3.41 \(\int \frac{\sin ^{-1}(a x)^4}{x^4} \, dx\)

Optimal. Leaf size=276 \[ 2 i a^3 \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (4,-e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (4,e^{i \sin ^{-1}(a x)}\right )-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^4}{3 x^3} \]

[Out]

(-2*a^2*ArcSin[a*x]^2)/x - (2*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*x^2) - ArcSin[a*x]^4/(3*x^3) - 8*a^3*ArcSi
n[a*x]*ArcTanh[E^(I*ArcSin[a*x])] - (4*a^3*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])])/3 + (4*I)*a^3*PolyLog[2,
-E^(I*ArcSin[a*x])] + (2*I)*a^3*ArcSin[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] - (4*I)*a^3*PolyLog[2, E^(I*ArcSi
n[a*x])] - (2*I)*a^3*ArcSin[a*x]^2*PolyLog[2, E^(I*ArcSin[a*x])] - 4*a^3*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a
*x])] + 4*a^3*ArcSin[a*x]*PolyLog[3, E^(I*ArcSin[a*x])] - (4*I)*a^3*PolyLog[4, -E^(I*ArcSin[a*x])] + (4*I)*a^3
*PolyLog[4, E^(I*ArcSin[a*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.410257, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4627, 4701, 4709, 4183, 2531, 6609, 2282, 6589, 2279, 2391} \[ 2 i a^3 \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text{PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text{PolyLog}\left (4,-e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text{PolyLog}\left (4,e^{i \sin ^{-1}(a x)}\right )-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^4}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^4/x^4,x]

[Out]

(-2*a^2*ArcSin[a*x]^2)/x - (2*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*x^2) - ArcSin[a*x]^4/(3*x^3) - 8*a^3*ArcSi
n[a*x]*ArcTanh[E^(I*ArcSin[a*x])] - (4*a^3*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])])/3 + (4*I)*a^3*PolyLog[2,
-E^(I*ArcSin[a*x])] + (2*I)*a^3*ArcSin[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] - (4*I)*a^3*PolyLog[2, E^(I*ArcSi
n[a*x])] - (2*I)*a^3*ArcSin[a*x]^2*PolyLog[2, E^(I*ArcSin[a*x])] - 4*a^3*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a
*x])] + 4*a^3*ArcSin[a*x]*PolyLog[3, E^(I*ArcSin[a*x])] - (4*I)*a^3*PolyLog[4, -E^(I*ArcSin[a*x])] + (4*I)*a^3
*PolyLog[4, E^(I*ArcSin[a*x])]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4701

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(d*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m
+ 1)), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^F
racPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1] && Inte
gerQ[m]

Rule 4709

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c^(m
+ 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2
*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]

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 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]

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]

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}(a x)^4}{x^4} \, dx &=-\frac{\sin ^{-1}(a x)^4}{3 x^3}+\frac{1}{3} (4 a) \int \frac{\sin ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{\sin ^{-1}(a x)^4}{3 x^3}+\left (2 a^2\right ) \int \frac{\sin ^{-1}(a x)^2}{x^2} \, dx+\frac{1}{3} \left (2 a^3\right ) \int \frac{\sin ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{\sin ^{-1}(a x)^4}{3 x^3}+\frac{1}{3} \left (2 a^3\right ) \operatorname{Subst}\left (\int x^3 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 a^3\right ) \int \frac{\sin ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{\sin ^{-1}(a x)^4}{3 x^3}-\frac{4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\left (2 a^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (2 a^3\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname{Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+\left (4 a^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text{Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text{Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+\left (4 a^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-\left (4 a^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text{Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text{Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+\left (4 i a^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac{2 a^2 \sin ^{-1}(a x)^2}{x}-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 x^2}-\frac{\sin ^{-1}(a x)^4}{3 x^3}-8 a^3 \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-\frac{4}{3} a^3 \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-2 i a^3 \sin ^{-1}(a x)^2 \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-4 a^3 \sin ^{-1}(a x) \text{Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+4 a^3 \sin ^{-1}(a x) \text{Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-4 i a^3 \text{Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+4 i a^3 \text{Li}_4\left (e^{i \sin ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 4.26995, size = 399, normalized size = 1.45 \[ \frac{1}{24} a^3 \left (48 i \sin ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-i \sin ^{-1}(a x)}\right )+96 \sin ^{-1}(a x) \text{PolyLog}\left (3,e^{-i \sin ^{-1}(a x)}\right )-96 \sin ^{-1}(a x) \text{PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+48 i \left (\sin ^{-1}(a x)^2+2\right ) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-96 i \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-96 i \text{PolyLog}\left (4,e^{-i \sin ^{-1}(a x)}\right )-96 i \text{PolyLog}\left (4,-e^{i \sin ^{-1}(a x)}\right )-\frac{8 \sin ^4\left (\frac{1}{2} \sin ^{-1}(a x)\right ) \sin ^{-1}(a x)^4}{a^3 x^3}+4 i \sin ^{-1}(a x)^4+16 \sin ^{-1}(a x)^3 \log \left (1-e^{-i \sin ^{-1}(a x)}\right )-16 \sin ^{-1}(a x)^3 \log \left (1+e^{i \sin ^{-1}(a x)}\right )+96 \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )-96 \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )-2 \sin ^{-1}(a x)^4 \tan \left (\frac{1}{2} \sin ^{-1}(a x)\right )-24 \sin ^{-1}(a x)^2 \tan \left (\frac{1}{2} \sin ^{-1}(a x)\right )-2 \sin ^{-1}(a x)^4 \cot \left (\frac{1}{2} \sin ^{-1}(a x)\right )-24 \sin ^{-1}(a x)^2 \cot \left (\frac{1}{2} \sin ^{-1}(a x)\right )-\frac{1}{2} a x \sin ^{-1}(a x)^4 \csc ^4\left (\frac{1}{2} \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x)^3 \csc ^2\left (\frac{1}{2} \sin ^{-1}(a x)\right )+4 \sin ^{-1}(a x)^3 \sec ^2\left (\frac{1}{2} \sin ^{-1}(a x)\right )-2 i \pi ^4\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSin[a*x]^4/x^4,x]

[Out]

(a^3*((-2*I)*Pi^4 + (4*I)*ArcSin[a*x]^4 - 24*ArcSin[a*x]^2*Cot[ArcSin[a*x]/2] - 2*ArcSin[a*x]^4*Cot[ArcSin[a*x
]/2] - 4*ArcSin[a*x]^3*Csc[ArcSin[a*x]/2]^2 - (a*x*ArcSin[a*x]^4*Csc[ArcSin[a*x]/2]^4)/2 + 16*ArcSin[a*x]^3*Lo
g[1 - E^((-I)*ArcSin[a*x])] + 96*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] - 96*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a
*x])] - 16*ArcSin[a*x]^3*Log[1 + E^(I*ArcSin[a*x])] + (48*I)*ArcSin[a*x]^2*PolyLog[2, E^((-I)*ArcSin[a*x])] +
(48*I)*(2 + ArcSin[a*x]^2)*PolyLog[2, -E^(I*ArcSin[a*x])] - (96*I)*PolyLog[2, E^(I*ArcSin[a*x])] + 96*ArcSin[a
*x]*PolyLog[3, E^((-I)*ArcSin[a*x])] - 96*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] - (96*I)*PolyLog[4, E^((-
I)*ArcSin[a*x])] - (96*I)*PolyLog[4, -E^(I*ArcSin[a*x])] + 4*ArcSin[a*x]^3*Sec[ArcSin[a*x]/2]^2 - (8*ArcSin[a*
x]^4*Sin[ArcSin[a*x]/2]^4)/(a^3*x^3) - 24*ArcSin[a*x]^2*Tan[ArcSin[a*x]/2] - 2*ArcSin[a*x]^4*Tan[ArcSin[a*x]/2
]))/24

________________________________________________________________________________________

Maple [A]  time = 0.145, size = 409, normalized size = 1.5 \begin{align*} -{\frac{2\,a \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{3\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-2\,{\frac{{a}^{2} \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{x}}-{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{3\,{x}^{3}}}-{\frac{2\,{a}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{3}\ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) }+2\,i{a}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -4\,{a}^{3}\arcsin \left ( ax \right ){\it polylog} \left ( 3,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -4\,i{a}^{3}{\it polylog} \left ( 4,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +{\frac{2\,{a}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{3}\ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) }-2\,i{a}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +4\,{a}^{3}\arcsin \left ( ax \right ){\it polylog} \left ( 3,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +4\,i{a}^{3}{\it polylog} \left ( 4,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) -4\,{a}^{3}\arcsin \left ( ax \right ) \ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +4\,i{a}^{3}{\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +4\,{a}^{3}\arcsin \left ( ax \right ) \ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -4\,i{a}^{3}{\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^4/x^4,x)

[Out]

-2/3*a*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/x^2-2*a^2*arcsin(a*x)^2/x-1/3*arcsin(a*x)^4/x^3-2/3*a^3*arcsin(a*x)^3*
ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+2*I*a^3*arcsin(a*x)^2*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-4*a^3*arcsin(a*x)*po
lylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))-4*I*a^3*polylog(4,-I*a*x-(-a^2*x^2+1)^(1/2))+2/3*a^3*arcsin(a*x)^3*ln(1-I*a
*x-(-a^2*x^2+1)^(1/2))-2*I*a^3*arcsin(a*x)^2*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))+4*a^3*arcsin(a*x)*polylog(3,I
*a*x+(-a^2*x^2+1)^(1/2))+4*I*a^3*polylog(4,I*a*x+(-a^2*x^2+1)^(1/2))-4*a^3*arcsin(a*x)*ln(1+I*a*x+(-a^2*x^2+1)
^(1/2))+4*I*a^3*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))+4*a^3*arcsin(a*x)*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-4*I*a^3*
polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \, a x^{3} \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3}}{a^{2} x^{5} - x^{3}}\,{d x} + \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4/x^4,x, algorithm="maxima")

[Out]

-1/3*(12*a*x^3*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3/(a^2*x^
5 - x^3), x) + arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^4)/x^3

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arcsin \left (a x\right )^{4}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4/x^4,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^4/x^4, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{4}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**4/x**4,x)

[Out]

Integral(asin(a*x)**4/x**4, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{4}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4/x^4,x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^4/x^4, x)