Integrand size = 42, antiderivative size = 300 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{117 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{195 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
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)-28/117*a^2*(g*cos(f*x+e))^(5/2)/c/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin( f*x+e))^(7/2)+14/195*a^2*(g*cos(f*x+e))^(5/2)/c^2/f/g/(a+a*sin(f*x+e))^(1/ 2)/(c-c*sin(f*x+e))^(5/2)+14/195*a^2*(g*cos(f*x+e))^(5/2)/c^3/f/g/(a+a*sin (f*x+e))^(1/2)/(c-c*sin(f*x+e))^(3/2)-14/195*a^2*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))/c^4/f/(a+a*sin(f*x+e) )^(1/2)/(c-c*sin(f*x+e))^(1/2)
Time = 10.41 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.55 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {14 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{3/2}}{195 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{9/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 \left (\frac {14}{195}+\frac {8}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {64}{117 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {14}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {16 \sin \left (\frac {1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {128 \sin \left (\frac {1}{2} (e+f x)\right )}{117 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{3/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{9/2}} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(9/2),x]
Output:
(-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 = 2.40 (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 |
| \(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}\right )}{3 c}\right )}{13 c}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x ])^(9/2),x]
Output:
(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)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1125\) vs. \(2(260)=520\).
Time = 28.22 (sec) , antiderivative size = 1126, normalized size of antiderivative = 3.75
Input:
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)
Output:
-1/585/f*g/(2*2^(1/2)-3)/(1+2^(1/2))*a/c^4*(124+21*(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)*(4*cos(1/2*f*x+1/2*e)^4-4*cos (1/2*f*x+1/2*e)^2-3)*sin(1/2*f*x+1/2*e)-12*cos(1/2*f*x+1/2*e)^4+12*cos(1/2 *f*x+1/2*e)^2+1)*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)*(-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)+42*(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)*(4*cos(1/2*f *x+1/2*e)^4-4*cos(1/2*f*x+1/2*e)^2-3)*sin(1/2*f*x+1/2*e)-12*cos(1/2*f*x+1/ 2*e)^4+12*cos(1/2*f*x+1/2*e)^2+1)*2^(1/2)-4*cos(1/2*f*x+1/2*e)*(4*cos(1/2* f*x+1/2*e)^4-4*cos(1/2*f*x+1/2*e)^2-3)*sin(1/2*f*x+1/2*e)+24*cos(1/2*f*x+1 /2*e)^4-24*cos(1/2*f*x+1/2*e)^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)*(-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)*El lipticF((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*(-62+3*(-7+168*cos(1/2*f*x+1/2*e)^5+84*cos(1/2*f*x+1/2*e)^4-168*cos(1/2* f*x+1/2*e)^3-148*cos(1/2*f*x+1/2*e)^2-78*cos(1/2*f*x+1/2*e))*sin(1/2*f*x+1 /2*e)+336*cos(1/2*f*x+1/2*e)^8+168*cos(1/2*f*x+1/2*e)^7-672*cos(1/2*f*x+1/ 2*e)^6+472*cos(1/2*f*x+1/2*e)^5+892*cos(1/2*f*x+1/2*e)^4-766*cos(1/2*f*x+1 /2*e)^3-556*cos(1/2*f*x+1/2*e)^2+64*cos(1/2*f*x+1/2*e))*2^(1/2)+6*(7-16...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.06 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {2 \, {\left (21 \, \sqrt {\frac {1}{2}} {\left (-i \, a g \cos \left (f x + e\right )^{4} + 8 i \, a g \cos \left (f x + e\right )^{2} - 8 i \, a g + 4 \, {\left (-i \, a g \cos \left (f x + e\right )^{2} + 2 i \, a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {\frac {1}{2}} {\left (i \, a g \cos \left (f x + e\right )^{4} - 8 i \, a g \cos \left (f x + e\right )^{2} + 8 i \, a g + 4 \, {\left (i \, a g \cos \left (f x + e\right )^{2} - 2 i \, a g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (84 \, a g \cos \left (f x + e\right )^{2} - 146 \, a g - {\left (21 \, a g \cos \left (f x + e\right )^{2} + 34 \, a g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{585 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
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")
Output:
-2/585*(21*sqrt(1/2)*(-I*a*g*cos(f*x + e)^4 + 8*I*a*g*cos(f*x + e)^2 - 8*I *a*g + 4*(-I*a*g*cos(f*x + e)^2 + 2*I*a*g)*sin(f*x + e))*sqrt(a*c*g)*weier strassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e) )) + 21*sqrt(1/2)*(I*a*g*cos(f*x + e)^4 - 8*I*a*g*cos(f*x + e)^2 + 8*I*a*g + 4*(I*a*g*cos(f*x + e)^2 - 2*I*a*g)*sin(f*x + e))*sqrt(a*c*g)*weierstras sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + (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*sin(f*x + e) + c))/(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. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
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)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
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")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(3/2)/(-c*sin(f*x + e) + c)^(9/2), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
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")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x ))^(9/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x ))^(9/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {too large to display} \] Input:
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)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*a*g*(64*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f* x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x) + 16*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x)) - 12*int((sqrt(sin(e + f*x) + 1)*sq rt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*sin(e + f*x)**4*f + 48*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 - 2*sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*sin(e + f*x)**3*f - 72*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sq rt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*si n(e + f*x) - 1),x)*sin(e + f*x)**2*f + 48*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)**4 - 2*sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*sin(e + f*x)*f - 12*int((sqrt(s in(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x) )/(sin(e + f*x)**4 - 2*sin(e + f*x)**3 + 2*sin(e + f*x) - 1),x)*f + 35*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)**5 - 3*cos(e + f*x)*sin(e + f*x)** 4 + 2*cos(e + f*x)*sin(e + f*x)**3 + 2*cos(e + f*x)*sin(e + f*x)**2 - 3*co s(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f*x)**4*f - 140*int((sq rt(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)**5 - 3*cos(e + f*x)*sin(e + f*x)**4...