\(\int \frac {\arcsin (a x)^4}{x^4} \, dx\) [41]

Optimal result

Integrand size = 10, antiderivative size = 276 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=-\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \]

[Out]

-2*a^2*arcsin(a*x)^2/x-1/3*arcsin(a*x)^4/x^3-8*a^3*arcsin(a*x)*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))-4/3*a^3*arcsi
n(a*x)^3*arctanh(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))+2*I*a^3*arcsin(a*x)^2*
polylog(2,-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))-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*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*I*a^3*polylog(4,I*a*x+(-a^2*x^2+1)^
(1/2))-2/3*a*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4723, 4789, 4803, 4268, 2611, 6744, 2320, 6724, 2317, 2438} \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right )-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {\arcsin (a x)^4}{3 x^3} \]

[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 2317

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 2320

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 2438

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 2611

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 4268

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 4723

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 4789

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/(f*(m + 1)))*Simp[(d + e*x
^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4803

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

Rule 6724

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 6744

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*} \text {integral}& = -\frac {\arcsin (a x)^4}{3 x^3}+\frac {1}{3} (4 a) \int \frac {\arcsin (a x)^3}{x^3 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}+\left (2 a^2\right ) \int \frac {\arcsin (a x)^2}{x^2} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}+\frac {1}{3} \left (2 a^3\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arcsin (a x)\right )+\left (4 a^3\right ) \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (2 a^3\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (4 a^3\right ) \text {Subst}(\int x \csc (x) \, dx,x,\arcsin (a x)) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (4 i a^3\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+\left (4 a^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )+\left (4 a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )-\left (4 a^3\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )+\left (4 i a^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arcsin (a x)}\right ) \\ & = -\frac {2 a^2 \arcsin (a x)^2}{x}-\frac {2 a \sqrt {1-a^2 x^2} \arcsin (a x)^3}{3 x^2}-\frac {\arcsin (a x)^4}{3 x^3}-8 a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )-\frac {4}{3} a^3 \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 i a^3 \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+4 a^3 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-4 i a^3 \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 i a^3 \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 3.42 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.45 \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\frac {1}{24} a^3 \left (-2 i \pi ^4+4 i \arcsin (a x)^4-24 \arcsin (a x)^2 \cot \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \cot \left (\frac {1}{2} \arcsin (a x)\right )-4 \arcsin (a x)^3 \csc ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {1}{2} a x \arcsin (a x)^4 \csc ^4\left (\frac {1}{2} \arcsin (a x)\right )+16 \arcsin (a x)^3 \log \left (1-e^{-i \arcsin (a x)}\right )+96 \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )-96 \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )-16 \arcsin (a x)^3 \log \left (1+e^{i \arcsin (a x)}\right )+48 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (a x)}\right )+48 i \left (2+\arcsin (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )+96 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (a x)}\right )-96 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,e^{-i \arcsin (a x)}\right )-96 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+4 \arcsin (a x)^3 \sec ^2\left (\frac {1}{2} \arcsin (a x)\right )-\frac {8 \arcsin (a x)^4 \sin ^4\left (\frac {1}{2} \arcsin (a x)\right )}{a^3 x^3}-24 \arcsin (a x)^2 \tan \left (\frac {1}{2} \arcsin (a x)\right )-2 \arcsin (a x)^4 \tan \left (\frac {1}{2} \arcsin (a x)\right )\right ) \]

[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] (verified)

Time = 0.10 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(377\)
default	\(a^{3} \left (-\frac {\arcsin \left (a x \right )^{2} \left (2 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+6 a^{2} x^{2}\right )}{3 a^{3} x^{3}}+\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-\frac {2 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+2 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+4 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(377\)

[In]

int(arcsin(a*x)^4/x^4,x,method=_RETURNVERBOSE)

[Out]

a^3*(-1/3/a^3/x^3*arcsin(a*x)^2*(2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x+arcsin(a*x)^2+6*a^2*x^2)+2/3*arcsin(a*x)
^3*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-2*I*arcsin(a*x)^2*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))+4*arcsin(a*x)*polylog(
3,I*a*x+(-a^2*x^2+1)^(1/2))+4*I*polylog(4,I*a*x+(-a^2*x^2+1)^(1/2))-2/3*arcsin(a*x)^3*ln(1+I*a*x+(-a^2*x^2+1)^
(1/2))+2*I*arcsin(a*x)^2*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-4*arcsin(a*x)*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2
))-4*I*polylog(4,-I*a*x-(-a^2*x^2+1)^(1/2))+4*arcsin(a*x)*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-4*I*polylog(2,I*a*x+(
-a^2*x^2+1)^(1/2))-4*arcsin(a*x)*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+4*I*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2)))

Fricas [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]

[In]

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

[Out]

integral(arcsin(a*x)^4/x^4, x)

Sympy [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{4}}\, dx \]

[In]

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

[Out]

Integral(asin(a*x)**4/x**4, x)

Maxima [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]

[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

Giac [F]

\[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate(arcsin(a*x)^4/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^4}{x^4} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x^4} \,d x \]

[In]

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

[Out]

int(asin(a*x)^4/x^4, x)