Optimal. Leaf size=43 \[ \frac {1}{2} \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \sinh ^{-1}(a x)^2+\sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5659, 3716, 2190, 2279, 2391} \[ \frac {1}{2} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \sinh ^{-1}(a x)^2+\sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)}{x} \, dx &=\operatorname {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{2} \sinh ^{-1}(a x)^2-2 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{2} \sinh ^{-1}(a x)^2+\sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {1}{2} \sinh ^{-1}(a x)^2+\sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )\\ &=-\frac {1}{2} \sinh ^{-1}(a x)^2+\sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 43, normalized size = 1.00 \[ \frac {1}{2} \text {Li}_2\left (e^{2 \sinh ^{-1}(a x)}\right )-\frac {1}{2} \sinh ^{-1}(a x)^2+\sinh ^{-1}(a x) \log \left (1-e^{2 \sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 94, normalized size = 2.19 \[ -\frac {\arcsinh \left (a x \right )^{2}}{2}+\arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+\polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )+\arcsinh \left (a x \right ) \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )+\polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )}{x}\,{d x} \]
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 \[ \int \frac {\mathrm {asinh}\left (a\,x\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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