Optimal. Leaf size=56 \[ \frac{1}{2} a^2 \text{PolyLog}(2,-a x)-\frac{1}{2} a^2 \text{PolyLog}(2,a x)+\frac{1}{2} a^2 \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x} \]
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Rubi [A] time = 0.0509202, antiderivative size = 56, 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, 5916, 325, 206, 5912} \[ \frac{1}{2} a^2 \text{PolyLog}(2,-a x)-\frac{1}{2} a^2 \text{PolyLog}(2,a x)+\frac{1}{2} a^2 \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x} \]
Antiderivative was successfully verified.
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Rule 6014
Rule 5916
Rule 325
Rule 206
Rule 5912
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}{x^3} \, dx &=-\left (a^2 \int \frac{\tanh ^{-1}(a x)}{x} \, dx\right )+\int \frac{\tanh ^{-1}(a x)}{x^3} \, dx\\ &=-\frac{\tanh ^{-1}(a x)}{2 x^2}+\frac{1}{2} a^2 \text{Li}_2(-a x)-\frac{1}{2} a^2 \text{Li}_2(a x)+\frac{1}{2} a \int \frac{1}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac{a}{2 x}-\frac{\tanh ^{-1}(a x)}{2 x^2}+\frac{1}{2} a^2 \text{Li}_2(-a x)-\frac{1}{2} a^2 \text{Li}_2(a x)+\frac{1}{2} a^3 \int \frac{1}{1-a^2 x^2} \, dx\\ &=-\frac{a}{2 x}+\frac{1}{2} a^2 \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)}{2 x^2}+\frac{1}{2} a^2 \text{Li}_2(-a x)-\frac{1}{2} a^2 \text{Li}_2(a x)\\ \end{align*}
Mathematica [A] time = 0.0313162, size = 68, normalized size = 1.21 \[ -\frac{1}{2} a^2 (\text{PolyLog}(2,a x)-\text{PolyLog}(2,-a x))-\frac{1}{4} a^2 \log (1-a x)+\frac{1}{4} a^2 \log (a x+1)-\frac{\tanh ^{-1}(a x)}{2 x^2}-\frac{a}{2 x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.044, size = 87, normalized size = 1.6 \begin{align*} -{a}^{2}{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) -{\frac{{\it Artanh} \left ( ax \right ) }{2\,{x}^{2}}}-{\frac{a}{2\,x}}-{\frac{{a}^{2}\ln \left ( ax-1 \right ) }{4}}+{\frac{{a}^{2}\ln \left ( ax+1 \right ) }{4}}+{\frac{{a}^{2}{\it dilog} \left ( ax \right ) }{2}}+{\frac{{a}^{2}{\it dilog} \left ( ax+1 \right ) }{2}}+{\frac{{a}^{2}\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 [A] time = 0.956987, size = 109, normalized size = 1.95 \begin{align*} \frac{1}{4} \,{\left (2 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )} a - 2 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )} a + a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac{2}{x}\right )} a - \frac{1}{2} \,{\left (a^{2} \log \left (x^{2}\right ) + \frac{1}{x^{2}}\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^{3}}, 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^{3}}\, dx - \int \frac{a^{2} \operatorname{atanh}{\left (a x \right )}}{x}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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