Optimal. Leaf size=32 \[ a^2 \log (x)-a^2 \log (1-a x)-\frac{a}{x}-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.090373, antiderivative size = 32, 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, 44} \[ a^2 \log (x)-a^2 \log (1-a x)-\frac{a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^3 \sqrt{1-a^2 x^2}} \, dx &=\int \frac{1}{x^3 (1-a x)} \, dx\\ &=\int \left (\frac{1}{x^3}+\frac{a}{x^2}+\frac{a^2}{x}-\frac{a^3}{-1+a x}\right ) \, dx\\ &=-\frac{1}{2 x^2}-\frac{a}{x}+a^2 \log (x)-a^2 \log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.0136628, size = 32, normalized size = 1. \[ a^2 \log (x)-a^2 \log (1-a x)-\frac{a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 30, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-{\frac{a}{x}}+{a}^{2}\ln \left ( x \right ) -{a}^{2}\ln \left ( ax-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955309, size = 39, normalized size = 1.22 \begin{align*} -a^{2} \log \left (a x - 1\right ) + a^{2} \log \left (x\right ) - \frac{2 \, a x + 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67278, size = 89, normalized size = 2.78 \begin{align*} -\frac{2 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) + 2 \, a x + 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.311324, size = 26, normalized size = 0.81 \begin{align*} - a^{2} \left (- \log{\left (x \right )} + \log{\left (x - \frac{1}{a} \right )}\right ) - \frac{2 a x + 1}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1655, size = 42, normalized size = 1.31 \begin{align*} -a^{2} \log \left ({\left | a x - 1 \right |}\right ) + a^{2} \log \left ({\left | x \right |}\right ) - \frac{2 \, a x + 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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