Integrand size = 42, antiderivative size = 352 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^3 c^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{15 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{11 f g} \]
-2/33*a*c^2*(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)+14/99*c^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*si n(f*x+e))^(1/2)-14/45*a^3*c^2*(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)+14/15*a^3*c^2*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/co s(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)/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-2/15*a^ 2*c^2*(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)+2/11*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/ 2)/f/g
Time = 7.52 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.54 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {c (g \cos (e+f x))^{3/2} (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^{5/2} \sqrt {c-c \sin (e+f x)} \left (-3696 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (450 \cos (e+f x)+225 \cos (3 (e+f x))+45 \cos (5 (e+f x))-836 \sin (2 (e+f x))-110 \sin (4 (e+f x)))\right )}{3960 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
(c*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2) *Sqrt[c - c*Sin[e + f*x]]*(-3696*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(450*Cos[e + f*x] + 225*Cos[3*(e + f*x)] + 45*Cos[5*(e + f*x)] - 836 *Sin[2*(e + f*x)] - 110*Sin[4*(e + f*x)])))/(3960*f*Cos[e + f*x]^(3/2)*(Co s[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) ^5)
Time = 2.70 (sec) , antiderivative size = 354, 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, 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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2} \sqrt {c-c \sin (e+f x)}dx+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} c \int (g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2} \sqrt {c-c \sin (e+f x)}dx+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} c \left (\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 c (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g}+\frac {7}{11} c \left (\frac {2 c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{3} c \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\right )\right )\) |
(2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(11*f*g) + (7*c*((2*c*(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]]) + (c*((-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*S in[e + f*x]])))/5))/7))/3))/11
3.2.7.3.1 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[(- 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]
Result contains complex when optimal does not.
Time = 6.84 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.41
| method | result | size |
| default | \(-\frac {2 \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {g \cos \left (f x +e \right )}\, c \,a^{2} g \left (45 \left (\cos ^{5}\left (f x +e \right )\right )+231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+45 \left (\cos ^{4}\left (f x +e \right )\right )-55 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+462 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-462 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-55 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )-231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )-77 \cos \left (f x +e \right ) \sin \left (f x +e \right )-77 \sin \left (f x +e \right )-231 \tan \left (f x +e \right )\right )}{495 f \left (1+\cos \left (f x +e \right )\right )}\) | \(497\) |
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2),x,m ethod=_RETURNVERBOSE)
-2/495/f*(-c*(sin(f*x+e)-1))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e)) ^(1/2)*c*a^2*g/(1+cos(f*x+e))*(45*cos(f*x+e)^5+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)-231*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellipt icE(I*(csc(f*x+e)-cot(f*x+e)),I)+45*cos(f*x+e)^4-55*cos(f*x+e)^3*sin(f*x+e )+462*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellipti cF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)-462*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)*s ec(f*x+e)-55*cos(f*x+e)^2*sin(f*x+e)+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)*sec(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)*Ell ipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)^2-77*cos(f*x+e)*sin(f*x+e)- 77*sin(f*x+e)-231*tan(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.47 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} a^{2} c g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a^{2} c g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (45 \, a^{2} c g \cos \left (f x + e\right )^{4} - 11 \, {\left (5 \, a^{2} c g \cos \left (f x + e\right )^{2} + 7 \, a^{2} c g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{495 \, f} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/ 2),x, algorithm="fricas")
1/495*(-231*I*sqrt(2)*sqrt(a*c*g)*a^2*c*g*weierstrassZeta(-4, 0, weierstra ssPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 231*I*sqrt(2)*sqrt(a*c *g)*a^2*c*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) - 2*(45*a^2*c*g*cos(f*x + e)^4 - 11*(5*a^2*c*g*cos(f*x + e)^2 + 7*a^2*c*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. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx=\text {Timed out} \]
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/ 2),x, algorithm="maxima")
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/ 2),x, algorithm="giac")
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]