Optimal. Leaf size=55 \[ -a \text {Li}_2\left (\frac {2}{a x+1}-1\right )+a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x) \]
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Rubi [A] time = 0.11, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5917, 5989, 5933, 2447} \[ -a \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 2447
Rule 5917
Rule 5933
Rule 5989
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx &=-\frac {\coth ^{-1}(a x)^2}{x}+(2 a) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+(2 a) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx\\ &=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\left (2 a^2\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=a \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 49, normalized size = 0.89 \[ -a \text {Li}_2\left (-e^{-2 \coth ^{-1}(a x)}\right )+\frac {(a x-1) \coth ^{-1}(a x)^2}{x}+2 a \coth ^{-1}(a x) \log \left (e^{-2 \coth ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 159, normalized size = 2.89 \[ -\frac {\mathrm {arccoth}\left (a x \right )^{2}}{x}+2 a \,\mathrm {arccoth}\left (a x \right ) \ln \left (a x \right )-a \,\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )-a \,\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )-\frac {a \ln \left (a x -1\right )^{2}}{4}+a \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )+\frac {a \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{2}+\frac {a \ln \left (a x +1\right )^{2}}{4}+\frac {a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{2}-\frac {a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{2}-a \dilog \left (a x \right )-a \dilog \left (a x +1\right )-a \ln \left (a x \right ) \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 146, normalized size = 2.65 \[ \frac {1}{4} \, a^{2} {\left (\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - a {\left (\log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{x} \]
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 \[ \int \frac {{\mathrm {acoth}\left (a\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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