Optimal. Leaf size=46 \[ \frac{a}{2 (1-a x)}+a \log (x)-\frac{5}{4} a \log (1-a x)+\frac{1}{4} a \log (a x+1)-\frac{1}{x} \]
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Rubi [A] time = 0.109848, antiderivative size = 46, 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, 88} \[ \frac{a}{2 (1-a x)}+a \log (x)-\frac{5}{4} a \log (1-a x)+\frac{1}{4} a \log (a x+1)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac{1}{x^2 (1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac{1}{x^2}+\frac{a}{x}+\frac{a^2}{2 (-1+a x)^2}-\frac{5 a^2}{4 (-1+a x)}+\frac{a^2}{4 (1+a x)}\right ) \, dx\\ &=-\frac{1}{x}+\frac{a}{2 (1-a x)}+a \log (x)-\frac{5}{4} a \log (1-a x)+\frac{1}{4} a \log (1+a x)\\ \end{align*}
Mathematica [A] time = 0.035168, size = 46, normalized size = 1. \[ \frac{a}{2 (1-a x)}+a \log (x)-\frac{5}{4} a \log (1-a x)+\frac{1}{4} a \log (a x+1)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 39, normalized size = 0.9 \begin{align*} -{x}^{-1}+a\ln \left ( x \right ) +{\frac{a\ln \left ( ax+1 \right ) }{4}}-{\frac{a}{2\,ax-2}}-{\frac{5\,a\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 = 0.962485, size = 57, normalized size = 1.24 \begin{align*} \frac{1}{4} \, a \log \left (a x + 1\right ) - \frac{5}{4} \, a \log \left (a x - 1\right ) + a \log \left (x\right ) - \frac{3 \, a x - 2}{2 \,{\left (a x^{2} - x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73207, size = 163, normalized size = 3.54 \begin{align*} -\frac{6 \, a x -{\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right ) + 5 \,{\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 4 \,{\left (a^{2} x^{2} - a x\right )} \log \left (x\right ) - 4}{4 \,{\left (a x^{2} - x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.525264, size = 42, normalized size = 0.91 \begin{align*} a \log{\left (x \right )} - \frac{5 a \log{\left (x - \frac{1}{a} \right )}}{4} + \frac{a \log{\left (x + \frac{1}{a} \right )}}{4} - \frac{3 a x - 2}{2 a x^{2} - 2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13994, size = 59, normalized size = 1.28 \begin{align*} \frac{1}{4} \, a \log \left ({\left | a x + 1 \right |}\right ) - \frac{5}{4} \, a \log \left ({\left | a x - 1 \right |}\right ) + a \log \left ({\left | x \right |}\right ) - \frac{3 \, a x - 2}{2 \,{\left (a x - 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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