Integrand size = 42, antiderivative size = 406 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {154 a^3 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^3 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{195 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^3 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{39 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{117 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{13 f g}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{13 f g} \] Output:
154/585*a^3*c^3*(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)+154/195*a^3*c^3*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*Ellip ticE(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)+22/195*a^3*c^2*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/2)/f/g/(a+a *sin(f*x+e))^(1/2)+2/39*a^3*c*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2)/ f/g/(a+a*sin(f*x+e))^(1/2)-14/117*a^3*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e) )^(5/2)/f/g/(a+a*sin(f*x+e))^(1/2)-2/13*a^2*(g*cos(f*x+e))^(5/2)*(a+a*sin( f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2)/f/g-2/13*a*(g*cos(f*x+e))^(5/2)*(a+a* sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2)/f/g
Time = 4.63 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.30 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {a^2 c^2 (g \cos (e+f x))^{3/2} \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} \left (7392 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (1897 \sin (2 (e+f x))+400 \sin (4 (e+f x))+45 \sin (6 (e+f x)))\right )}{9360 f \cos ^{\frac {5}{2}}(e+f x)} \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2),x]
Output:
(a^2*c^2*(g*Cos[e + f*x])^(3/2)*Sqrt[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]]*(1897*Sin[2 *(e + f*x)] + 400*Sin[4*(e + f*x)] + 45*Sin[6*(e + f*x)])))/(9360*f*Cos[e + f*x]^(5/2))
Time = 3.36 (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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{13} a \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2} (c-c \sin (e+f x))^{5/2}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2} (c-c \sin (e+f x))^{5/2}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \int (g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \int (g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {11}{13} a \left (\frac {7}{11} a \left (\frac {1}{3} a \left (\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {11}{7} c \left (\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}+\frac {7}{5} c \left (\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )\right )\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{13 f g}\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x]) ^(5/2),x]
Output:
(-2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x ])^(5/2))/(13*f*g) + (11*a*((-2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))/(11*f*g) + (7*a*((-2*a*(g*Cos[e + f*x] )^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]) + (a* ((2*c*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a *Sin[e + f*x]]) + (11*c*((2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f* x]])/(5*f*g*Sqrt[a + a*Sin[e + f*x]]) + (7*c*((2*c*(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*c*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. \(890\) vs. \(2(350)=700\).
Time = 39.38 (sec) , antiderivative size = 891, normalized size of antiderivative = 2.19
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2),x,m ethod=_RETURNVERBOSE)
Output:
-1/585*a^2*c^2*g/f/(2*2^(1/2)-3)/(1+2^(1/2))*(2*sin(1/2*f*x+1/2*e)*(231+(5 760*cos(1/2*f*x+1/2*e)^14+5760*cos(1/2*f*x+1/2*e)^13-17280*cos(1/2*f*x+1/2 *e)^12-17280*cos(1/2*f*x+1/2*e)^11+23360*cos(1/2*f*x+1/2*e)^10+23360*cos(1 /2*f*x+1/2*e)^9-17920*cos(1/2*f*x+1/2*e)^8-17920*cos(1/2*f*x+1/2*e)^7+8656 *cos(1/2*f*x+1/2*e)^6+8656*cos(1/2*f*x+1/2*e)^5-2576*cos(1/2*f*x+1/2*e)^4- 2576*cos(1/2*f*x+1/2*e)^3+816*cos(1/2*f*x+1/2*e)^2+354*cos(1/2*f*x+1/2*e)- 231)*2^(1/2)-5760*cos(1/2*f*x+1/2*e)^14-5760*cos(1/2*f*x+1/2*e)^13+17280*c os(1/2*f*x+1/2*e)^12+17280*cos(1/2*f*x+1/2*e)^11-23360*cos(1/2*f*x+1/2*e)^ 10-23360*cos(1/2*f*x+1/2*e)^9+17920*cos(1/2*f*x+1/2*e)^8+17920*cos(1/2*f*x +1/2*e)^7-8656*cos(1/2*f*x+1/2*e)^6-8656*cos(1/2*f*x+1/2*e)^5+2576*cos(1/2 *f*x+1/2*e)^4+2576*cos(1/2*f*x+1/2*e)^3-816*cos(1/2*f*x+1/2*e)^2-354*cos(1 /2*f*x+1/2*e))+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 /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)*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)+231*(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*2^(1/2)*Ell ipticE((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)*(-2*(2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.42 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{2} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{2} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (45 \, a^{2} c^{2} g \cos \left (f x + e\right )^{4} + 55 \, a^{2} c^{2} g \cos \left (f x + e\right )^{2} + 77 \, a^{2} c^{2} g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )\right )}}{585 \, f} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="fricas")
Output:
-2/585*(231*I*sqrt(1/2)*sqrt(a*c*g)*a^2*c^2*g*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*I*sqrt(1/2)*sq rt(a*c*g)*a^2*c^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( f*x + e) - I*sin(f*x + e))) - (45*a^2*c^2*g*cos(f*x + e)^4 + 55*a^2*c^2*g* cos(f*x + e)^2 + 77*a^2*c^2*g)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/f
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(5/2)*(c-c*sin(f*x+e))** (5/2),x)
Output:
Timed out
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(5/2)*(-c*sin(f*x + e) + c)^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="giac")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(5/2)*(-c*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x)) ^(5/2),x)
Output:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x)) ^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, a^{2} c^{2} g \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}d x -2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )d x \right ) \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(g)*sqrt(c)*sqrt(a)*a**2*c**2*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)**2,x) + int(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f* x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x),x))