2*e)^9+6240*cos(1/2*f*x+1/2*e)^8-6240*cos(1/2 *f*x+1/2*e)^7-6240*cos(1/2*f*x+1/2*e)^6+3120*cos(1/2*f*x+1/2*e)^5+3120*cos (1/2*f*x+1/2*e)^4-780*cos(1/2*f*x+1/2*e)^3-780*cos(1/2*f*x+1/2*e)^2+78*cos (1/2*f*x+1/2*e)+78)*2^(1/2)+22*(192*cos(1/2*f*x+1/2*e)^12+192*cos(1/2*f*x+ 1/2*e)^11-480*cos(1/2*f*x+1/2*e)^10-480*cos(1/2*f*x+1/2*e)^9+400*cos(1/2*f *x+1/2*e)^8+400*cos(1/2*f*x+1/2*e)^7-120*cos(1/2*f*x+1/2*e)^6-120*cos(1/2* f*x+1/2*e)^5-28*cos(1/2*f*x+1/2*e)^4-28*cos(1/2*f*x+1/2*e)^3+18*cos(1/2*f* x+1/2*e)^2+18*cos(1/2*f*x+1/2*e)-21)*(-1+2*cos(1/2*f*x+1/2*e)^2)*sin(1/2*f *x+1/2*e)+156*(32*cos(1/2*f*x+1/2*e)^11+32*cos(1/2*f*x+1/2*e)^10-80*cos(1/ 2*f*x+1/2*e)^9-80*cos(1/2*f*x+1/2*e)^8+80*cos(1/2*f*x+1/2*e)^7+80*cos(1/2* f*x+1/2*e)^6-40*cos(1/2*f*x+1/2*e)^5-40*cos(1/2*f*x+1/2*e)^4+10*cos(1/2*f* x+1/2*e)^3+10*cos(1/2*f*x+1/2*e)^2-cos(1/2*f*x+1/2*e)-1)*(-1+2*cos(1/2*f*x +1/2*e)^2)+462*(-2+2^(1/2)*(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)^2-4*cos(1/2*f*x+1/2*e))*(-2*(2^(1/2)*cos(1/2*f*x+1/...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.44 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {2 \, {\left (231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{3} c g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{3} c g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (78 \, a^{3} c g \cos \left (f x + e\right )^{4} + 11 \, {\left (3 \, a^{3} c g \cos \left (f x + e\right )^{4} - 5 \, a^{3} c g \cos \left (f x + e\right )^{2} - 7 \, a^{3} c g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{429 \, f} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/ 2),x, algorithm="fricas")
Output:
-2/429*(231*I*sqrt(1/2)*sqrt(a*c*g)*a^3*c*g*weierstrassZeta(-4, 0, weierst rassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*I*sqrt(1/2)*sqrt (a*c*g)*a^3*c*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + (78*a^3*c*g*cos(f*x + e)^4 + 11*(3*a^3*c*g*cos(f *x + e)^4 - 5*a^3*c*g*cos(f*x + e)^2 - 7*a^3*c*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))/f
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))** (3/2),x)
Output:
Timed out
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/ 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)^(3/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/ 2),x, algorithm="giac")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(3/2), x)
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x)) ^(3/2),x)
Output:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x)) ^(3/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx=\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, a^{3} c g \left (-\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}d x \right )-2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}d x \right )+2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )d x \right ) \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2),x)
Output:
sqrt(g)*sqrt(c)*sqrt(a)*a**3*c*g*( - 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)**4,x) - 2*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)**3,x) + 2*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),x) + int(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x),x))