3.126 \(\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx\)

Optimal. Leaf size=471 \[ \frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5525 c^7 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}} \]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(25*f*g*(c - c*Sin[e + f*x])^(15/2)) - (4*a^2*(g*Cos[e
+ f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(35*c*f*g*(c - c*Sin[e + f*x])^(13/2)) + (44*a^3*(g*Cos[e + f*x])^(
5/2)*Sqrt[a + a*Sin[e + f*x]])/(595*c^2*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^4*(g*Cos[e + f*x])^(5/2))/(11
05*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (22*a^4*(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])^(7/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^5*f*g*Sqrt[a
+ a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^6*f*g*Sqrt[a + a*Sin[e
+ f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2
, 2])/(5525*c^7*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 2.48679, antiderivative size = 471, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5525 c^7 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}} \]

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])^(15/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(25*f*g*(c - c*Sin[e + f*x])^(15/2)) - (4*a^2*(g*Cos[e
+ f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(35*c*f*g*(c - c*Sin[e + f*x])^(13/2)) + (44*a^3*(g*Cos[e + f*x])^(
5/2)*Sqrt[a + a*Sin[e + f*x]])/(595*c^2*f*g*(c - c*Sin[e + f*x])^(11/2)) - (44*a^4*(g*Cos[e + f*x])^(5/2))/(11
05*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) + (22*a^4*(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])^(7/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^5*f*g*Sqrt[a
+ a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(5525*c^6*f*g*Sqrt[a + a*Sin[e
+ f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2
, 2])/(5525*c^7*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))^{15/2}} \, dx &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{(3 a) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{5 c}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{\left (11 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))^{11/2}} \, dx}{35 c^2}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{\left (11 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))^{9/2}} \, dx}{85 c^3}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^4 (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))^{9/2}}+\frac{\left (33 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))^{7/2}} \, dx}{1105 c^4}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^4 (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))^{9/2}}+\frac{22 a^4 (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))^{7/2}}+\frac{\left (11 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))^{5/2}} \, dx}{1105 c^5}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^4 (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))^{9/2}}+\frac{22 a^4 (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))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{\left (11 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}{5525 c^6}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^4 (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))^{9/2}}+\frac{22 a^4 (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))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (11 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}{5525 c^7}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^4 (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))^{9/2}}+\frac{22 a^4 (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))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (11 a^4 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{5525 c^7 \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}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^4 (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))^{9/2}}+\frac{22 a^4 (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))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (11 a^4 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{5525 c^7 \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}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac{4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac{44 a^4 (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))^{9/2}}+\frac{22 a^4 (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))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{22 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 )}{5525 c^7 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.88652, size = 668, normalized size = 1.42 \[ \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 )^{15} \left (\frac{44 \sin \left (\frac{1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}+\frac{44 \sin \left (\frac{1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{44 \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 )^5}-\frac{4288 \sin \left (\frac{1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{9312 \sin \left (\frac{1}{2} (e+f x)\right )}{2975 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}-\frac{832 \sin \left (\frac{1}{2} (e+f x)\right )}{175 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}+\frac{64 \sin \left (\frac{1}{2} (e+f x)\right )}{25 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{13}}+\frac{22}{5525 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}+\frac{22}{3315 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}-\frac{2144}{5525 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}+\frac{4656}{2975 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8}-\frac{416}{175 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{10}}+\frac{32}{25 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{12}}+\frac{22}{5525}\right )}{f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{22 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 )^{15}}{5525 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{15/2} \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])^(15/2),x]

[Out]

(-22*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^15*(a*(1 + Sin[e +
f*x]))^(7/2))/(5525*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2))
+ ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^15*(22/5525 + 32/(25*(Cos[(e + f
*x)/2] - Sin[(e + f*x)/2])^12) - 416/(175*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10) + 4656/(2975*(Cos[(e + f*x
)/2] - Sin[(e + f*x)/2])^8) - 2144/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 22/(3315*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^4) + 22/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(25*(Cos[(
e + f*x)/2] - Sin[(e + f*x)/2])^13) - (832*Sin[(e + f*x)/2])/(175*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) +
(9312*Sin[(e + f*x)/2])/(2975*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (4288*Sin[(e + f*x)/2])/(5525*(Cos[(e
+ f*x)/2] - Sin[(e + f*x)/2])^7) + (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (44
*Sin[(e + f*x)/2])/(5525*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (44*Sin[(e + f*x)/2])/(5525*(Cos[(e + f*x)
/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[
e + f*x])^(15/2))

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Maple [C]  time = 0.486, size = 1644, normalized size = 3.5 \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))^(7/2)/(c-c*sin(f*x+e))^(15/2),x)

[Out]

2/116025/f*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(7/2)*(sin(f*x+e)*cos(f*x+e)-sin(f*x+e)-cos(f*x+e)+1)*(-742
56-74256*sin(f*x+e)+81648*cos(f*x+e)+66864*sin(f*x+e)*cos(f*x+e)+112324*cos(f*x+e)^2+50003*cos(f*x+e)^5-130804
*cos(f*x+e)^3-85052*sin(f*x+e)*cos(f*x+e)^3+20895*sin(f*x+e)*cos(f*x+e)^5-29673*sin(f*x+e)*cos(f*x+e)^4-7392*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)+7392*I*E
llipticE(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)-385*cos(f*
x+e)^7+1386*cos(f*x+e)^6*sin(f*x+e)+99836*cos(f*x+e)^2*sin(f*x+e)-4004*cos(f*x+e)^6-34757*cos(f*x+e)^4+231*cos
(f*x+e)^8+23562*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^4*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/
(cos(f*x+e)+1))^(1/2)+22176*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^2*(1/(cos(f*x+e)+1))^(1/2)*
(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-22176*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^2*(1/(cos(f*x+e
)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+7392*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*(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)*sin(f*
x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+231*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I
)*sin(f*x+e)*cos(f*x+e)^8*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-231*I*EllipticE(I*(-1+cos
(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^8*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-1386
*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^8*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))
^(1/2)+1386*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^8*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos
(f*x+e)+1))^(1/2)+10164*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^6*(1/(cos(f*x+e)+1))^(1/2)*(cos
(f*x+e)/(cos(f*x+e)+1))^(1/2)-10164*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^6*(1/(cos(f*x+e)+1)
)^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-23562*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^4*(1/(c
os(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-4389*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x
+e)*cos(f*x+e)^6*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+4389*I*EllipticE(I*(-1+cos(f*x+e))
/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^6*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+15246*I*Elli
pticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^4*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)
+1))^(1/2)-15246*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^4*(1/(cos(f*x+e)+1))^(1/2)*
(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-18480*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*cos(f*x+e)^2*(1
/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+18480*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin
(f*x+e)*cos(f*x+e)^2*(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)^4+4*cos(f*x+e)^2*sin(f*x+e)+8*cos(f*x+e)^2-8*sin(f*x+e)-8)/(-c*(-1+sin(f*x+e)))^(15/2)/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{7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{15}{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))^(15/2),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)^(15/2), 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^{8} \cos \left (f x + e\right )^{8} - 32 \, c^{8} \cos \left (f x + e\right )^{6} + 160 \, c^{8} \cos \left (f x + e\right )^{4} - 256 \, c^{8} \cos \left (f x + e\right )^{2} + 128 \, c^{8} + 8 \,{\left (c^{8} \cos \left (f x + e\right )^{6} - 10 \, c^{8} \cos \left (f x + e\right )^{4} + 24 \, c^{8} \cos \left (f x + e\right )^{2} - 16 \, c^{8}\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))^(15/2),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^8*cos(f*x + e)^8 - 32*c^8*
cos(f*x + e)^6 + 160*c^8*cos(f*x + e)^4 - 256*c^8*cos(f*x + e)^2 + 128*c^8 + 8*(c^8*cos(f*x + e)^6 - 10*c^8*co
s(f*x + e)^4 + 24*c^8*cos(f*x + e)^2 - 16*c^8)*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))**(15/2),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{15}{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))^(15/2),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)^(15/2), x)