Optimal. Leaf size=12 \[ \log (x)-\log (1-a x) \]
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Rubi [A] time = 0.0795024, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6150, 36, 29, 31} \[ \log (x)-\log (1-a x) \]
Antiderivative was successfully verified.
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Rule 6150
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x \sqrt{1-a^2 x^2}} \, dx &=\int \frac{1}{x (1-a x)} \, dx\\ &=a \int \frac{1}{1-a x} \, dx+\int \frac{1}{x} \, dx\\ &=\log (x)-\log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.006461, size = 12, normalized size = 1. \[ \log (x)-\log (1-a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 12, normalized size = 1. \begin{align*} \ln \left ( x \right ) -\ln \left ( ax-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95007, size = 15, normalized size = 1.25 \begin{align*} -\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 = 1.76315, size = 32, normalized size = 2.67 \begin{align*} -\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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Sympy [A] time = 0.111181, size = 8, normalized size = 0.67 \begin{align*} \log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15031, size = 18, normalized size = 1.5 \begin{align*} -\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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