Optimal. Leaf size=54 \[ \frac{\text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^2}-\frac{\tanh ^{-1}(a x)^2}{2 a^2}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^2} \]
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Rubi [A] time = 0.0706783, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5984, 5918, 2402, 2315} \[ \frac{\text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{2 a^2}-\frac{\tanh ^{-1}(a x)^2}{2 a^2}+\frac{\log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx &=-\frac{\tanh ^{-1}(a x)^2}{2 a^2}+\frac{\int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(a x)^2}{2 a^2}+\frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{a^2}-\frac{\int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a}\\ &=-\frac{\tanh ^{-1}(a x)^2}{2 a^2}+\frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )}{a^2}\\ &=-\frac{\tanh ^{-1}(a x)^2}{2 a^2}+\frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{a^2}+\frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0485874, size = 44, normalized size = 0.81 \[ -\frac{\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x) \left (\tanh ^{-1}(a x)+2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.046, size = 125, normalized size = 2.3 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{2\,{a}^{2}}}-{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{2\,{a}^{2}}}-{\frac{ \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{8\,{a}^{2}}}+{\frac{1}{2\,{a}^{2}}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{\ln \left ( ax-1 \right ) }{4\,{a}^{2}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{\ln \left ( ax+1 \right ) }{4\,{a}^{2}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{1}{4\,{a}^{2}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{ \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{8\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.967308, size = 169, normalized size = 3.13 \begin{align*} -\frac{1}{8} \, a{\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^{3}} - \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^{3}}\right )} + \frac{{\left (\frac{\log \left (a x + 1\right )}{a} - \frac{\log \left (a x - 1\right )}{a}\right )} \log \left (a^{2} x^{2} - 1\right )}{4 \, a} - \frac{\operatorname{artanh}\left (a x\right ) \log \left (a^{2} x^{2} - 1\right )}{2 \, a^{2}} \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{x \operatorname{artanh}\left (a x\right )}{a^{2} x^{2} - 1}, 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{x \operatorname{atanh}{\left (a x \right )}}{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{x \operatorname{artanh}\left (a x\right )}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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