3.124 \(\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\)
Optimal. Leaf size=357 \[ \frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{221 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{60 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]
[Out]
(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (60*a^2*(g*Cos[
e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (220*a^3*(g*Cos[e + f*x])
^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(663*c^2*f*g*(c - c*Sin[e + f*x])^(7/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(
663*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(221*c^4*f
*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*E
llipticE[(e + f*x)/2, 2])/(221*c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])
________________________________________________________________________________________
Rubi [A] time = 1.83963, 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} \[ \frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{221 c^5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{60 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]
Antiderivative was successfully verified.
[In]
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(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])^(5/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (60*a^2*(g*Cos[
e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (220*a^3*(g*Cos[e + f*x])
^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(663*c^2*f*g*(c - c*Sin[e + f*x])^(7/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(
663*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(221*c^4*f
*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*E
llipticE[(e + f*x)/2, 2])/(221*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))^{7/2}}{(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))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{(15 a) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(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))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{\left (165 a^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(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))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{\left (385 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))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{\left (77 a^4\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}{221 c^4}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (77 a^4\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}{221 c^5}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (77 a^4 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{221 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))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (77 a^4 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{221 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))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac{60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{154 a^4 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{221 c^5 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.76856, size = 532, normalized size = 1.49 \[ \frac{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11} \left (\frac{308 \sin \left (\frac{1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}-\frac{576 \sin \left (\frac{1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{4192 \sin \left (\frac{1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}-\frac{1728 \sin \left (\frac{1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{64 \sin \left (\frac{1}{2} (e+f x)\right )}{17 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}-\frac{288}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}+\frac{2096}{663 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}-\frac{864}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}+\frac{32}{17 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8}+\frac{154}{221}\right )}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{154 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}{221 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
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])^(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]))^(7/2))/(221*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(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*(154/221 + 32/(17*(Cos[(e + f
*x)/2] - Sin[(e + f*x)/2])^8) - 864/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 2096/(663*(Cos[(e + f*x)/2
] - Sin[(e + f*x)/2])^4) - 288/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(17*(Cos[
(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (1728*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) +
(4192*Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (576*Sin[(e + f*x)/2])/(221*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^3) + (308*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + S
in[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2))
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Maple [C] time = 0.363, size = 3910, normalized size = 11. \begin{align*} \text{output 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))^(7/2)/(c-c*sin(f*x+e))^(11/2),x)
[Out]
1/1326/f*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(7/2)*(-1+cos(f*x+e))^4*(-1+sin(f*x+e))*(cos(f*x+e)+1)*(-2652
*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)*cos(f*x+e)^5-663*cos(f*x+e)^5*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x
+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*
sin(f*x+e)+663*cos(f*x+e)^5*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)
-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*sin(f*x+e)-9888*sin(f*x+e)*cos(f*x+e)*(-cos(f*x+e)/(c
os(f*x+e)+1)^2)^(1/2)+4896*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-3696*I*EllipticF(I*(-1+cos(f*x+e))/
sin(f*x+e),I)*cos(f*x+e)^6*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+
e)+1))^(1/2)-7956*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos
(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^3+7956*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e
)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^3+5
304*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f
*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)-5304*cos(f*x+e)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^
2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-2496*(-cos(f*x+e)/(
cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-7856*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-2496*(-cos(f*x+e)/(co
s(f*x+e)+1)^2)^(1/2)+6928*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-5304*ln(-2*(2*cos(f*x+e)^2*(-cos(f
*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2
)*cos(f*x+e)*sin(f*x+e)+5304*cos(f*x+e)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+
2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*sin(f*x+e)+752*sin(f*x+e)*cos(f*x+e)^2*(-
cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+3696*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^6*(-cos(f*x+e)/
(cos(f*x+e)+1)^2)^(1/2)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+5304*sin(f*x+e)*cos(f*x+e)^
3*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x
+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-5304*sin(f*x+e)*cos(f*x+e)^3*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^
2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+11840*sin(f*x+e)*co
s(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+1044*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)*cos(f*x+e
)^4+2652*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(
cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*cos(f*x+e)^5-2652*cos(f*x+e)^5*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f
*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-5356*cos(f
*x+e)^4*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+2960*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)^5+924*(-cos(
f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)^6-7392*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)/(cos(f*x+e)+1)^2)^(1/2)+7392*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)/(cos(f*x+e)+1)^2)^(1
/2)-7392*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*Ell
ipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^5+7392*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)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)^5+7392*I*Ell
ipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^4*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1/(cos(f*x+e)+1))^(1
/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-7392*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^4*(-cos(f*x+
e)/(cos(f*x+e)+1)^2)^(1/2)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+22176*I*(1/(cos(f*x+e)+1
))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*EllipticF(I*(-1+cos(f*x+e))/si
n(f*x+e),I)*cos(f*x+e)^3-22176*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+co
s(f*x+e))/sin(f*x+e),I)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)^3+3696*I*EllipticF(I*(-1+cos(f*x+e))/s
in(f*x+e),I)*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e
)+1))^(1/2)-3696*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)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)^2+7392*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)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-1478
4*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*EllipticF(
I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)-7392*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*E
llipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+14784*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)/(cos(f*x+
e)+1)^2)^(1/2)*cos(f*x+e)+924*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(-cos(f*x+e)/(cos(f
*x+e)+1)^2)^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^6-924*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)/(cos(f*x+e)+1)^2
)^(1/2)*sin(f*x+e)*cos(f*x+e)^6+1848*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(-cos(f*x+e)
/(cos(f*x+e)+1)^2)^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^5-1848*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)/(cos(f*x
+e)+1)^2)^(1/2)*sin(f*x+e)*cos(f*x+e)^5-6468*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(-co
s(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^4+6468*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)/
(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)*cos(f*x+e)^4-14784*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(
1/2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^3+14
784*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)*(-c
os(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)*cos(f*x+e)^3+14784*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*
x+e)+1))^(1/2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f
*x+e)-14784*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(
1/2)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2))/(-cos(f*x+e)^4+4*cos(f*x+e)^2*sin(f*x+e)+8*co
s(f*x+e)^2-8*sin(f*x+e)-8)/(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)/(-c*(-1+sin(f*x+e)))^(11/2)/sin(f*x+e)^9
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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{7}{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))^(7/2)/(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)^(7/2)/(-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 (3 \, a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right ) +{\left (a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right )\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}}{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))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="fricas")
[Out]
integral((3*a^3*g*cos(f*x + e)^3 - 4*a^3*g*cos(f*x + e) + (a^3*g*cos(f*x + e)^3 - 4*a^3*g*cos(f*x + e))*sin(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*c
os(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))**(7/2)/(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{7}{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))^(7/2)/(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)^(7/2)/(-c*sin(f*x + e) + c)^(11/2), x)