Optimal. Leaf size=28 \[ \frac {1}{2} \text {Li}_2\left (-\frac {1}{a x}\right )-\frac {1}{2} \text {Li}_2\left (\frac {1}{a x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ \frac {1}{2} \left (\text {Li}_2\left (-\frac {1}{a x}\right )-\text {Li}_2\left (\frac {1}{a x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 1.32 \[ \ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 86, normalized size = 3.07 \[ -\frac {1}{2} \, a {\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \log \relax (x) - \frac {1}{2} \, a {\left (\frac {\log \left (a x - 1\right ) \log \left (a x\right ) + {\rm Li}_2\left (-a x + 1\right )}{a} - \frac {\log \left (a x + 1\right ) \log \left (-a x\right ) + {\rm Li}_2\left (a x + 1\right )}{a}\right )} + \operatorname {arcoth}\left (a x\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {acoth}\left (a\,x\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acoth}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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