Optimal. Leaf size=45 \[ -\frac {1}{2} \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{2} \tanh ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.09, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5988, 5932, 2447} \[ -\frac {1}{2} \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )+\frac {1}{2} \tanh ^{-1}(a x)^2+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 2447
Rule 5932
Rule 5988
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx &=\frac {1}{2} \tanh ^{-1}(a x)^2+\int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 42, normalized size = 0.93 \[ \frac {1}{2} \left (\tanh ^{-1}(a x) \left (\tanh ^{-1}(a x)+2 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-\text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )}{a^{2} x^{3} - x}, x\right ) \]
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 \[ \int -\frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 130, normalized size = 2.89 \[ \arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {1}{2}+\frac {a x}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 132, normalized size = 2.93 \[ \frac {1}{8} \, a {\left (\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - \frac {1}{2} \, {\left (\log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ -\int \frac {\mathrm {atanh}\left (a\,x\right )}{x\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {\operatorname {atanh}{\left (a x \right )}}{a^{2} x^{3} - x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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