Optimal. Leaf size=48 \[ -\frac{1}{2} \text{PolyLog}(2,-a x)+\frac{1}{2} \text{PolyLog}(2,a x)-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac{a x}{2}+\frac{1}{2} \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.0473091, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6014, 5912, 5916, 321, 206} \[ -\frac{1}{2} \text{PolyLog}(2,-a x)+\frac{1}{2} \text{PolyLog}(2,a x)-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac{a x}{2}+\frac{1}{2} \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6014
Rule 5912
Rule 5916
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{x} \, dx &=-\left (a^2 \int x \tanh ^{-1}(a x) \, dx\right )+\int \frac{\tanh ^{-1}(a x)}{x} \, dx\\ &=-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac{\text{Li}_2(-a x)}{2}+\frac{\text{Li}_2(a x)}{2}+\frac{1}{2} a^3 \int \frac{x^2}{1-a^2 x^2} \, dx\\ &=-\frac{a x}{2}-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac{\text{Li}_2(-a x)}{2}+\frac{\text{Li}_2(a x)}{2}+\frac{1}{2} a \int \frac{1}{1-a^2 x^2} \, dx\\ &=-\frac{a x}{2}+\frac{1}{2} \tanh ^{-1}(a x)-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac{\text{Li}_2(-a x)}{2}+\frac{\text{Li}_2(a x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0177563, size = 60, normalized size = 1.25 \[ \frac{1}{2} (\text{PolyLog}(2,a x)-\text{PolyLog}(2,-a x))-\frac{1}{2} a^2 x^2 \tanh ^{-1}(a x)-\frac{a x}{2}-\frac{1}{4} \log (1-a x)+\frac{1}{4} \log (a x+1) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 69, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) }{2}}+{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) -{\frac{ax}{2}}-{\frac{\ln \left ( ax-1 \right ) }{4}}+{\frac{\ln \left ( ax+1 \right ) }{4}}-{\frac{{\it dilog} \left ( ax \right ) }{2}}-{\frac{{\it dilog} \left ( ax+1 \right ) }{2}}-{\frac{\ln \left ( ax \right ) \ln \left ( ax+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.952142, size = 120, normalized size = 2.5 \begin{align*} -\frac{1}{4} \, a{\left (2 \, x + \frac{2 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )}}{a} - \frac{2 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\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 (a^{2} x^{2} - \log \left (x^{2}\right )\right )} \operatorname{artanh}\left (a x\right ) \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{{\left (a^{2} x^{2} - 1\right )} \operatorname{artanh}\left (a x\right )}{x}, 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{\operatorname{atanh}{\left (a x \right )}}{x}\, dx - \int a^{2} x \operatorname{atanh}{\left (a x \right )}\, 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{{\left (a^{2} x^{2} - 1\right )} \operatorname{artanh}\left (a x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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