3.139 \(\int \frac{x^3}{(a+a \sin (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=691 \[ \frac{3 i x^2 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{3 i x^2 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{12 x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (3,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{12 x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (3,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (4,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}+\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (4,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{3 x^2}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{24 x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^3 \sqrt{a \sin (e+f x)+a}}-\frac{x^3 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f \sqrt{a \sin (e+f x)+a}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{2 a f \sqrt{a \sin (e+f x)+a}} \]

[Out]

(-3*x^2)/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - (x^3*Cot[e/2 + Pi/4 + (f*x)/2])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) -
(24*x*ArcTanh[E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) - (x^
3*ArcTanh[E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + ((24*I)*Po
lyLog[2, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]) + ((3*I)*x
^2*PolyLog[2, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - ((2
4*I)*PolyLog[2, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]) - ((
3*I)*x^2*PolyLog[2, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]])
- (12*x*PolyLog[3, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]])
+ (12*x*PolyLog[3, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) -
((24*I)*PolyLog[4, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]])
+ ((24*I)*PolyLog[4, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]
)

________________________________________________________________________________________

Rubi [A]  time = 0.353678, antiderivative size = 691, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3319, 4186, 4183, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac{3 i x^2 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{3 i x^2 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{12 x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (3,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{12 x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (3,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^3 \sqrt{a \sin (e+f x)+a}}+\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (2,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (4,-e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}+\frac{24 i \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \text{PolyLog}\left (4,e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^4 \sqrt{a \sin (e+f x)+a}}-\frac{3 x^2}{a f^2 \sqrt{a \sin (e+f x)+a}}-\frac{24 x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f^3 \sqrt{a \sin (e+f x)+a}}-\frac{x^3 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+2 f x+\pi )}\right )}{a f \sqrt{a \sin (e+f x)+a}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{2 a f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(-3*x^2)/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - (x^3*Cot[e/2 + Pi/4 + (f*x)/2])/(2*a*f*Sqrt[a + a*Sin[e + f*x]]) -
(24*x*ArcTanh[E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) - (x^
3*ArcTanh[E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f*Sqrt[a + a*Sin[e + f*x]]) + ((24*I)*Po
lyLog[2, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]) + ((3*I)*x
^2*PolyLog[2, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]]) - ((2
4*I)*PolyLog[2, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]) - ((
3*I)*x^2*PolyLog[2, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^2*Sqrt[a + a*Sin[e + f*x]])
- (12*x*PolyLog[3, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]])
+ (12*x*PolyLog[3, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^3*Sqrt[a + a*Sin[e + f*x]]) -
((24*I)*PolyLog[4, -E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]])
+ ((24*I)*PolyLog[4, E^((I/4)*(2*e + Pi + 2*f*x))]*Sin[e/2 + Pi/4 + (f*x)/2])/(a*f^4*Sqrt[a + a*Sin[e + f*x]]
)

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 4186

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{x^3}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac{\sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \int x^3 \csc ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{2 a \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{3 x^2}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+a \sin (e+f x)}}+\frac{\sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \int x^3 \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{4 a \sqrt{a+a \sin (e+f x)}}+\frac{\left (6 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int x \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{a f^2 \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{3 x^2}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+a \sin (e+f x)}}-\frac{24 x \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+a \sin (e+f x)}}-\frac{\left (12 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int \log \left (1-e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{a f^3 \sqrt{a+a \sin (e+f x)}}+\frac{\left (12 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int \log \left (1+e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{\left (3 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int x^2 \log \left (1-e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{2 a f \sqrt{a+a \sin (e+f x)}}+\frac{\left (3 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int x^2 \log \left (1+e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{2 a f \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{3 x^2}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+a \sin (e+f x)}}-\frac{24 x \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+a \sin (e+f x)}}+\frac{3 i x^2 \text{Li}_2\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{3 i x^2 \text{Li}_2\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}+\frac{\left (24 i \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}-\frac{\left (24 i \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}-\frac{\left (6 i \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int x \text{Li}_2\left (-e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{a f^2 \sqrt{a+a \sin (e+f x)}}+\frac{\left (6 i \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int x \text{Li}_2\left (e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{a f^2 \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{3 x^2}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+a \sin (e+f x)}}-\frac{24 x \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+a \sin (e+f x)}}+\frac{24 i \text{Li}_2\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}+\frac{3 i x^2 \text{Li}_2\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{24 i \text{Li}_2\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}-\frac{3 i x^2 \text{Li}_2\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{12 x \text{Li}_3\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}+\frac{12 x \text{Li}_3\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}+\frac{\left (12 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int \text{Li}_3\left (-e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{\left (12 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \int \text{Li}_3\left (e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right ) \, dx}{a f^3 \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{3 x^2}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+a \sin (e+f x)}}-\frac{24 x \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+a \sin (e+f x)}}+\frac{24 i \text{Li}_2\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}+\frac{3 i x^2 \text{Li}_2\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{24 i \text{Li}_2\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}-\frac{3 i x^2 \text{Li}_2\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{12 x \text{Li}_3\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}+\frac{12 x \text{Li}_3\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{\left (24 i \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}+\frac{\left (24 i \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{3 x^2}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{2 a f \sqrt{a+a \sin (e+f x)}}-\frac{24 x \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{x^3 \tanh ^{-1}\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f \sqrt{a+a \sin (e+f x)}}+\frac{24 i \text{Li}_2\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}+\frac{3 i x^2 \text{Li}_2\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{24 i \text{Li}_2\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}-\frac{3 i x^2 \text{Li}_2\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^2 \sqrt{a+a \sin (e+f x)}}-\frac{12 x \text{Li}_3\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}+\frac{12 x \text{Li}_3\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^3 \sqrt{a+a \sin (e+f x)}}-\frac{24 i \text{Li}_4\left (-e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}+\frac{24 i \text{Li}_4\left (e^{\frac{1}{4} i (2 e+\pi +2 f x)}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f^4 \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 2.80731, size = 455, normalized size = 0.66 \[ -\frac{x^2 \sqrt{a (\sin (e+f x)+1)} \left ((6-f x) \sin \left (\frac{1}{2} (e+f x)\right )+(f x+6) \cos \left (\frac{1}{2} (e+f x)\right )\right )}{2 a^2 f^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}-\frac{(-1)^{3/4} e^{-\frac{3}{2} i (e+f x)} \left (e^{i (e+f x)}+i\right )^3 \left (6 \left (f^2 x^2+8\right ) \text{PolyLog}\left (2,-\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )-6 \left (f^2 x^2+8\right ) \text{PolyLog}\left (2,\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )-i \left (-24 f x \text{PolyLog}\left (3,-\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )+24 f x \text{PolyLog}\left (3,\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )-48 i \text{PolyLog}\left (4,-\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )+48 i \text{PolyLog}\left (4,\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )+f^3 x^3 \log \left (1-\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )-f^3 x^3 \log \left (1+\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )+24 f x \log \left (1-\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )-24 f x \log \left (1+\sqrt [4]{-1} e^{\frac{1}{2} i (e+f x)}\right )\right )\right )}{2 \sqrt{2} f^4 \left (-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + a*Sin[e + f*x])^(3/2),x]

[Out]

-((-1)^(3/4)*(I + E^(I*(e + f*x)))^3*(6*(8 + f^2*x^2)*PolyLog[2, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] - 6*(8 + f
^2*x^2)*PolyLog[2, (-1)^(1/4)*E^((I/2)*(e + f*x))] - I*(24*f*x*Log[1 - (-1)^(1/4)*E^((I/2)*(e + f*x))] + f^3*x
^3*Log[1 - (-1)^(1/4)*E^((I/2)*(e + f*x))] - 24*f*x*Log[1 + (-1)^(1/4)*E^((I/2)*(e + f*x))] - f^3*x^3*Log[1 +
(-1)^(1/4)*E^((I/2)*(e + f*x))] - 24*f*x*PolyLog[3, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] + 24*f*x*PolyLog[3, (-1
)^(1/4)*E^((I/2)*(e + f*x))] - (48*I)*PolyLog[4, -((-1)^(1/4)*E^((I/2)*(e + f*x)))] + (48*I)*PolyLog[4, (-1)^(
1/4)*E^((I/2)*(e + f*x))])))/(2*Sqrt[2]*E^(((3*I)/2)*(e + f*x))*(((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*
x)))^(3/2)*f^4) - (x^2*((6 + f*x)*Cos[(e + f*x)/2] + (6 - f*x)*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])])/(
2*a^2*f^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+a*sin(f*x+e))^(3/2),x)

[Out]

int(x^3/(a+a*sin(f*x+e))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3/(a*sin(f*x + e) + a)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+a*sin(f*x+e))**(3/2),x)

[Out]

Integral(x**3/(a*(sin(e + f*x) + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(x^3/(a*sin(f*x + e) + a)^(3/2), x)