Optimal. Leaf size=93 \[ -a^2 x \tanh ^{-1}(a x)^2+a \text {Li}_2\left (1-\frac {2}{1-a x}\right )-a \text {Li}_2\left (\frac {2}{a x+1}-1\right )-\frac {\tanh ^{-1}(a x)^2}{x}+2 a \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)+2 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.22, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6014, 5916, 5988, 5932, 2447, 5910, 5984, 5918, 2402, 2315} \[ a \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )-a \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-a^2 x \tanh ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)^2}{x}+2 a \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)+2 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 2447
Rule 5910
Rule 5916
Rule 5918
Rule 5932
Rule 5984
Rule 5988
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{x^2} \, dx &=-\left (a^2 \int \tanh ^{-1}(a x)^2 \, dx\right )+\int \frac {\tanh ^{-1}(a x)^2}{x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{x}-a^2 x \tanh ^{-1}(a x)^2+(2 a) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\left (2 a^3\right ) \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{x}-a^2 x \tanh ^{-1}(a x)^2+(2 a) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\left (2 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{x}-a^2 x \tanh ^{-1}(a x)^2+2 a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\left (2 a^2\right ) \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx-\left (2 a^2\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^2}{x}-a^2 x \tanh ^{-1}(a x)^2+2 a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+(2 a) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )\\ &=-\frac {\tanh ^{-1}(a x)^2}{x}-a^2 x \tanh ^{-1}(a x)^2+2 a \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+a \text {Li}_2\left (1-\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 = 0.14, size = 102, normalized size = 1.10 \[ -a \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+a \left (\tanh ^{-1}(a x) \left (-\frac {\tanh ^{-1}(a x)}{a x}+\tanh ^{-1}(a x)+2 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-\text {Li}_2\left (e^{-2 \tanh ^{-1}(a x)}\right )\right )-a \tanh ^{-1}(a x) \left (a x \tanh ^{-1}(a x)-\tanh ^{-1}(a x)-2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 170, normalized size = 1.83 \[ -a^{2} x \arctanh \left (a x \right )^{2}-\frac {\arctanh \left (a x \right )^{2}}{x}+2 a \arctanh \left (a x \right ) \ln \left (a x \right )-2 a \arctanh \left (a x \right ) \ln \left (a x -1\right )-2 a \arctanh \left (a x \right ) \ln \left (a x +1\right )-a \dilog \left (a x \right )-a \dilog \left (a x +1\right )-a \ln \left (a x \right ) \ln \left (a x +1\right )-\frac {a \ln \left (a x -1\right )^{2}}{2}+2 a \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )+a \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )+\frac {a \ln \left (a x +1\right )^{2}}{2}-a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )+a \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 152, normalized size = 1.63 \[ \frac {1}{2} \, a^{2} {\left (\frac {\log \left (a x + 1\right )^{2} - 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a} + \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a} - \frac {2 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{a} + \frac {2 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{a}\right )} - 2 \, a {\left (\log \left (a x + 1\right ) + \log \left (a x - 1\right ) - \log \relax (x)\right )} \operatorname {artanh}\left (a x\right ) - {\left (a^{2} x + \frac {1}{x}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
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 \[ \int -\frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )}{x^2} \,d x \]
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 \[ - \int a^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx - \int \left (- \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{2}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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