Optimal. Leaf size=52 \[ -\frac {1}{2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} \sinh ^{-1}\left (\frac {a}{x}\right )^2-\sinh ^{-1}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {a}{x}\right )}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5890
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*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 1.00 \[ -\frac {1}{2} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {a}{x}\right )}\right )+\frac {1}{2} \sinh ^{-1}\left (\frac {a}{x}\right )^2-\sinh ^{-1}\left (\frac {a}{x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {a}{x}\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (\frac {a}{x}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 114, normalized size = 2.19 \[ \frac {\arcsinh \left (\frac {a}{x}\right )^{2}}{2}-\arcsinh \left (\frac {a}{x}\right ) \ln \left (1+\frac {a}{x}+\sqrt {1+\frac {a^{2}}{x^{2}}}\right )-\polylog \left (2, -\frac {a}{x}-\sqrt {1+\frac {a^{2}}{x^{2}}}\right )-\arcsinh \left (\frac {a}{x}\right ) \ln \left (1-\frac {a}{x}-\sqrt {1+\frac {a^{2}}{x^{2}}}\right )-\polylog \left (2, \frac {a}{x}+\sqrt {1+\frac {a^{2}}{x^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \int \frac {x \log \relax (x)}{a^{3} + a x^{2} + {\left (a^{2} + x^{2}\right )}^{\frac {3}{2}}}\,{d x} + \log \left (a + \sqrt {a^{2} + x^{2}}\right ) \log \relax (x) - \frac {1}{2} \, \log \relax (x)^{2} - \frac {1}{2} \, \log \relax (x) \log \left (\frac {x^{2}}{a^{2}} + 1\right ) - \frac {1}{4} \, {\rm Li}_2\left (-\frac {x^{2}}{a^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {asinh}\left (\frac {a}{x}\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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