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

Optimal. Leaf size=691 \[ \frac {24 i \text {Li}_2\left (-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^4 \sqrt {a \sin (e+f x)+a}}-\frac {24 i \text {Li}_2\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^4 \sqrt {a \sin (e+f x)+a}}-\frac {24 i \text {Li}_4\left (-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^4 \sqrt {a \sin (e+f x)+a}}+\frac {24 i \text {Li}_4\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^4 \sqrt {a \sin (e+f x)+a}}-\frac {12 x \text {Li}_3\left (-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^3 \sqrt {a \sin (e+f x)+a}}+\frac {12 x \text {Li}_3\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^3 \sqrt {a \sin (e+f x)+a}}-\frac {24 x \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )}{a f^3 \sqrt {a \sin (e+f x)+a}}+\frac {3 i x^2 \text {Li}_2\left (-e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {3 i x^2 \text {Li}_2\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right ) \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {3 x^2}{a f^2 \sqrt {a \sin (e+f x)+a}}-\frac {x^3 \sin \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \tanh ^{-1}\left (e^{\frac {1}{4} i (2 e+2 f x+\pi )}\right )}{a f \sqrt {a \sin (e+f x)+a}}-\frac {x^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{2 a f \sqrt {a \sin (e+f x)+a}} \]

[Out]

-3*x^2/a/f^2/(a+a*sin(f*x+e))^(1/2)-1/2*x^3*cot(1/2*e+1/4*Pi+1/2*f*x)/a/f/(a+a*sin(f*x+e))^(1/2)-24*x*arctanh(
exp(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)/a/f^3/(a+a*sin(f*x+e))^(1/2)-x^3*arctanh(exp(1/4*I*(2*f*x
+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)/a/f/(a+a*sin(f*x+e))^(1/2)+24*I*polylog(2,-exp(1/4*I*(2*f*x+Pi+2*e)))*sin
(1/2*e+1/4*Pi+1/2*f*x)/a/f^4/(a+a*sin(f*x+e))^(1/2)+3*I*x^2*polylog(2,-exp(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/
4*Pi+1/2*f*x)/a/f^2/(a+a*sin(f*x+e))^(1/2)-24*I*polylog(2,exp(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)
/a/f^4/(a+a*sin(f*x+e))^(1/2)-3*I*x^2*polylog(2,exp(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)/a/f^2/(a+
a*sin(f*x+e))^(1/2)-12*x*polylog(3,-exp(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)/a/f^3/(a+a*sin(f*x+e)
)^(1/2)+12*x*polylog(3,exp(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)/a/f^3/(a+a*sin(f*x+e))^(1/2)-24*I*
polylog(4,-exp(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)/a/f^4/(a+a*sin(f*x+e))^(1/2)+24*I*polylog(4,ex
p(1/4*I*(2*f*x+Pi+2*e)))*sin(1/2*e+1/4*Pi+1/2*f*x)/a/f^4/(a+a*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.35, antiderivative size = 691, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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 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 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 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 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 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 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]

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]

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*}

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Mathematica [A]  time = 2.93, size = 455, normalized size = 0.66 \[ -\frac {x^2 \sqrt {a (\sin (e+f x)+1)} \left ((6-f x) \sin \left (\frac {1}{2} (e+f x)\right )+(f x+6) \cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 a^2 f^2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {(-1)^{3/4} e^{-\frac {3}{2} i (e+f x)} \left (e^{i (e+f x)}+i\right )^3 \left (-i \left (f^3 x^3 \log \left (1-\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )-f^3 x^3 \log \left (1+\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )-24 f x \text {Li}_3\left (-\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )+24 f x \text {Li}_3\left (\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )-48 i \text {Li}_4\left (-\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )+48 i \text {Li}_4\left (\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )+24 f x \log \left (1-\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )-24 f x \log \left (1+\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )\right )+6 \left (f^2 x^2+8\right ) \text {Li}_2\left (-\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )-6 \left (f^2 x^2+8\right ) \text {Li}_2\left (\sqrt [4]{-1} e^{\frac {1}{2} i (e+f x)}\right )\right )}{2 \sqrt {2} f^4 \left (-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((-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))])))/(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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fricas [F]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right ) + a} x^{3}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

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)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + a*sin(e + f*x))^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

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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