Optimal. Leaf size=12 \[ \log (x)-\log (1-a x) \]
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Rubi [A] time = 0.08, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 6150
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*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \log (x)-\log (1-a x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 11, normalized size = 0.92 \[ -\log \left (a x - 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 13, normalized size = 1.08 \[ -\log \left ({\left | a x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 12, normalized size = 1.00 \[ \ln \relax (x )-\ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 11, normalized size = 0.92 \[ -\log \left (a x - 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 9, normalized size = 0.75 \[ 2\,\mathrm {atanh}\left (2\,a\,x-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 8, normalized size = 0.67 \[ \log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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