\(\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx\) [116]

Optimal result

Integrand size = 42, antiderivative size = 463 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^4 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g} \]

[Out]

2/15*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2)/f/g-2/39*a^2*c^3*(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)-14/585*a*c^3*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f
/g/(c-c*sin(f*x+e))^(1/2)+14/195*c^3*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(7/2)/f/g/(c-c*sin(f*x+e))^(1/2)-15
4/585*a^4*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)+154/195*a^4*c^3*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)/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-22/195*a^3*c^3*(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)+22/195*c^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2)/f
/g

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2930, 2921, 2721, 2719} \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^4 c^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{195 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g}+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g} \]

[In]

Int[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(-154*a^4*c^3*(g*Cos[e + f*x])^(5/2))/(585*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (154*a^4*c
^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(195*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c
- c*Sin[e + f*x]]) - (22*a^3*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(195*f*g*Sqrt[c - c*Sin[e +
f*x]]) - (2*a^2*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(39*f*g*Sqrt[c - c*Sin[e + f*x]]) - (1
4*a*c^3*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(585*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^3*(g*Cos
[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(7/2))/(195*f*g*Sqrt[c - c*Sin[e + f*x]]) + (22*c^2*(g*Cos[e + f*x])^(5/
2)*(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]])/(195*f*g) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e +
f*x])^(7/2)*(c - c*Sin[e + f*x])^(3/2))/(15*f*g)

Rule 2719

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]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2921

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)]]), x_Symbol] :> Dist[g*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), In
t[(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]

Rule 2930

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {1}{15} (11 c) \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2} \, dx \\ & = \frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {1}{195} \left (77 c^2\right ) \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {1}{65} \left (7 c^3\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {1}{39} \left (7 a c^3\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {1}{39} \left (11 a^2 c^3\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {1}{195} \left (77 a^3 c^3\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {1}{195} \left (77 a^4 c^3\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {\left (77 a^4 c^3 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{195 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g}+\frac {\left (77 a^4 c^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{195 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^4 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 )}{195 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 f g \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {14 c^3 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)}}{195 f g}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{3/2}}{15 f g} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.47 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.49 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {a^3 c^2 (g \cos (e+f x))^{3/2} (-1+\sin (e+f x))^2 (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} \left (-14784 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (1365 \cos (e+f x)+819 \cos (3 (e+f x))+273 \cos (5 (e+f x))+39 \cos (7 (e+f x))-3794 \sin (2 (e+f x))-800 \sin (4 (e+f x))-90 \sin (6 (e+f x)))\right )}{18720 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]

[In]

Integrate[(g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

-1/18720*(a^3*c^2*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]
*Sqrt[c - c*Sin[e + f*x]]*(-14784*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(1365*Cos[e + f*x] + 819*Cos[
3*(e + f*x)] + 273*Cos[5*(e + f*x)] + 39*Cos[7*(e + f*x)] - 3794*Sin[2*(e + f*x)] - 800*Sin[4*(e + f*x)] - 90*
Sin[6*(e + f*x)])))/(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])^7)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 11.98 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.15

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^{2} a^{3} g \left (39 \left (\cos ^{7}\left (f x +e \right )\right )+39 \left (\cos ^{6}\left (f x +e \right )\right )-45 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-45 \left (\cos ^{4}\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 )-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 )-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 )}{585 f \left (1+\cos \left (f x +e \right )\right )}\)	\(531\)

[In]

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/585/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^2*a^3*g/(1+cos(f*x+e))*(39*
cos(f*x+e)^7+39*cos(f*x+e)^6-45*cos(f*x+e)^5*sin(f*x+e)-45*cos(f*x+e)^4*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)-231*I*(1/(1+cos(f*x+e)))^(1/2)*(c
os(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)-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)*EllipticF(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)*sec(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)*EllipticE(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))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.41 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} a^{3} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a^{3} c^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - 2 \, {\left (39 \, a^{3} c^{2} g \cos \left (f x + e\right )^{6} - {\left (45 \, a^{3} c^{2} g \cos \left (f x + e\right )^{4} + 55 \, a^{3} c^{2} g \cos \left (f x + e\right )^{2} + 77 \, a^{3} c^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{585 \, f} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/585*(-231*I*sqrt(2)*sqrt(a*c*g)*a^3*c^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I
*sin(f*x + e))) + 231*I*sqrt(2)*sqrt(a*c*g)*a^3*c^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*
x + e) - I*sin(f*x + e))) - 2*(39*a^3*c^2*g*cos(f*x + e)^6 - (45*a^3*c^2*g*cos(f*x + e)^4 + 55*a^3*c^2*g*cos(f
*x + e)^2 + 77*a^3*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

Sympy [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(5/2), x)

Giac [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{5/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

[In]

int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(5/2),x)

[Out]

int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(5/2), x)