Integrand size = 42, antiderivative size = 300 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {154 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{13 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
4/13*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(9 /2)-20/39*a^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/c/f/g/(c-c*sin(f *x+e))^(7/2)-308/39*a^4*(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)+44/39*a^3*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^( 1/2)/c^2/f/g/(c-c*sin(f*x+e))^(5/2)+154/13*a^4*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 = 15.29 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.55 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {154 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{13 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{9/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 \left (-\frac {128}{13}+\frac {32}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {224}{39 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {80}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {448 \sin \left (\frac {1}{2} (e+f x)\right )}{39 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {160 \sin \left (\frac {1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {256 \sin \left (\frac {1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{9/2}} \]
(154*(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]))^(7/2))/(13*f*Cos[e + f*x]^(3/2) *(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(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* (-128/13 + 32/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) - 224/(39*(Cos[ (e + f*x)/2] - Sin[(e + f*x)/2])^4) + 80/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/ 2])^7) - (448*Sin[(e + f*x)/2])/(39*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^ 5) + (160*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (256*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c* Sin[e + f*x])^(9/2))
Time = 2.26 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.02, 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, 3329, 3042, 3329, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{7/2}}dx}{13 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{5/2}}dx}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{5/2}}dx}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{3/2}}dx}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{3/2}}dx}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {3 a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {3 a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {3 a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {3 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {3 a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {15 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {7 a \left (\frac {4 a (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {6 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{5 c}\right )}{9 c}\right )}{13 c}\) |
(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(13*f*g*(c - c*Sin [e + f*x])^(9/2)) - (15*a*((4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x] )^(3/2))/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (11*a*((4*a*(g*Cos[e + f*x]) ^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (7*a *((4*a*(g*Cos[e + f*x])^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (6*a*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)))/(9*c)))/(13*c)
3.2.23.3.1 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]
Result contains complex when optimal does not.
Time = 2.95 (sec) , antiderivative size = 3799, normalized size of antiderivative = 12.66
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/2),x,m ethod=_RETURNVERBOSE)
2/39/f*(g*cos(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*g*a^3/(cos(f*x+e)^2+c os(f*x+e)*sin(f*x+e)-cos(f*x+e)+2*sin(f*x+e)-2)/(-c*(sin(f*x+e)-1))^(1/2)/ c^4*(-86-231*I*(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)*cos(f*x+e)^2+231*I*(1/(1+cos(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)*cos(f*x+e)^2+78*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f *x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e))) *(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)*sin(f*x+e)-78*ln((2*(-cos (f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2 )^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2) *cos(f*x+e)*sin(f*x+e)-1092*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*c os(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+ e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)*tan(f*x+e)+1092*ln((2 *(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x +e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2) ^(3/2)*sec(f*x+e)*tan(f*x+e)+39*cos(f*x+e)^2-78*ln(2*(2*(-cos(f*x+e)/(1+co s(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f *x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)^3 -812*tan(f*x+e)-48*sec(f*x+e)*tan(f*x+e)-231*I*EllipticE(I*(csc(f*x+e)-cot (f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.23 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {16 \, {\left (57 \, a^{3} g \cos \left (f x + e\right )^{2} - 74 \, a^{3} g - 8 \, {\left (3 \, a^{3} g \cos \left (f x + e\right )^{2} - 10 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 231 \, {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 8 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 8 i \, \sqrt {2} a^{3} g + 4 \, {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 \, {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 8 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 8 i \, \sqrt {2} a^{3} g + 4 \, {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{39 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="fricas")
1/39*(16*(57*a^3*g*cos(f*x + e)^2 - 74*a^3*g - 8*(3*a^3*g*cos(f*x + e)^2 - 10*a^3*g)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqr t(-c*sin(f*x + e) + c) - 231*(I*sqrt(2)*a^3*g*cos(f*x + e)^4 - 8*I*sqrt(2) *a^3*g*cos(f*x + e)^2 + 8*I*sqrt(2)*a^3*g + 4*(I*sqrt(2)*a^3*g*cos(f*x + e )^2 - 2*I*sqrt(2)*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*(-I*sqrt( 2)*a^3*g*cos(f*x + e)^4 + 8*I*sqrt(2)*a^3*g*cos(f*x + e)^2 - 8*I*sqrt(2)*a ^3*g + 4*(-I*sqrt(2)*a^3*g*cos(f*x + e)^2 + 2*I*sqrt(2)*a^3*g)*sin(f*x + e ))*sqrt(a*c*g)*weierstrassZeta(-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. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="maxima")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/ 2),x, algorithm="giac")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]