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

Optimal. Leaf size=357 \[ \frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{1105 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (28*a^2*(g*Cos[e
+ f*x])^(5/2))/(221*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2
))/(663*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c
^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c^4*f*g*Sq
rt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*Ellipti
cE[(e + f*x)/2, 2])/(1105*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 1.8083, 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} \[ \frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{14 a^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{1105 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{4 a \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (28*a^2*(g*Cos[e
+ f*x])^(5/2))/(221*c*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2
))/(663*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c
^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*a^2*(g*Cos[e + f*x])^(5/2))/(1105*c^4*f*g*Sq
rt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*a^2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*Ellipti
cE[(e + f*x)/2, 2])/(1105*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))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx &=\frac{4 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{(7 a) \int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx}{17 c}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac{\left (21 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} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{\left (7 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))^{5/2}} \, dx}{221 c^3}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{\left (7 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))^{3/2}} \, dx}{1105 c^4}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (7 a^2\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}{1105 c^5}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (7 a^2 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{1105 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} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (7 a^2 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{1105 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} \sqrt{a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{14 a^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 )}{1105 c^5 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.53852, size = 532, normalized size = 1.49 \[ \frac{\sec (e+f x) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11} \left (\frac{28 \sin \left (\frac{1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}+\frac{28 \sin \left (\frac{1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{28 \sin \left (\frac{1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}-\frac{160 \sin \left (\frac{1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{16 \sin \left (\frac{1}{2} (e+f x)\right )}{17 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}+\frac{14}{1105 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}+\frac{14}{663 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}-\frac{80}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}+\frac{8}{17 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8}+\frac{14}{1105}\right )}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}-\frac{14 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}{1105 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14*(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]))^(3/2))/(1105*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(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*(14/1105 + 8/(17*(Cos[(e + f*
x)/2] - Sin[(e + f*x)/2])^8) - 80/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 14/(663*(Cos[(e + f*x)/2] -
Sin[(e + f*x)/2])^4) + 14/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (16*Sin[(e + f*x)/2])/(17*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^9) - (160*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) + (28*S
in[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (28*Sin[(e + f*x)/2])/(1105*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^3) + (28*Sin[(e + f*x)/2])/(1105*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e +
f*x]))^(3/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(11/2))

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Maple [C]  time = 0.41, size = 1298, normalized size = 3.6 \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))^(3/2)/(c-c*sin(f*x+e))^(11/2),x)

[Out]

-2/3315/f*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(3/2)*(sin(f*x+e)*cos(f*x+e)-sin(f*x+e)-cos(f*x+e)+1)*(-780-
780*sin(f*x+e)+948*cos(f*x+e)+612*sin(f*x+e)*cos(f*x+e)+605*cos(f*x+e)^2+35*cos(f*x+e)^5-941*cos(f*x+e)^3-523*
sin(f*x+e)*cos(f*x+e)^3-84*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*
x+e))/sin(f*x+e),I)*cos(f*x+e)^6+84*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(
-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^6+336*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ell
ipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^4-336*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))
^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^4-420*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(
f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^2+420*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*
x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^2-168*I*(1/(cos(f*x+e)+1))^(1/
2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)+168*I*(1/(cos(f*x+e)
+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)-84*sin(f*x+e
)*cos(f*x+e)^4+775*cos(f*x+e)^2*sin(f*x+e)-21*cos(f*x+e)^6+154*cos(f*x+e)^4-21*I*(1/(cos(f*x+e)+1))^(1/2)*(cos
(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^6+21*I*(1/(cos(f
*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x
+e)^6+168*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),
I)-168*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-
189*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin
(f*x+e)*cos(f*x+e)^4+189*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+
e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^4+336*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ell
ipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^2-336*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos
(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^2)*(cos(f*x+e)^2+2*cos(f*x+e
)+1)/(-cos(f*x+e)^2+2*sin(f*x+e)+2)/(-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{3}{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))^(3/2)/(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)^(3/2)/(-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 g \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a 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))^(3/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

integral(-(a*g*cos(f*x + e)*sin(f*x + e) + a*g*cos(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqr
t(-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))**(3/2)/(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{3}{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))^(3/2)/(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)^(3/2)/(-c*sin(f*x + e) + c)^(11/2), x)