Optimal. Leaf size=34 \[ -\text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+\text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.0916102, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5760, 4182, 2279, 2391} \[ -\text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+\text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 5760
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx &=\operatorname{Subst}\left (\int x \text{csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+\operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-2 \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-\text{Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+\text{Li}_2\left (e^{\sinh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.096044, size = 57, normalized size = 1.68 \[ \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )+\sinh ^{-1}(a x) \left (\log \left (1-e^{-\sinh ^{-1}(a x)}\right )-\log \left (e^{-\sinh ^{-1}(a x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 42, normalized size = 1.2 \begin{align*} 2\,{\it dilog} \left ( \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{-1} \right ) -{\frac{1}{2}{\it dilog} \left ( \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{-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*} \int \frac{\operatorname{arsinh}\left (a x\right )}{\sqrt{a^{2} x^{2} + 1} x}\,{d x} \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{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}{a^{2} x^{3} + 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 (a x \right )}}{x \sqrt{a^{2} x^{2} + 1}}\, 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 (a x\right )}{\sqrt{a^{2} x^{2} + 1} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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