\(\int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx\) [349]

Optimal result

Integrand size = 23, antiderivative size = 210 \[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=-\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}-6 a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right ) \]

[Out]

-3/2*a*arcsinh(a*x)^2/x-6*a^2*arcsinh(a*x)*arctanh(a*x+(a^2*x^2+1)^(1/2))+a^2*arcsinh(a*x)^3*arctanh(a*x+(a^2*
x^2+1)^(1/2))-3*a^2*polylog(2,-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*polylog(2,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*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*polylog(4,a*x+(a^2*x^2+1)^(1/2))-1/2*arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5809, 5816, 4267, 2611, 6744, 2320, 6724, 5776, 2317, 2438} \[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-6 a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right )-\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 x^2}-\frac {3 a \text {arcsinh}(a x)^2}{2 x} \]

[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 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 4267

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 5776

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 5809

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/(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*ArcSinh[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5816

Int[(((a_.) + ArcSinh[(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*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 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 {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\text {arcsinh}(a x)^2}{x^2} \, dx-\frac {1}{2} a^2 \int \frac {\text {arcsinh}(a x)^3}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}-\frac {1}{2} a^2 \text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\text {arcsinh}(a x)\right )+\left (3 a^2\right ) \int \frac {\text {arcsinh}(a x)}{x \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}+a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )-\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+\left (3 a^2\right ) \text {Subst}(\int x \text {csch}(x) \, dx,x,\text {arcsinh}(a x)) \\ & = -\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}-6 a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\left (3 a^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+\left (3 a^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )-\left (3 a^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )+\left (3 a^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}-6 a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )-\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-\left (3 a^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx,x,\text {arcsinh}(a x)\right )-\left (3 a^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^x\right ) \, dx,x,\text {arcsinh}(a x)\right ) \\ & = -\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}-6 a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right )-\left (3 a^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{\text {arcsinh}(a x)}\right ) \\ & = -\frac {3 a \text {arcsinh}(a x)^2}{2 x}-\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{2 x^2}-6 a^2 \text {arcsinh}(a x) \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )+a^2 \text {arcsinh}(a x)^3 \text {arctanh}\left (e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-\frac {3}{2} a^2 \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(a x)}\right )+3 a^2 \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(a x)}\right )-3 a^2 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 3.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.45 \[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=\frac {a \left (-a \pi ^4 x+2 a x \text {arcsinh}(a x)^4-12 a x \text {arcsinh}(a x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(a x)\right )-2 a x \text {arcsinh}(a x)^3 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(a x)\right )+48 a x \text {arcsinh}(a x) \log \left (1-e^{-\text {arcsinh}(a x)}\right )-48 a x \text {arcsinh}(a x) \log \left (1+e^{-\text {arcsinh}(a x)}\right )+8 a x \text {arcsinh}(a x)^3 \log \left (1+e^{-\text {arcsinh}(a x)}\right )-8 a x \text {arcsinh}(a x)^3 \log \left (1-e^{\text {arcsinh}(a x)}\right )-24 a x \left (-2+\text {arcsinh}(a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(a x)}\right )-48 a x \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(a x)}\right )-24 a x \text {arcsinh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(a x)}\right )-48 a x \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(a x)}\right )+48 a x \text {arcsinh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(a x)}\right )-48 a x \operatorname {PolyLog}\left (4,-e^{-\text {arcsinh}(a x)}\right )-48 a x \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(a x)}\right )+12 a x \text {arcsinh}(a x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )-4 \text {arcsinh}(a x)^3 \tanh \left (\frac {1}{2} \text {arcsinh}(a x)\right )\right )}{16 x} \]

[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)

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.80

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

[In]

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

[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))

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{x^{3} \sqrt {a^{2} x^{2} + 1}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} x^{2} + 1} x^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{x^3 \sqrt {1+a^2 x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{x^3\,\sqrt {a^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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