3.29 \(\int \frac{\sinh ^{-1}(a x)^3}{x^3} \, dx\)

Optimal. Leaf size=93 \[ \frac{3}{2} a^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac{3 a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x}-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac{\sinh ^{-1}(a x)^3}{2 x^2} \]

[Out]

(-3*a^2*ArcSinh[a*x]^2)/2 - (3*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x) - ArcSinh[a*x]^3/(2*x^2) + 3*a^2*ArcS
inh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] + (3*a^2*PolyLog[2, E^(2*ArcSinh[a*x])])/2

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Rubi [A]  time = 0.168133, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5661, 5723, 5659, 3716, 2190, 2279, 2391} \[ \frac{3}{2} a^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac{3 a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x}-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac{\sinh ^{-1}(a x)^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^3/x^3,x]

[Out]

(-3*a^2*ArcSinh[a*x]^2)/2 - (3*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x) - ArcSinh[a*x]^3/(2*x^2) + 3*a^2*ArcS
inh[a*x]*Log[1 - E^(2*ArcSinh[a*x])] + (3*a^2*PolyLog[2, E^(2*ArcSinh[a*x])])/2

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 5723

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[(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*Arc
Sinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 5659

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tanh[x], x], x, ArcSinh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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} \, dx &=-\frac{\sinh ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} (3 a) \int \frac{\sinh ^{-1}(a x)^2}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac{\sinh ^{-1}(a x)}{x} \, dx\\ &=-\frac{3 a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2-\frac{3 a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{2 x^2}-\left (6 a^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2-\frac{3 a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{2 x^2}+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2-\frac{3 a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{2 x^2}+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{2} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2-\frac{3 a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{2 x^2}+3 a^2 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac{3}{2} a^2 \text{Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.317221, size = 80, normalized size = 0.86 \[ -\frac{\sinh ^{-1}(a x)^3-3 a x \left (\sinh ^{-1}(a x) \left (\left (a x-\sqrt{a^2 x^2+1}\right ) \sinh ^{-1}(a x)+2 a x \log \left (1-e^{-2 \sinh ^{-1}(a x)}\right )\right )-a x \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(a x)}\right )\right )}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSinh[a*x]^3/x^3,x]

[Out]

-(ArcSinh[a*x]^3 - 3*a*x*(ArcSinh[a*x]*((a*x - Sqrt[1 + a^2*x^2])*ArcSinh[a*x] + 2*a*x*Log[1 - E^(-2*ArcSinh[a
*x])]) - a*x*PolyLog[2, E^(-2*ArcSinh[a*x])]))/(2*x^2)

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Maple [A]  time = 0.068, size = 149, normalized size = 1.6 \begin{align*} -{\frac{3\,{a}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2}}-{\frac{3\,a \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2\,x}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{2\,{x}^{2}}}+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,x)

[Out]

-3/2*a^2*arcsinh(a*x)^2-3/2*a*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/x-1/2*arcsinh(a*x)^3/x^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*} -\frac{\log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3}}{2 \, x^{2}} + \int \frac{3 \,{\left (a^{3} x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{2 \,{\left (a^{3} x^{5} + a x^{3} +{\left (a^{2} x^{4} + x^{2}\right )} \sqrt{a^{2} x^{2} + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsinh(a*x)^3/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}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3/x**3,x)

[Out]

Integral(asinh(a*x)**3/x**3, 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}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x)^3/x^3, x)