3.304 \(\int \frac{\sinh ^{-1}(\frac{a}{x})}{x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{\sqrt{\frac{a^2}{x^2}+1}}{a}-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{x} \]

[Out]

Sqrt[1 + a^2/x^2]/a - ArcCsch[x/a]/x

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Rubi [A]  time = 0.0241453, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5892, 6284, 261} \[ \frac{\sqrt{\frac{a^2}{x^2}+1}}{a}-\frac{\text{csch}^{-1}\left (\frac{x}{a}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a/x]/x^2,x]

[Out]

Sqrt[1 + a^2/x^2]/a - ArcCsch[x/a]/x

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 261

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

Rubi steps

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

Mathematica [A]  time = 0.0154129, size = 29, normalized size = 1. \[ \frac{\sqrt{\frac{a^2}{x^2}+1}}{a}-\frac{\sinh ^{-1}\left (\frac{a}{x}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a/x]/x^2,x]

[Out]

Sqrt[1 + a^2/x^2]/a - ArcSinh[a/x]/x

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Maple [A]  time = 0.003, size = 31, normalized size = 1.1 \begin{align*} -{\frac{1}{a} \left ({\frac{a}{x}{\it Arcsinh} \left ({\frac{a}{x}} \right ) }-\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^2,x)

[Out]

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

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Maxima [A]  time = 1.11267, size = 41, normalized size = 1.41 \begin{align*} -\frac{\frac{a \operatorname{arsinh}\left (\frac{a}{x}\right )}{x} - \sqrt{\frac{a^{2}}{x^{2}} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a/x)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.68268, size = 101, normalized size = 3.48 \begin{align*} -\frac{a \log \left (\frac{x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) - x \sqrt{\frac{a^{2} + x^{2}}{x^{2}}}}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a/x)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.66167, size = 20, normalized size = 0.69 \begin{align*} \begin{cases} - \frac{\operatorname{asinh}{\left (\frac{a}{x} \right )}}{x} + \frac{\sqrt{\frac{a^{2}}{x^{2}} + 1}}{a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a/x)/x**2,x)

[Out]

Piecewise((-asinh(a/x)/x + sqrt(a**2/x**2 + 1)/a, Ne(a, 0)), (0, True))

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Giac [A]  time = 1.36858, size = 53, normalized size = 1.83 \begin{align*} -\frac{\log \left (\sqrt{\frac{a^{2}}{x^{2}} + 1} + \frac{a}{x}\right )}{x} + \frac{\sqrt{\frac{a^{2}}{x^{2}} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a/x)/x^2,x, algorithm="giac")

[Out]

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