Optimal. Leaf size=52 \[ -\frac{1}{2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )+\frac{1}{2} \sinh ^{-1}\left (\frac{a}{x}\right )^2-\sinh ^{-1}\left (\frac{a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right ) \]
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Rubi [A] time = 0.0619257, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5890, 3716, 2190, 2279, 2391} \[ -\frac{1}{2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )+\frac{1}{2} \sinh ^{-1}\left (\frac{a}{x}\right )^2-\sinh ^{-1}\left (\frac{a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right ) \]
Antiderivative was successfully verified.
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Rule 5890
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}\left (\frac{a}{x}\right )}{x} \, dx &=-\operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{a}{x}\right )\right )\\ &=\frac{1}{2} \sinh ^{-1}\left (\frac{a}{x}\right )^2+2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{a}{x}\right )\right )\\ &=\frac{1}{2} \sinh ^{-1}\left (\frac{a}{x}\right )^2-\sinh ^{-1}\left (\frac{a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )+\operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{a}{x}\right )\right )\\ &=\frac{1}{2} \sinh ^{-1}\left (\frac{a}{x}\right )^2-\sinh ^{-1}\left (\frac{a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )\\ &=\frac{1}{2} \sinh ^{-1}\left (\frac{a}{x}\right )^2-\sinh ^{-1}\left (\frac{a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )-\frac{1}{2} \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )\\ \end{align*}
Mathematica [A] time = 0.0073577, size = 52, normalized size = 1. \[ -\frac{1}{2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right )+\frac{1}{2} \sinh ^{-1}\left (\frac{a}{x}\right )^2-\sinh ^{-1}\left (\frac{a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{a}{x}\right )}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 114, normalized size = 2.2 \begin{align*}{\frac{1}{2} \left ({\it Arcsinh} \left ({\frac{a}{x}} \right ) \right ) ^{2}}-{\it Arcsinh} \left ({\frac{a}{x}} \right ) \ln \left ( 1+{\frac{a}{x}}+\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}} \right ) -{\it polylog} \left ( 2,-{\frac{a}{x}}-\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}} \right ) -{\it Arcsinh} \left ({\frac{a}{x}} \right ) \ln \left ( 1-{\frac{a}{x}}-\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}} \right ) -{\it polylog} \left ( 2,{\frac{a}{x}}+\sqrt{1+{\frac{{a}^{2}}{{x}^{2}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \int \frac{x \log \left (x\right )}{a^{3} + a x^{2} +{\left (a^{2} + x^{2}\right )}^{\frac{3}{2}}}\,{d x} + \log \left (a + \sqrt{a^{2} + x^{2}}\right ) \log \left (x\right ) - \frac{1}{2} \, \log \left (x\right )^{2} - \frac{1}{2} \, \log \left (x\right ) \log \left (\frac{x^{2}}{a^{2}} + 1\right ) - \frac{1}{4} \,{\rm Li}_2\left (-\frac{x^{2}}{a^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arsinh}\left (\frac{a}{x}\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (\frac{a}{x}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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