3.349 \(\int \frac{\sinh ^{-1}(a x)^3}{x^3 \sqrt{1+a^2 x^2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-3 a^2 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \sinh ^{-1}(a x) \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 x^2}+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-6 a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac{3 a \sinh ^{-1}(a x)^2}{2 x} \]

[Out]

(-3*a*ArcSinh[a*x]^2)/(2*x) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2*x^2) - 6*a^2*ArcSinh[a*x]*ArcTanh[E^ArcSin
h[a*x]] + a^2*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 3*a^2*PolyLog[2, -E^ArcSinh[a*x]] + (3*a^2*ArcSinh[a*x]
^2*PolyLog[2, -E^ArcSinh[a*x]])/2 + 3*a^2*PolyLog[2, E^ArcSinh[a*x]] - (3*a^2*ArcSinh[a*x]^2*PolyLog[2, E^ArcS
inh[a*x]])/2 - 3*a^2*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] + 3*a^2*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]]
+ 3*a^2*PolyLog[4, -E^ArcSinh[a*x]] - 3*a^2*PolyLog[4, E^ArcSinh[a*x]]

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Rubi [A]  time = 0.362582, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {5747, 5760, 4182, 2531, 6609, 2282, 6589, 5661, 2279, 2391} \[ \frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-3 a^2 \sinh ^{-1}(a x) \text{PolyLog}\left (3,-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \sinh ^{-1}(a x) \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (4,-e^{\sinh ^{-1}(a x)}\right )-3 a^2 \text{PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 x^2}+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-6 a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\frac{3 a \sinh ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^3/(x^3*Sqrt[1 + a^2*x^2]),x]

[Out]

(-3*a*ArcSinh[a*x]^2)/(2*x) - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2*x^2) - 6*a^2*ArcSinh[a*x]*ArcTanh[E^ArcSin
h[a*x]] + a^2*ArcSinh[a*x]^3*ArcTanh[E^ArcSinh[a*x]] - 3*a^2*PolyLog[2, -E^ArcSinh[a*x]] + (3*a^2*ArcSinh[a*x]
^2*PolyLog[2, -E^ArcSinh[a*x]])/2 + 3*a^2*PolyLog[2, E^ArcSinh[a*x]] - (3*a^2*ArcSinh[a*x]^2*PolyLog[2, E^ArcS
inh[a*x]])/2 - 3*a^2*ArcSinh[a*x]*PolyLog[3, -E^ArcSinh[a*x]] + 3*a^2*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]]
+ 3*a^2*PolyLog[4, -E^ArcSinh[a*x]] - 3*a^2*PolyLog[4, E^ArcSinh[a*x]]

Rule 5747

Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[c*x])^n, x], x] - Dist[(b*c*n*d^IntPart[p]*(d + e*x^
2)^FracPart[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
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1] && Int
egerQ[m]

Rule 5760

Int[(((a_.) + ArcSinh[(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*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[
e, c^2*d] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[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 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{\sinh ^{-1}(a x)^3}{x^3 \sqrt{1+a^2 x^2}} \, dx &=-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} (3 a) \int \frac{\sinh ^{-1}(a x)^2}{x^2} \, dx-\frac{1}{2} a^2 \int \frac{\sinh ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 a \sinh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 x^2}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int x^3 \text{csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )+\left (3 a^2\right ) \int \frac{\sinh ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 a \sinh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 x^2}+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{3 a \sinh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 x^2}-6 a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{3 a \sinh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 x^2}-6 a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left (e^{\sinh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{3 a \sinh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 x^2}-6 a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-3 a^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left (e^{\sinh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac{3 a \sinh ^{-1}(a x)^2}{2 x}-\frac{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{2 x^2}-6 a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-3 a^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left (-e^{\sinh ^{-1}(a x)}\right )+3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left (e^{\sinh ^{-1}(a x)}\right )+3 a^2 \text{Li}_4\left (-e^{\sinh ^{-1}(a x)}\right )-3 a^2 \text{Li}_4\left (e^{\sinh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 4.46347, size = 304, normalized size = 1.45 \[ \frac{a \left (-24 a x \sinh ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-48 a x \sinh ^{-1}(a x) \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(a x)}\right )+48 a x \sinh ^{-1}(a x) \text{PolyLog}\left (3,e^{\sinh ^{-1}(a x)}\right )-24 a x \left (\sinh ^{-1}(a x)^2-2\right ) \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-48 a x \text{PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )-48 a x \text{PolyLog}\left (4,-e^{-\sinh ^{-1}(a x)}\right )-48 a x \text{PolyLog}\left (4,e^{\sinh ^{-1}(a x)}\right )-\pi ^4 a x+2 a x \sinh ^{-1}(a x)^4+8 a x \sinh ^{-1}(a x)^3 \log \left (e^{-\sinh ^{-1}(a x)}+1\right )-8 a x \sinh ^{-1}(a x)^3 \log \left (1-e^{\sinh ^{-1}(a x)}\right )+48 a x \sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-48 a x \sinh ^{-1}(a x) \log \left (e^{-\sinh ^{-1}(a x)}+1\right )-4 \sinh ^{-1}(a x)^3 \tanh \left (\frac{1}{2} \sinh ^{-1}(a x)\right )+12 a x \sinh ^{-1}(a x)^2 \tanh \left (\frac{1}{2} \sinh ^{-1}(a x)\right )-12 a x \sinh ^{-1}(a x)^2 \coth \left (\frac{1}{2} \sinh ^{-1}(a x)\right )-2 a x \sinh ^{-1}(a x)^3 \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(a x)\right )\right )}{16 x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSinh[a*x]^3/(x^3*Sqrt[1 + a^2*x^2]),x]

[Out]

(a*(-(a*Pi^4*x) + 2*a*x*ArcSinh[a*x]^4 - 12*a*x*ArcSinh[a*x]^2*Coth[ArcSinh[a*x]/2] - 2*a*x*ArcSinh[a*x]^3*Csc
h[ArcSinh[a*x]/2]^2 + 48*a*x*ArcSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] - 48*a*x*ArcSinh[a*x]*Log[1 + E^(-ArcSinh
[a*x])] + 8*a*x*ArcSinh[a*x]^3*Log[1 + E^(-ArcSinh[a*x])] - 8*a*x*ArcSinh[a*x]^3*Log[1 - E^ArcSinh[a*x]] - 24*
a*x*(-2 + ArcSinh[a*x]^2)*PolyLog[2, -E^(-ArcSinh[a*x])] - 48*a*x*PolyLog[2, E^(-ArcSinh[a*x])] - 24*a*x*ArcSi
nh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]] - 48*a*x*ArcSinh[a*x]*PolyLog[3, -E^(-ArcSinh[a*x])] + 48*a*x*ArcSinh[a*x
]*PolyLog[3, E^ArcSinh[a*x]] - 48*a*x*PolyLog[4, -E^(-ArcSinh[a*x])] - 48*a*x*PolyLog[4, E^ArcSinh[a*x]] + 12*
a*x*ArcSinh[a*x]^2*Tanh[ArcSinh[a*x]/2] - 4*ArcSinh[a*x]^3*Tanh[ArcSinh[a*x]/2]))/(16*x)

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Maple [A]  time = 0.133, size = 377, normalized size = 1.8 \begin{align*} -{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2\,{x}^{2}} \left ({a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) +3\,ax\sqrt{{a}^{2}{x}^{2}+1}+{\it Arcsinh} \left ( ax \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{a}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{2}\ln \left ( 1+ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) }+{\frac{3\,{a}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2}{\it polylog} \left ( 2,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) }-3\,{a}^{2}{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 3,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +3\,{a}^{2}{\it polylog} \left ( 4,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) -{\frac{{a}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{2}\ln \left ( 1-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) }-{\frac{3\,{a}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2}{\it polylog} \left ( 2,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) }+3\,{a}^{2}{\it Arcsinh} \left ( ax \right ){\it polylog} \left ( 3,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -3\,{a}^{2}{\it polylog} \left ( 4,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -3\,{a}^{2}{\it Arcsinh} \left ( ax \right ) \ln \left ( 1+ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -3\,{a}^{2}{\it polylog} \left ( 2,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +3\,{a}^{2}{\it Arcsinh} \left ( ax \right ) \ln \left ( 1-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) +3\,{a}^{2}{\it polylog} \left ( 2,ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^3/x^3/(a^2*x^2+1)^(1/2),x)

[Out]

-1/2/(a^2*x^2+1)^(1/2)/x^2*arcsinh(a*x)^2*(a^2*x^2*arcsinh(a*x)+3*a*x*(a^2*x^2+1)^(1/2)+arcsinh(a*x))+1/2*a^2*
arcsinh(a*x)^3*ln(1+a*x+(a^2*x^2+1)^(1/2))+3/2*a^2*arcsinh(a*x)^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))-3*a^2*arcs
inh(a*x)*polylog(3,-a*x-(a^2*x^2+1)^(1/2))+3*a^2*polylog(4,-a*x-(a^2*x^2+1)^(1/2))-1/2*a^2*arcsinh(a*x)^3*ln(1
-a*x-(a^2*x^2+1)^(1/2))-3/2*a^2*arcsinh(a*x)^2*polylog(2,a*x+(a^2*x^2+1)^(1/2))+3*a^2*arcsinh(a*x)*polylog(3,a
*x+(a^2*x^2+1)^(1/2))-3*a^2*polylog(4,a*x+(a^2*x^2+1)^(1/2))-3*a^2*arcsinh(a*x)*ln(1+a*x+(a^2*x^2+1)^(1/2))-3*
a^2*polylog(2,-a*x-(a^2*x^2+1)^(1/2))+3*a^2*arcsinh(a*x)*ln(1-a*x-(a^2*x^2+1)^(1/2))+3*a^2*polylog(2,a*x+(a^2*
x^2+1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/x^3/(a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^3/(sqrt(a^2*x^2 + 1)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{3}}{a^{2} x^{5} + x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/x^3/(a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*x^2 + 1)*arcsinh(a*x)^3/(a^2*x^5 + x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3/x**3/(a**2*x**2+1)**(1/2),x)

[Out]

Integral(asinh(a*x)**3/(x**3*sqrt(a**2*x**2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/x^3/(a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^3/(sqrt(a^2*x^2 + 1)*x^3), x)