Optimal. Leaf size=28 \[ \frac{1}{2} \text{PolyLog}\left (2,-\frac{1}{a x}\right )-\frac{1}{2} \text{PolyLog}\left (2,\frac{1}{a x}\right ) \]
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Rubi [A] time = 0.0097539, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5913} \[ \frac{1}{2} \text{PolyLog}\left (2,-\frac{1}{a x}\right )-\frac{1}{2} \text{PolyLog}\left (2,\frac{1}{a x}\right ) \]
Antiderivative was successfully verified.
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Rule 5913
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a x)}{x} \, dx &=\frac{1}{2} \text{Li}_2\left (-\frac{1}{a x}\right )-\frac{1}{2} \text{Li}_2\left (\frac{1}{a x}\right )\\ \end{align*}
Mathematica [A] time = 0.0081032, size = 26, normalized size = 0.93 \[ \frac{1}{2} \left (\text{PolyLog}\left (2,-\frac{1}{a x}\right )-\text{PolyLog}\left (2,\frac{1}{a x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 37, normalized size = 1.3 \begin{align*} \ln \left ( ax \right ){\rm arccoth} \left (ax\right )-{\frac{{\it dilog} \left ( ax \right ) }{2}}-{\frac{{\it dilog} \left ( ax+1 \right ) }{2}}-{\frac{\ln \left ( ax \right ) \ln \left ( ax+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.997304, size = 116, normalized size = 4.14 \begin{align*} -\frac{1}{2} \, a{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \log \left (x\right ) - \frac{1}{2} \, a{\left (\frac{\log \left (a x - 1\right ) \log \left (a x\right ) +{\rm Li}_2\left (-a x + 1\right )}{a} - \frac{\log \left (a x + 1\right ) \log \left (-a x\right ) +{\rm Li}_2\left (a x + 1\right )}{a}\right )} + \operatorname{arcoth}\left (a x\right ) \log \left (x\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{arcoth}\left (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{acoth}{\left (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{arcoth}\left (a x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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