Optimal. Leaf size=82 \[ \frac{1}{4 a^2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac{x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^2}{4 a^2} \]
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Rubi [A] time = 0.0679388, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5994, 5956, 261} \[ \frac{1}{4 a^2 \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac{x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^2}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 5994
Rule 5956
Rule 261
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx &=\frac{\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}-\frac{\int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{a}\\ &=-\frac{x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^2}{4 a^2}+\frac{\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}+\frac{1}{2} \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac{1}{4 a^2 \left (1-a^2 x^2\right )}-\frac{x \tanh ^{-1}(a x)}{2 a \left (1-a^2 x^2\right )}-\frac{\tanh ^{-1}(a x)^2}{4 a^2}+\frac{\tanh ^{-1}(a x)^2}{2 a^2 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0512578, size = 43, normalized size = 0.52 \[ \frac{\left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+1}{4 a^2-4 a^4 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 191, normalized size = 2.3 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2\,{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+{\frac{{\it Artanh} \left ( ax \right ) }{4\,{a}^{2} \left ( ax-1 \right ) }}+{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{4\,{a}^{2}}}+{\frac{{\it Artanh} \left ( ax \right ) }{4\,{a}^{2} \left ( ax+1 \right ) }}-{\frac{{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{4\,{a}^{2}}}+{\frac{ \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{16\,{a}^{2}}}+{\frac{1}{8\,{a}^{2}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{\ln \left ( ax+1 \right ) }{8\,{a}^{2}}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{ \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{16\,{a}^{2}}}-{\frac{\ln \left ( ax-1 \right ) }{8\,{a}^{2}}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{1}{8\,{a}^{2} \left ( ax-1 \right ) }}+{\frac{1}{8\,{a}^{2} \left ( ax+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.966322, size = 197, normalized size = 2.4 \begin{align*} \frac{{\left (\frac{2 \, x}{a^{2} x^{2} - 1} - \frac{\log \left (a x + 1\right )}{a} + \frac{\log \left (a x - 1\right )}{a}\right )} \operatorname{artanh}\left (a x\right )}{4 \, a} + \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} - 4}{16 \,{\left (a^{4} x^{2} - a^{2}\right )}} - \frac{\operatorname{artanh}\left (a x\right )^{2}}{2 \,{\left (a^{2} x^{2} - 1\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09665, size = 140, normalized size = 1.71 \begin{align*} \frac{4 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) -{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 4}{16 \,{\left (a^{4} x^{2} - a^{2}\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{x \operatorname{atanh}^{2}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, 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{x \operatorname{artanh}\left (a x\right )^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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