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

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

[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

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Rubi [A]  time = 2.39, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac {154 a^4 c^3 (g \cos (e+f x))^{5/2}}{585 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 c^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{195 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 c^3 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 f g \sqrt {c-c \sin (e+f x)}}+\frac {154 a^4 c^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{195 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {14 a c^3 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{585 f g \sqrt {c-c \sin (e+f x)}}+\frac {22 c^2 (a \sin (e+f x)+a)^{7/2} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{195 f g}+\frac {14 c^3 (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{195 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{15 f g} \]

Antiderivative was successfully verified.

[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 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2842

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 2851

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] && Eq
Q[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*} \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 {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*}

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Mathematica [A]  time = 3.52, size = 226, normalized size = 0.49 \[ -\frac {a^3 c^2 (\sin (e+f x)-1)^2 (\sin (e+f x)+1)^3 \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (\sqrt {\cos (e+f x)} (-3794 \sin (2 (e+f x))-800 \sin (4 (e+f x))-90 \sin (6 (e+f x))+1365 \cos (e+f x)+819 \cos (3 (e+f x))+273 \cos (5 (e+f x))+39 \cos (7 (e+f x)))-14784 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right )}{18720 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7} \]

Antiderivative was successfully verified.

[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)

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{3} c^{2} g \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) + a^{3} c^{2} g \cos \left (f x + e\right )^{5}\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

integral((a^3*c^2*g*cos(f*x + e)^5*sin(f*x + e) + a^3*c^2*g*cos(f*x + e)^5)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*
x + e) + a)*sqrt(-c*sin(f*x + e) + c), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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maple [C]  time = 0.86, size = 392, normalized size = 0.85 \[ -\frac {2 \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} \left (39 \left (\cos ^{8}\left (f x +e \right )\right ) \sin \left (f x +e \right )+45 \left (\cos ^{8}\left (f x +e \right )\right )+10 \left (\cos ^{6}\left (f x +e \right )\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+231 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+22 \left (\cos ^{4}\left (f x +e \right )\right )+154 \left (\cos ^{2}\left (f x +e \right )\right )-231 \cos \left (f x +e \right )\right ) \left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}}{585 f \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{7} \sin \left (f x +e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

-2/585/f*(-c*(sin(f*x+e)-1))^(5/2)*(39*cos(f*x+e)^8*sin(f*x+e)+45*cos(f*x+e)^8+10*cos(f*x+e)^6-231*I*(1/(cos(f
*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e)
,I)+231*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*x+e)*EllipticE(I*(-1+cos
(f*x+e))/sin(f*x+e),I)-231*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*EllipticF(I
*(-1+cos(f*x+e))/sin(f*x+e),I)+231*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*Ell
ipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+22*cos(f*x+e)^4+154*cos(f*x+e)^2-231*cos(f*x+e))*(g*cos(f*x+e))^(3/2)*(
a*(1+sin(f*x+e)))^(7/2)/(1+sin(f*x+e))/cos(f*x+e)^7/sin(f*x+e)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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