Optimal. Leaf size=82 \[ -\frac{a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.170999, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6030, 6008, 266, 63, 208, 5958} \[ -\frac{a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 6030
Rule 6008
Rule 266
Rule 63
Rule 208
Rule 5958
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac{\tanh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+a \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a}\\ &=-\frac{a}{\sqrt{1-a^2 x^2}}+\frac{a^2 x \tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.109833, size = 89, normalized size = 1.09 \[ \frac{a x \left (\sqrt{1-a^2 x^2} \log (x)-\sqrt{1-a^2 x^2} \log \left (\sqrt{1-a^2 x^2}+1\right )-1\right )+\left (2 a^2 x^2-1\right ) \tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.253, size = 132, normalized size = 1.6 \begin{align*} -{\frac{a \left ({\it Artanh} \left ( ax \right ) -1 \right ) }{2\,ax-2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{ \left ({\it Artanh} \left ( ax \right ) +1 \right ) a}{2\,ax+2}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{\it Artanh} \left ( ax \right ) }{x}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+a\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-1 \right ) -a\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957291, size = 113, normalized size = 1.38 \begin{align*} -a{\left (\frac{1}{\sqrt{-a^{2} x^{2} + 1}} + \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )\right )} +{\left (\frac{2 \, a^{2} x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} x}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06518, size = 223, normalized size = 2.72 \begin{align*} -\frac{2 \, a^{3} x^{3} - 2 \, a x - 2 \,{\left (a^{3} x^{3} - a x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x -{\left (2 \, a^{2} x^{2} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )}}{2 \,{\left (a^{2} 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^{2} \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 [B] time = 1.26088, size = 209, normalized size = 2.55 \begin{align*} -\frac{1}{2} \, a \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) + \frac{1}{2} \, a \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right ) + \frac{1}{4} \,{\left (\frac{a^{4} x}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2} x}{a^{2} x^{2} - 1} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{x{\left | a \right |}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - \frac{a}{\sqrt{-a^{2} x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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