Optimal. Leaf size=40 \[ \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{1}{a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0252848, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {5958} \[ \frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{1}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5958
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=-\frac{1}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0312152, size = 27, normalized size = 0.68 \[ \frac{a x \tanh ^{-1}(a x)-1}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.22, size = 38, normalized size = 1. \begin{align*} -{\frac{ax{\it Artanh} \left ( ax \right ) -1}{a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957387, size = 49, normalized size = 1.22 \begin{align*} \frac{x \operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98146, size = 101, normalized size = 2.52 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - 2\right )}}{2 \,{\left (a^{3} x^{2} - a\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 )}}{\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 = 1.22904, size = 80, normalized size = 2. \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x \log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \,{\left (a^{2} x^{2} - 1\right )}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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