Optimal. Leaf size=12 \[ \log (x)-\log (1-a x) \]
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Rubi [A]
time = 0.05, 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 = {6285, 36, 29,
31} \begin {gather*} \log (x)-\log (1-a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 6285
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 \begin {gather*} \log (x)-\log (1-a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 12, normalized size = 1.00
method | result | size |
default | \(\ln \left (x \right )-\ln \left (a x -1\right )\) | \(12\) |
norman | \(\ln \left (x \right )-\ln \left (a x -1\right )\) | \(12\) |
risch | \(\ln \left (-x \right )-\ln \left (a x -1\right )\) | \(14\) |
meijerg | \(\arctanh \left (a x \right )-\frac {\ln \left (-a^{2} x^{2}+1\right )}{2}+\ln \left (x \right )+\frac {\ln \left (-a^{2}\right )}{2}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 11, normalized size = 0.92 \begin {gather*} -\log \left (a x - 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 11, normalized size = 0.92 \begin {gather*} -\log \left (a x - 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 8, normalized size = 0.67 \begin {gather*} \log {\left (x \right )} - \log {\left (x - \frac {1}{a} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 13, normalized size = 1.08 \begin {gather*} -\log \left ({\left | a x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
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 \begin {gather*} 2\,\mathrm {atanh}\left (2\,a\,x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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