Optimal. Leaf size=43 \[ \frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{x}{a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0506343, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5994, 191} \[ \frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{x}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5994
Rule 191
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a}\\ &=-\frac{x}{a \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)}{a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0357683, size = 27, normalized size = 0.63 \[ \frac{\tanh ^{-1}(a x)-a x}{a^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.205, size = 66, normalized size = 1.5 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) -1}{2\,{a}^{2} \left ( ax-1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{\it Artanh} \left ( ax \right ) +1}{2\,{a}^{2} \left ( ax+1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955062, size = 53, normalized size = 1.23 \begin{align*} -\frac{x}{\sqrt{-a^{2} x^{2} + 1} a} + \frac{\operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0054, size = 103, normalized size = 2.4 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x - \log \left (-\frac{a x + 1}{a x - 1}\right )\right )}}{2 \,{\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}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23861, size = 82, normalized size = 1.91 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} a} + \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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