Integrand size = 42, antiderivative size = 352 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {14 a^2 c^3 (g \cos (e+f x))^{5/2}}{45 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^2 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{15 f g \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{33 f g \sqrt {a+a \sin (e+f x)}}-\frac {14 a^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{99 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}{11 f g} \] Output:
14/45*a^2*c^3*(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^2*c^3*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))/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)*(c-c*sin(f*x+e))^(1/2)/f/g/(a+a*sin(f *x+e))^(1/2)+2/33*a^2*c*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2)/f/g/(a +a*sin(f*x+e))^(1/2)-14/99*a^2*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2) /f/g/(a+a*sin(f*x+e))^(1/2)-2/11*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^( 1/2)*(c-c*sin(f*x+e))^(5/2)/f/g
Time = 7.94 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.55 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^2 (g \cos (e+f x))^{3/2} (-1+\sin (e+f x))^2 (a (1+\sin (e+f x)))^{3/2} \sqrt {c-c \sin (e+f x)} \left (3696 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (450 \cos (e+f x)+225 \cos (3 (e+f x))+45 \cos (5 (e+f x))+836 \sin (2 (e+f x))+110 \sin (4 (e+f x)))\right )}{3960 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \] Input:
Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(5/2),x]
Output:
(c^2*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])^2*(a*(1 + Sin[e + f*x]))^( 3/2)*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)* (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/ 2])^3)
Time = 2.73 (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.
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} a \int (g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} a \int (g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {11}{7} c \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )+\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {7}{11} a \left (\frac {1}{3} a \left (\frac {2 c (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}+\frac {11}{7} c \left (\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}+\frac {7}{5} c \left (\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )\right )\right )-\frac {2 a (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{11 f g}\) |
Input:
Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x]) ^(5/2),x]
Output:
(-2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x]) ^(5/2))/(11*f*g) + (7*a*((-2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x]) ^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]) + (a*((2*c*(g*Cos[e + f*x])^(5/2) *(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a*Sin[e + f*x]]) + (11*c*((2* c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*f*g*Sqrt[a + a*Sin[e + f*x]]) + (7*c*((2*c*(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*c*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*Si n[e + f*x]])))/5))/7))/3))/11
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1410\) vs. \(2(304)=608\).
Time = 25.00 (sec) , antiderivative size = 1411, normalized size of antiderivative = 4.01
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2),x,m ethod=_RETURNVERBOSE)
Output:
1/1980/g*a*c^2/f/(2*2^(1/2)-3)/(1+2^(1/2))*(1659735*(-2+2^(1/2)*(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)^2-4*cos(1/2*f*x+1 /2*e))*g^(3/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1 /2)*ln(4*g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^ (1/2)*2^(1/2)*cos(1/2*f*x+1/2*e)+4*2^(1/2)*g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/ 2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)+8*cos(1/2*f*x+1/2*e)*g)+1659735*(2 +(-cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e)-1)*2^(1/2)+2*cos(1/2*f*x+1/2* e)^2+4*cos(1/2*f*x+1/2*e))*g^(3/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2 *f*x+1/2*e)+1)^2)^(1/2)*arctanh(2^(1/2)*g^(1/2)*cos(1/2*f*x+1/2*e)/(cos(1/ 2*f*x+1/2*e)+1)/(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^( 1/2))+1848*(2+(-cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e)-1)*2^(1/2)+2*cos (1/2*f*x+1/2*e)^2+4*cos(1/2*f*x+1/2*e))*((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)*EllipticF((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)*g^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)+924*(-cos(1/2*f*x+1/2*e)^2-2*cos(1/2*f*x+1/2*e)-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)*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)*g^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)+8*(-1+2*cos(1/2*f*x+1/2*e)^2)*((-1...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.47 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {2 \, {\left (231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (45 \, a c^{2} g \cos \left (f x + e\right )^{4} + 11 \, {\left (5 \, a c^{2} g \cos \left (f x + e\right )^{2} + 7 \, a c^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{495 \, f} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="fricas")
Output:
-2/495*(231*I*sqrt(1/2)*sqrt(a*c*g)*a*c^2*g*weierstrassZeta(-4, 0, weierst rassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*I*sqrt(1/2)*sqrt (a*c*g)*a*c^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) - (45*a*c^2*g*cos(f*x + e)^4 + 11*(5*a*c^2*g*cos(f *x + e)^2 + 7*a*c^2*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. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(3/2)*(c-c*sin(f*x+e))** (5/2),x)
Output:
Timed out
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/ 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)^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/ 2),x, algorithm="giac")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(3/2)*(-c*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x)) ^(5/2),x)
Output:
int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2)*(c - c*sin(e + f*x)) ^(5/2), x)
\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2} \, dx=\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, a \,c^{2} g \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}d x -\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}d x \right )-\left (\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )d x \right ) \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2),x)
Output:
sqrt(g)*sqrt(c)*sqrt(a)*a*c**2*g*(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)**3,x) - 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)**2,x) - 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),x) + int(sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x),x))