Integrand size = 42, antiderivative size = 463 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^4 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g} \] Output:
-154/585*a^4*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^4*c^3*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*Elli pticE(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^3*(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)-2/39*a^2*c^3*(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)-14/585*a*c^3*(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)+14/195*c^3*(g*cos(f*x+e))^(5/2)*(a +a*sin(f*x+e))^(7/2)/f/g/(c-c*sin(f*x+e))^(1/2)+22/195*c^2*(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+2/15*c*(g*cos(f*x+ e))^(5/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2)/f/g
Time = 10.51 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.49 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {a^3 c^2 (g \cos (e+f x))^{3/2} (-1+\sin (e+f x))^2 (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} \left (-14784 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (1365 \cos (e+f x)+819 \cos (3 (e+f x))+273 \cos (5 (e+f x))+39 \cos (7 (e+f x))-3794 \sin (2 (e+f x))-800 \sin (4 (e+f x))-90 \sin (6 (e+f x)))\right )}{18720 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]
Output:
-1/18720*(a^3*c^2*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-14784*Elli pticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(1365*Cos[e + f*x] + 819*Cos[3* (e + f*x)] + 273*Cos[5*(e + f*x)] + 39*Cos[7*(e + f*x)] - 3794*Sin[2*(e + f*x)] - 800*Sin[4*(e + f*x)] - 90*Sin[6*(e + f*x)])))/(f*Cos[e + f*x]^(3/2 )*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x )/2])^7)
Time = 3.82 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.02, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3330, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{15} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2} (c-c \sin (e+f x))^{3/2}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2} (c-c \sin (e+f x))^{3/2}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2} \sqrt {c-c \sin (e+f x)}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2} \sqrt {c-c \sin (e+f x)}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2}}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{7/2}}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {11}{15} c \left (\frac {7}{13} c \left (\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}\right )+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g}+\frac {11}{15} c \left (\frac {2 c (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{13 f g}+\frac {7}{13} c \left (\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}}+\frac {3}{11} c \left (\frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )\right )\right )\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x]) ^(5/2),x]
Output:
(2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x] )^(3/2))/(15*f*g) + (11*c*((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*Sin[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[Cos[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))/15
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. \(1615\) vs. \(2(399)=798\).
Time = 41.44 (sec) , antiderivative size = 1616, normalized size of antiderivative = 3.49
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x,m ethod=_RETURNVERBOSE)
Output:
-1/37440*a^3*c^2/g/f/(2*2^(1/2)-3)/(1+2^(1/2))*(975538395*(-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))*g^(3/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1) ^2)^(1/2)*ln(4*g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+ 1)^2)^(1/2)*2^(1/2)*cos(1/2*f*x+1/2*e)+4*2^(1/2)*g^(1/2)*(g*(-1+2*cos(1/2* f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)+8*cos(1/2*f*x+1/2*e)*g)+9755 38395*(2+(-cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e)-1)*2^(1/2)+2*cos(1/2* f*x+1/2*e)^2+4*cos(1/2*f*x+1/2*e))*g^(3/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/ (cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*arctanh(2^(1/2)*g^(1/2)*cos(1/2*f*x+1/2*e) /(cos(1/2*f*x+1/2*e)+1)/(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e) +1)^2)^(1/2))+29568*(-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^(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)*Elliptic F((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)*g^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)+14784*(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+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^(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)*g^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)+128*(-1+2*cos(1/2*f*x+1/...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.41 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{3} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{3} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (39 \, a^{3} c^{2} g \cos \left (f x + e\right )^{6} - {\left (45 \, a^{3} c^{2} g \cos \left (f x + e\right )^{4} + 55 \, a^{3} c^{2} g \cos \left (f x + e\right )^{2} + 77 \, a^{3} c^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{585 \, f} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="fricas")
Output:
-2/585*(231*I*sqrt(1/2)*sqrt(a*c*g)*a^3*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^3*c^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( f*x + e) - I*sin(f*x + e))) + (39*a^3*c^2*g*cos(f*x + e)^6 - (45*a^3*c^2*g *cos(f*x + e)^4 + 55*a^3*c^2*g*cos(f*x + e)^2 + 77*a^3*c^2*g)*sin(f*x + e) )*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)) /f
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))** (5/2),x)
Output:
Timed out
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(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)^(7/2)*(-c*sin(f*x + e) + c)^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(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)^(7/2)*(-c*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x)) ^(5/2),x)
Output:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x)) ^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, a^{3} c^{2} g \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}d x +\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}d x -2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}d x \right )-2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )d x +\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )d x \right ) \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(g)*sqrt(c)*sqrt(a)*a**3*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)**5,x) + int(s qrt(sin(e + f*x) + 1)*sqrt( - sin(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(si n(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)*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))