Optimal. Leaf size=91 \[ -\frac {a x}{4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}-\frac {1}{2} \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{4} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.16, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6030, 5988, 5932, 2447, 5994, 199, 206} \[ -\frac {1}{2} \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {a x}{4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{4} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 2447
Rule 5932
Rule 5988
Rule 5994
Rule 6030
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2-\frac {1}{2} a \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a x}{4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{4} a \int \frac {1}{1-a^2 x^2} \, dx-a \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a x}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)}{2 \left (1-a^2 x^2\right )}+\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 = 0.18, size = 63, normalized size = 0.69 \[ \frac {1}{8} \left (-4 \text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )+4 \tanh ^{-1}(a x)^2-\sinh \left (2 \tanh ^{-1}(a x)\right )+2 \tanh ^{-1}(a x) \left (4 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )+\cosh \left (2 \tanh ^{-1}(a x)\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )}{a^{4} x^{5} - 2 \, a^{2} x^{3} + x}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )}{{\left (a^{2} x^{2} - 1\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 190, normalized size = 2.09 \[ \arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right )}{4 \left (a x -1\right )}-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}+\frac {\arctanh \left (a x \right )}{4 a x +4}-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {1}{2}+\frac {a x}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {1}{8 a x -8}+\frac {\ln \left (a x -1\right )}{8}+\frac {1}{8 a x +8}-\frac {\ln \left (a x +1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 206, normalized size = 2.26 \[ \frac {1}{8} \, a {\left (\frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )^{2} + 2 \, a x}{a^{3} x^{2} - a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {4 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {4 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a} - \frac {\log \left (a x + 1\right )}{a} + \frac {\log \left (a x - 1\right )}{a}\right )} - \frac {1}{2} \, {\left (\frac {1}{a^{2} x^{2} - 1} + \log \left (a^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{x\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\left (a x \right )}}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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