Integrand size = 42, antiderivative size = 350 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 a^2 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
-2/5*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2 )-14/5*(g*cos(f*x+e))^(5/2)/a/f/g/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^ (7/2)+14/9*(g*cos(f*x+e))^(5/2)/a^2/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f* x+e))^(7/2)+14/15*(g*cos(f*x+e))^(5/2)/a^2/c/f/g/(a+a*sin(f*x+e))^(1/2)/(c -c*sin(f*x+e))^(5/2)+14/15*(g*cos(f*x+e))^(5/2)/a^2/c^2/f/g/(a+a*sin(f*x+e ))^(1/2)/(c-c*sin(f*x+e))^(3/2)-14/15*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))/a^2/c^3/f/(a+a*sin(f*x+e))^(1/2) /(c-c*sin(f*x+e))^(1/2)
Time = 8.68 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.49 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {\cos (e+f x)} (g \cos (e+f x))^{3/2} \left (42 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (-3 \cos (e+f x)-\cos (3 (e+f x))+4 \cos ^3(e+f x) \sin (e+f x)\right )+\sqrt {\cos (e+f x)} (-9+28 \cos (2 (e+f x))+21 \cos (4 (e+f x))+98 \sin (e+f x)+42 \sin (3 (e+f x)))\right )}{180 c^3 f (-1+\sin (e+f x))^3 (a (1+\sin (e+f x)))^{5/2} \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)),x]
Output:
-1/180*(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(42*EllipticE[(e + f*x)/ 2, 2]*(-3*Cos[e + f*x] - Cos[3*(e + f*x)] + 4*Cos[e + f*x]^3*Sin[e + f*x]) + Sqrt[Cos[e + f*x]]*(-9 + 28*Cos[2*(e + f*x)] + 21*Cos[4*(e + f*x)] + 98 *Sin[e + f*x] + 42*Sin[3*(e + f*x)])))/(c^3*f*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])
Time = 2.93 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3331, 3042, 3331, 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.
| \(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {7 \int \frac {(g \cos (e+f x))^{3/2}}{(\sin (e+f x) a+a)^{3/2} (c-c \sin (e+f x))^{7/2}}dx}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \int \frac {(g \cos (e+f x))^{3/2}}{(\sin (e+f x) a+a)^{3/2} (c-c \sin (e+f x))^{7/2}}dx}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {7 \left (\frac {5 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{7/2}}dx}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}}{3 c}+\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {7 \left (\frac {5 \left (\frac {2 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}}{3 c}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}\right )}{5 a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}}\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x] )^(7/2)),x]
Output:
(-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)) + (7*((-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x] )^(3/2)*(c - c*Sin[e + f*x])^(7/2)) + (5*((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)))/a))/(5*a)
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*((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. \(1336\) vs. \(2(302)=604\).
Time = 32.73 (sec) , antiderivative size = 1337, normalized size of antiderivative = 3.82
Input:
int((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2),x,m ethod=_RETURNVERBOSE)
Output:
1/45/f*g/(1+2^(1/2))/(2*2^(1/2)-3)/a^2/c^3*(-10+21*(-1+2*cos(1/2*f*x+1/2*e )*(4*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/2*e)^5-8*cos(1/2*f*x+1/2*e)^3-3* cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*sin(1/2*f*x+1/2*e)-4*cos(1/2* f*x+1/2*e)^6-8*cos(1/2*f*x+1/2*e)^5+8*cos(1/2*f*x+1/2*e)^3+3*cos(1/2*f*x+1 /2*e)^2-2*cos(1/2*f*x+1/2*e))*2^(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)*EllipticE((1+2^(1/2 ))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3)*(-2*(2^(1/2)*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)+42*(2+(-1+2*cos(1/2*f*x+1/2*e)*(4*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/ 2*e)^5-8*cos(1/2*f*x+1/2*e)^3-3*cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+ 1)*sin(1/2*f*x+1/2*e)-4*cos(1/2*f*x+1/2*e)^6-8*cos(1/2*f*x+1/2*e)^5+8*cos( 1/2*f*x+1/2*e)^3+3*cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e))*2^(1/2)-4*co s(1/2*f*x+1/2*e)*(4*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/2*e)^5-8*cos(1/2* f*x+1/2*e)^3-3*cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*sin(1/2*f*x+1/ 2*e)+8*cos(1/2*f*x+1/2*e)^6+16*cos(1/2*f*x+1/2*e)^5-16*cos(1/2*f*x+1/2*e)^ 3-6*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)*Ellipti cF((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*(5+7*(3+24*cos(1/2*f*x+1/2*e)^5+12*cos(1/2*f*x+1/2*e)^4-2...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.71 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left (21 \, \sqrt {\frac {1}{2}} {\left (-i \, g \cos \left (f x + e\right )^{4} \sin \left (f x + e\right ) + i \, g \cos \left (f x + e\right )^{4}\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {\frac {1}{2}} {\left (i \, g \cos \left (f x + e\right )^{4} \sin \left (f x + e\right ) - i \, g \cos \left (f x + e\right )^{4}\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (21 \, g \cos \left (f x + e\right )^{4} - 14 \, g \cos \left (f x + e\right )^{2} + 7 \, {\left (3 \, g \cos \left (f x + e\right )^{2} + g\right )} \sin \left (f x + e\right ) - 2 \, g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{45 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{4} \sin \left (f x + e\right ) - a^{3} c^{4} f \cos \left (f x + e\right )^{4}\right )}} \] Input:
integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/ 2),x, algorithm="fricas")
Output:
-2/45*(21*sqrt(1/2)*(-I*g*cos(f*x + e)^4*sin(f*x + e) + I*g*cos(f*x + e)^4 )*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*sqrt(1/2)*(I*g*cos(f*x + e)^4*sin(f*x + e) - I* g*cos(f*x + e)^4)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(- 4, 0, cos(f*x + e) - I*sin(f*x + e))) + (21*g*cos(f*x + e)^4 - 14*g*cos(f* x + e)^2 + 7*(3*g*cos(f*x + e)^2 + g)*sin(f*x + e) - 2*g)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a^3*c^4*f*cos(f* x + e)^4*sin(f*x + e) - a^3*c^4*f*cos(f*x + e)^4)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)/(a+a*sin(f*x+e))**(5/2)/(c-c*sin(f*x+e))** (7/2),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/ 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)^(7/2)), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/ 2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x) )^(7/2)),x)
Output:
int((g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x) )^(7/2)), x)
\[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\text {too large to display} \] Input:
int((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*g*( - 16*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f *x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**4 + 16*sqrt(sin(e + f*x) + 1)*sq rt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3 + 24*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**2 - 24*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*s in(e + f*x) - 6*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos( e + f*x)) - 8*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(c os(e + f*x))*sin(e + f*x)**3)/(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x) *sin(e + f*x)**2 - cos(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f* x)**5*f + 8*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos (e + f*x))*sin(e + f*x)**3)/(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*s in(e + f*x)**2 - cos(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f*x) **4*f + 16*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos( e + f*x))*sin(e + f*x)**3)/(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*si n(e + f*x)**2 - cos(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f*x)* *3*f - 16*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3)/(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*sin (e + f*x)**2 - cos(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f*x)** 2*f - 8*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3)/(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*si...