Integrand size = 42, antiderivative size = 300 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{117 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{195 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
-28/117*a^2*(g*cos(f*x+e))^(5/2)/c/f/g/(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x +e))^(1/2)+14/195*a^2*(g*cos(f*x+e))^(5/2)/c^2/f/g/(c-c*sin(f*x+e))^(5/2)/ (a+a*sin(f*x+e))^(1/2)+14/195*a^2*(g*cos(f*x+e))^(5/2)/c^3/f/g/(c-c*sin(f* x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)+4/13*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f* x+e))^(1/2)/f/g/(c-c*sin(f*x+e))^(9/2)-14/195*a^2*g*(cos(1/2*f*x+1/2*e)^2) ^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e) ^(1/2)*(g*cos(f*x+e))^(1/2)/c^4/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^ (1/2)
Time = 10.35 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.55 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {14 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{3/2}}{195 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{9/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 \left (\frac {14}{195}+\frac {8}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {64}{117 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {14}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {16 \sin \left (\frac {1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {128 \sin \left (\frac {1}{2} (e+f x)\right )}{117 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{3/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{9/2}} \]
(-14*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(3/2))/(195*f*Cos[e + f*x]^(3/2 )*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(9/2)) + (( g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9 *(14/195 + 8/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) - 64/(117*(Cos[( e + f*x)/2] - Sin[(e + f*x)/2])^4) + 14/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (16*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/ 2])^7) - (128*Sin[(e + f*x)/2])/(117*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) ^5) + (28*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (28*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(3/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c *Sin[e + f*x])^(9/2))
Time = 2.28 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3329, 3042, 3329, 3042, 3331, 3042, 3331, 3042, 3321, 3042, 3121, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3331 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{3 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}\right )}{3 c}\right )}{13 c}\) |
(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (7*a*((4*a*(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)) - (a*((2*(g*Cos[e + f*x])^(5/2))/(5 *f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + ((2*(g*Cos[e + f*x])^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c *f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))/(5*c)))/(3*c)))/(13 *c)
3.2.4.3.1 Defintions of rubi rules used
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* (Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) Int[(g *Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2 *b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* x])^n/(f*g*(2*n + p + 1))), x] - Simp[b*((2*m + p - 1)/(d*(2*n + p + 1))) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^( n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] & & EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && In tegersQ[2*m, 2*n, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b* (g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(a* f*g*(2*m + p + 1))), x] + Simp[(m + n + p + 1)/(a*(2*m + p + 1)) Int[(g*C os[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && !LtQ[m, n, -1] && Integers Q[2*m, 2*n, 2*p]
Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 1389, normalized size of antiderivative = 4.63
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/2),x,m ethod=_RETURNVERBOSE)
-2/585*I/f*(g*cos(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*g*a/(cos(f*x+e)^2 +2*sin(f*x+e)-2)/(-c*(sin(f*x+e)-1))^(1/2)/c^4/(1+cos(f*x+e))*(-21*(1/(1+c os(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e )-cot(f*x+e)),I)+21*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^( 1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)-21*sin(f*x+e)*(cos(f*x+e)/(1+c os(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)) )^(1/2)*cos(f*x+e)^2+21*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellip ticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-42 *(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)* (1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)*sin(f*x+e)+42*(cos(f*x+e)/(1+cos(f*x+e )))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)* cos(f*x+e)*sin(f*x+e)+63*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(cs c(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-63*(cos(f*x+ e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos( f*x+e)))^(1/2)*cos(f*x+e)^2+126*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Elliptic E(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)+63*sin( f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e )),I)*(1/(1+cos(f*x+e)))^(1/2)-126*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellip ticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)-63*s in(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.14 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {2 \, {\left (84 \, a g \cos \left (f x + e\right )^{2} - 146 \, a g - {\left (21 \, a g \cos \left (f x + e\right )^{2} + 34 \, a g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 21 \, {\left (-i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} + 8 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} - 8 i \, \sqrt {2} a g + 4 \, {\left (-i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, {\left (i \, \sqrt {2} a g \cos \left (f x + e\right )^{4} - 8 i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} + 8 i \, \sqrt {2} a g + 4 \, {\left (i \, \sqrt {2} a g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{585 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="fricas")
-1/585*(2*(84*a*g*cos(f*x + e)^2 - 146*a*g - (21*a*g*cos(f*x + e)^2 + 34*a *g)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*si n(f*x + e) + c) + 21*(-I*sqrt(2)*a*g*cos(f*x + e)^4 + 8*I*sqrt(2)*a*g*cos( f*x + e)^2 - 8*I*sqrt(2)*a*g + 4*(-I*sqrt(2)*a*g*cos(f*x + e)^2 + 2*I*sqrt (2)*a*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInve rse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*(I*sqrt(2)*a*g*cos(f*x + e )^4 - 8*I*sqrt(2)*a*g*cos(f*x + e)^2 + 8*I*sqrt(2)*a*g + 4*(I*sqrt(2)*a*g* cos(f*x + e)^2 - 2*I*sqrt(2)*a*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZet a(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(c^5* f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^2 + 8*c^5*f + 4*(c^5*f*cos(f*x + e )^2 - 2*c^5*f)*sin(f*x + e))
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="maxima")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="giac")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]