3.114 \(\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\)

Optimal. Leaf size=357 \[ -\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{3315 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^2*(g*Cos[
e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (308*a^3*(g*Cos[e + f*x])^(
5/2))/(1989*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3
315*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^4*
f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*
EllipticE[(e + f*x)/2, 2])/(3315*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 1.78659, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {2850, 2852, 2842, 2640, 2639} \[ -\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{154 a^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{3315 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{44 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^2*(g*Cos[
e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (308*a^3*(g*Cos[e + f*x])^(
5/2))/(1989*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3
315*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^4*
f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*
EllipticE[(e + f*x)/2, 2])/(3315*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 2850

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] - Dist[(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] && IntegersQ[2*m, 2*n, 2*p]

Rule 2852

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*(c + d*Sin[e + f*x])^
n)/(a*f*g*(2*m + p + 1)), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), 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] && E
qQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] &&  !LtQ[m, n, -1] && IntegersQ[2*m, 2*n, 2*p]

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 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 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]

Rubi steps

\begin{align*} \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{(11 a) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{17 c}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{\left (77 a^2\right ) \int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx}{221 c^2}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{\left (77 a^3\right ) \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{663 c^3}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{\left (77 a^3\right ) \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{3315 c^4}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (77 a^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}{3315 c^5}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (77 a^3 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{3315 c^5 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (77 a^3 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{3315 c^5 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{308 a^3 (g \cos (e+f x))^{5/2}}{1989 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{154 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{154 a^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 )}{3315 c^5 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.71596, size = 532, normalized size = 1.49 \[ \frac{154 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}{3315 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}+\frac{\sec (e+f x) (a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left (-\frac{308 \sin \left (\frac{1}{2} (e+f x)\right )}{3315 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}-\frac{308 \sin \left (\frac{1}{2} (e+f x)\right )}{3315 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{2344 \sin \left (\frac{1}{2} (e+f x)\right )}{1989 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}-\frac{592 \sin \left (\frac{1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{32 \sin \left (\frac{1}{2} (e+f x)\right )}{17 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}-\frac{154}{3315 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}+\frac{1172}{1989 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}-\frac{296}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}+\frac{16}{17 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8}-\frac{154}{3315}\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

(154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e +
f*x]))^(5/2))/(3315*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(11/2))
+ ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(-154/3315 + 16/(17*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^8) - 296/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 1172/(1989*(Cos[(e + f*x
)/2] - Sin[(e + f*x)/2])^4) - 154/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (32*Sin[(e + f*x)/2])/(17*(
Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (592*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7)
+ (2344*Sin[(e + f*x)/2])/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (308*Sin[(e + f*x)/2])/(3315*(Cos[
(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (308*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*
(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(11/2))

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Maple [C]  time = 0.385, size = 1313, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

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))^(5/2)/(c-c*sin(f*x+e))^(11/2),x)

[Out]

2/9945/f*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(5/2)*(sin(f*x+e)*cos(f*x+e)-sin(f*x+e)-cos(f*x+e)+1)*(-4680+
1848*I*sin(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1
))^(1/2)-924*I*cos(f*x+e)^6*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(co
s(f*x+e)+1))^(1/2)+924*I*cos(f*x+e)^6*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(
f*x+e)/(cos(f*x+e)+1))^(1/2)+3696*I*cos(f*x+e)^4*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^
(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-3696*I*cos(f*x+e)^4*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(
f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-4620*I*cos(f*x+e)^2*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),
I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+4620*I*cos(f*x+e)^2*EllipticE(I*(-1+cos(f*x+e))/
sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-1848*I*sin(f*x+e)*EllipticF(I*(-1+cos
(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+1848*I*EllipticF(I*(-1+cos(f
*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-1848*I*EllipticE(I*(-1+cos(f*x
+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-4680*sin(f*x+e)+2832*cos(f*x+e)+
6528*sin(f*x+e)*cos(f*x+e)+9920*cos(f*x+e)^2+2930*cos(f*x+e)^5-6224*cos(f*x+e)^3-4192*sin(f*x+e)*cos(f*x+e)^3+
231*I*sin(f*x+e)*cos(f*x+e)^6*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(
cos(f*x+e)+1))^(1/2)-231*I*sin(f*x+e)*cos(f*x+e)^6*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1)
)^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-2079*I*sin(f*x+e)*cos(f*x+e)^4*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e
),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+2079*I*sin(f*x+e)*cos(f*x+e)^4*EllipticE(I*(-1
+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+924*sin(f*x+e)*cos(f*x+e
)^4+1420*cos(f*x+e)^2*sin(f*x+e)+231*cos(f*x+e)^6-5009*cos(f*x+e)^4+3696*I*sin(f*x+e)*cos(f*x+e)^2*EllipticF(I
*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-3696*I*sin(f*x+e)*co
s(f*x+e)^2*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2
))*(cos(f*x+e)^2+2*cos(f*x+e)+1)/(cos(f*x+e)^2*sin(f*x+e)+3*cos(f*x+e)^2-4*sin(f*x+e)-4)/(-c*(-1+sin(f*x+e)))^
(11/2)/sin(f*x+e)^5/cos(f*x+e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

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))^(5/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(5/2)/(-c*sin(f*x + e) + c)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{2} g \cos \left (f x + e\right )^{3} - 2 \, a^{2} g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{2} g \cos \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}}{c^{6} \cos \left (f x + e\right )^{6} - 18 \, c^{6} \cos \left (f x + e\right )^{4} + 48 \, c^{6} \cos \left (f x + e\right )^{2} - 32 \, c^{6} + 2 \,{\left (3 \, c^{6} \cos \left (f x + e\right )^{4} - 16 \, c^{6} \cos \left (f x + e\right )^{2} + 16 \, c^{6}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}

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))^(5/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

integral((a^2*g*cos(f*x + e)^3 - 2*a^2*g*cos(f*x + e)*sin(f*x + e) - 2*a^2*g*cos(f*x + e))*sqrt(g*cos(f*x + e)
)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^6*cos(f*x + e)^6 - 18*c^6*cos(f*x + e)^4 + 48*c^6*cos(
f*x + e)^2 - 32*c^6 + 2*(3*c^6*cos(f*x + e)^4 - 16*c^6*cos(f*x + e)^2 + 16*c^6)*sin(f*x + e)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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))**(5/2)/(c-c*sin(f*x+e))**(11/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

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))^(5/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(5/2)/(-c*sin(f*x + e) + c)^(11/2), x)