3.31 \(\int \frac{\sinh ^{-1}(a x)^3}{x^5} \, dx\)

Optimal. Leaf size=159 \[ -\frac{1}{2} a^4 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac{a^3 \sqrt{a^2 x^2+1}}{4 x}+\frac{a^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x}-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}-\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{1}{2} a^4 \sinh ^{-1}(a x)^2-a^4 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac{\sinh ^{-1}(a x)^3}{4 x^4} \]

[Out]

-(a^3*Sqrt[1 + a^2*x^2])/(4*x) - (a^2*ArcSinh[a*x])/(4*x^2) + (a^4*ArcSinh[a*x]^2)/2 - (a*Sqrt[1 + a^2*x^2]*Ar
cSinh[a*x]^2)/(4*x^3) + (a^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x) - ArcSinh[a*x]^3/(4*x^4) - a^4*ArcSinh[a*
x]*Log[1 - E^(2*ArcSinh[a*x])] - (a^4*PolyLog[2, E^(2*ArcSinh[a*x])])/2

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Rubi [A]  time = 0.285884, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {5661, 5747, 5723, 5659, 3716, 2190, 2279, 2391, 264} \[ -\frac{1}{2} a^4 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac{a^3 \sqrt{a^2 x^2+1}}{4 x}+\frac{a^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x}-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}-\frac{a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{1}{2} a^4 \sinh ^{-1}(a x)^2-a^4 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac{\sinh ^{-1}(a x)^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^3/x^5,x]

[Out]

-(a^3*Sqrt[1 + a^2*x^2])/(4*x) - (a^2*ArcSinh[a*x])/(4*x^2) + (a^4*ArcSinh[a*x]^2)/2 - (a*Sqrt[1 + a^2*x^2]*Ar
cSinh[a*x]^2)/(4*x^3) + (a^3*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*x) - ArcSinh[a*x]^3/(4*x^4) - a^4*ArcSinh[a*
x]*Log[1 - E^(2*ArcSinh[a*x])] - (a^4*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 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 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]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}(a x)^3}{x^5} \, dx &=-\frac{\sinh ^{-1}(a x)^3}{4 x^4}+\frac{1}{4} (3 a) \int \frac{\sinh ^{-1}(a x)^2}{x^4 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 x^3}-\frac{\sinh ^{-1}(a x)^3}{4 x^4}+\frac{1}{2} a^2 \int \frac{\sinh ^{-1}(a x)}{x^3} \, dx-\frac{1}{2} a^3 \int \frac{\sinh ^{-1}(a x)^2}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{4 x^4}+\frac{1}{4} a^3 \int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx-a^4 \int \frac{\sinh ^{-1}(a x)}{x} \, dx\\ &=-\frac{a^3 \sqrt{1+a^2 x^2}}{4 x}-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{4 x^4}-a^4 \operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{a^3 \sqrt{1+a^2 x^2}}{4 x}-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \sinh ^{-1}(a x)^2-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{4 x^4}+\left (2 a^4\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{a^3 \sqrt{1+a^2 x^2}}{4 x}-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \sinh ^{-1}(a x)^2-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{4 x^4}-a^4 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+a^4 \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac{a^3 \sqrt{1+a^2 x^2}}{4 x}-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \sinh ^{-1}(a x)^2-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{4 x^4}-a^4 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-\frac{a^3 \sqrt{1+a^2 x^2}}{4 x}-\frac{a^2 \sinh ^{-1}(a x)}{4 x^2}+\frac{1}{2} a^4 \sinh ^{-1}(a x)^2-\frac{a \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{4 x^3}+\frac{a^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{2 x}-\frac{\sinh ^{-1}(a x)^3}{4 x^4}-a^4 \sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac{1}{2} a^4 \text{Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.606866, size = 107, normalized size = 0.67 \[ \frac{1}{4} \left (a^4 \left (2 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(a x)}\right )-\frac{\sqrt{a^2 x^2+1} \left (\left (\frac{1}{a^2 x^2}-2\right ) \sinh ^{-1}(a x)^2+1\right )}{a x}-\sinh ^{-1}(a x) \left (\frac{1}{a^2 x^2}+2 \sinh ^{-1}(a x)+4 \log \left (1-e^{-2 \sinh ^{-1}(a x)}\right )\right )\right )-\frac{\sinh ^{-1}(a x)^3}{x^4}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSinh[a*x]^3/x^5,x]

[Out]

(-(ArcSinh[a*x]^3/x^4) + a^4*(-((Sqrt[1 + a^2*x^2]*(1 + (-2 + 1/(a^2*x^2))*ArcSinh[a*x]^2))/(a*x)) - ArcSinh[a
*x]*(1/(a^2*x^2) + 2*ArcSinh[a*x] + 4*Log[1 - E^(-2*ArcSinh[a*x])]) + 2*PolyLog[2, E^(-2*ArcSinh[a*x])]))/4

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Maple [A]  time = 0.106, size = 210, normalized size = 1.3 \begin{align*}{\frac{{a}^{4} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2}}+{\frac{{a}^{3} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{2\,x}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{{a}^{4}}{4}}-{\frac{{a}^{3}}{4\,x}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{a \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{4\,{x}^{3}}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}{\it Arcsinh} \left ( ax \right ) }{4\,{x}^{2}}}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}}{4\,{x}^{4}}}-{a}^{4}{\it Arcsinh} \left ( ax \right ) \ln \left ( 1+ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) -{a}^{4}{\it polylog} \left ( 2,-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) -{a}^{4}{\it Arcsinh} \left ( ax \right ) \ln \left ( 1-ax-\sqrt{{a}^{2}{x}^{2}+1} \right ) -{a}^{4}{\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^5,x)

[Out]

1/2*a^4*arcsinh(a*x)^2+1/2*a^3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/x+1/4*a^4-1/4*a^3*(a^2*x^2+1)^(1/2)/x-1/4*a*ar
csinh(a*x)^2*(a^2*x^2+1)^(1/2)/x^3-1/4*a^2*arcsinh(a*x)/x^2-1/4*arcsinh(a*x)^3/x^4-a^4*arcsinh(a*x)*ln(1+a*x+(
a^2*x^2+1)^(1/2))-a^4*polylog(2,-a*x-(a^2*x^2+1)^(1/2))-a^4*arcsinh(a*x)*ln(1-a*x-(a^2*x^2+1)^(1/2))-a^4*polyl
og(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}}{4 \, x^{4}} + \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}}{4 \,{\left (a^{3} x^{7} + a x^{5} +{\left (a^{2} x^{6} + x^{4}\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^5,x, algorithm="maxima")

[Out]

-1/4*log(a*x + sqrt(a^2*x^2 + 1))^3/x^4 + integrate(3/4*(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^7 + a*x^5 + (a^2*x^6 + x^4)*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^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/x^5,x, algorithm="fricas")

[Out]

integral(arcsinh(a*x)^3/x^5, 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^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3/x**5,x)

[Out]

Integral(asinh(a*x)**3/x**5, 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/x^5,x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^3/x^5, x)