Optimal. Leaf size=36 \[ \frac{1}{2 (1-a x)}-\frac{3}{4} \log (1-a x)-\frac{1}{4} \log (a x+1)+\log (x) \]
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Rubi [A] time = 0.108868, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 72} \[ \frac{1}{2 (1-a x)}-\frac{3}{4} \log (1-a x)-\frac{1}{4} \log (a x+1)+\log (x) \]
Antiderivative was successfully verified.
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Rule 6150
Rule 72
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac{1}{x (1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac{1}{x}+\frac{a}{2 (-1+a x)^2}-\frac{3 a}{4 (-1+a x)}-\frac{a}{4 (1+a x)}\right ) \, dx\\ &=\frac{1}{2 (1-a x)}+\log (x)-\frac{3}{4} \log (1-a x)-\frac{1}{4} \log (1+a x)\\ \end{align*}
Mathematica [A] time = 0.0308583, size = 32, normalized size = 0.89 \[ \frac{1}{2-2 a x}-\frac{3}{4} \log (1-a x)-\frac{1}{4} \log (a x+1)+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 29, normalized size = 0.8 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ( ax+1 \right ) }{4}}-{\frac{1}{2\,ax-2}}-{\frac{3\,\ln \left ( ax-1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944727, size = 38, normalized size = 1.06 \begin{align*} -\frac{1}{2 \,{\left (a x - 1\right )}} - \frac{1}{4} \, \log \left (a x + 1\right ) - \frac{3}{4} \, \log \left (a x - 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83539, size = 126, normalized size = 3.5 \begin{align*} -\frac{{\left (a x - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 4 \,{\left (a x - 1\right )} \log \left (x\right ) + 2}{4 \,{\left (a x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.419124, size = 29, normalized size = 0.81 \begin{align*} \log{\left (x \right )} - \frac{3 \log{\left (x - \frac{1}{a} \right )}}{4} - \frac{\log{\left (x + \frac{1}{a} \right )}}{4} - \frac{1}{2 a x - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13694, size = 42, normalized size = 1.17 \begin{align*} -\frac{1}{2 \,{\left (a x - 1\right )}} - \frac{1}{4} \, \log \left ({\left | a x + 1 \right |}\right ) - \frac{3}{4} \, \log \left ({\left | a x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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