3.306 \(\int \frac{\sinh ^{-1}(\frac{a}{x})}{x^4} \, dx\)

Optimal. Leaf size=54 \[ \frac{\left (\frac{a^2}{x^2}+1\right )^{3/2}}{9 a^3}-\frac{\sqrt{\frac{a^2}{x^2}+1}}{3 a^3}-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]

[Out]

-Sqrt[1 + a^2/x^2]/(3*a^3) + (1 + a^2/x^2)^(3/2)/(9*a^3) - ArcCsch[x/a]/(3*x^3)

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Rubi [A]  time = 0.0399537, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5892, 6284, 266, 43} \[ \frac{\left (\frac{a^2}{x^2}+1\right )^{3/2}}{9 a^3}-\frac{\sqrt{\frac{a^2}{x^2}+1}}{3 a^3}-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a/x]/x^4,x]

[Out]

-Sqrt[1 + a^2/x^2]/(3*a^3) + (1 + a^2/x^2)^(3/2)/(9*a^3) - ArcCsch[x/a]/(3*x^3)

Rule 5892

Int[ArcSinh[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcCsch[a/c + (b*x^n)/c]^m, x] /
; FreeQ[{a, b, c, n, m}, x]

Rule 6284

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsch[c*
x]))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}\left (\frac{a}{x}\right )}{x^4} \, dx &=\int \frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{x^4} \, dx\\ &=-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{3 x^3}-\frac{1}{3} a \int \frac{1}{\sqrt{1+\frac{a^2}{x^2}} x^5} \, dx\\ &=-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{3 x^3}+\frac{1}{6} a \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+a^2 x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{3 x^3}+\frac{1}{6} a \operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \sqrt{1+a^2 x}}+\frac{\sqrt{1+a^2 x}}{a^2}\right ) \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{\sqrt{1+\frac{a^2}{x^2}}}{3 a^3}+\frac{\left (1+\frac{a^2}{x^2}\right )^{3/2}}{9 a^3}-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0281997, size = 48, normalized size = 0.89 \[ \left (\frac{1}{9 a x^2}-\frac{2}{9 a^3}\right ) \sqrt{\frac{a^2+x^2}{x^2}}-\frac{\sinh ^{-1}\left (\frac{a}{x}\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a/x]/x^4,x]

[Out]

(-2/(9*a^3) + 1/(9*a*x^2))*Sqrt[(a^2 + x^2)/x^2] - ArcSinh[a/x]/(3*x^3)

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Maple [A]  time = 0.006, size = 53, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}}{3\,{x}^{3}}{\it Arcsinh} \left ({\frac{a}{x}} \right ) }-{\frac{{a}^{2}}{9\,{x}^{2}}\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}}}+{\frac{2}{9}\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a/x)/x^4,x)

[Out]

-1/a^3*(1/3*a^3/x^3*arcsinh(a/x)-1/9*a^2/x^2*(1+a^2/x^2)^(1/2)+2/9*(1+a^2/x^2)^(1/2))

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Maxima [A]  time = 1.20935, size = 63, normalized size = 1.17 \begin{align*} \frac{1}{9} \, a{\left (\frac{{\left (\frac{a^{2}}{x^{2}} + 1\right )}^{\frac{3}{2}}}{a^{4}} - \frac{3 \, \sqrt{\frac{a^{2}}{x^{2}} + 1}}{a^{4}}\right )} - \frac{\operatorname{arsinh}\left (\frac{a}{x}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a/x)/x^4,x, algorithm="maxima")

[Out]

1/9*a*((a^2/x^2 + 1)^(3/2)/a^4 - 3*sqrt(a^2/x^2 + 1)/a^4) - 1/3*arcsinh(a/x)/x^3

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Fricas [A]  time = 2.75567, size = 136, normalized size = 2.52 \begin{align*} -\frac{3 \, a^{3} \log \left (\frac{x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) -{\left (a^{2} x - 2 \, x^{3}\right )} \sqrt{\frac{a^{2} + x^{2}}{x^{2}}}}{9 \, a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(3*a^3*log((x*sqrt((a^2 + x^2)/x^2) + a)/x) - (a^2*x - 2*x^3)*sqrt((a^2 + x^2)/x^2))/(a^3*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (\frac{a}{x} \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asinh(a/x)/x**4, x)

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Giac [A]  time = 1.39379, size = 101, normalized size = 1.87 \begin{align*} -\frac{\log \left (\sqrt{\frac{a^{2}}{x^{2}} + 1} + \frac{a}{x}\right )}{3 \, x^{3}} - \frac{4 \,{\left (a^{2} - 3 \,{\left (x - \sqrt{a^{2} + x^{2}}\right )}^{2}\right )} a}{9 \,{\left (a^{2} -{\left (x - \sqrt{a^{2} + x^{2}}\right )}^{2}\right )}^{3} \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*log(sqrt(a^2/x^2 + 1) + a/x)/x^3 - 4/9*(a^2 - 3*(x - sqrt(a^2 + x^2))^2)*a/((a^2 - (x - sqrt(a^2 + x^2))^
2)^3*sgn(x))