Optimal. Leaf size=12 \[ -\frac {\log (1-a x)}{a} \]
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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6275, 31}
\begin {gather*} -\frac {\log (1-a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 6275
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx &=\int \frac {1}{1-a x} \, dx\\ &=-\frac {\log (1-a x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log (1-a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 12, normalized size = 1.00
method | result | size |
default | \(-\frac {\ln \left (a x -1\right )}{a}\) | \(12\) |
norman | \(-\frac {\ln \left (a x -1\right )}{a}\) | \(12\) |
risch | \(-\frac {\ln \left (a x -1\right )}{a}\) | \(12\) |
meijerg | \(-\frac {\ln \left (-a^{2} x^{2}+1\right )}{2 a}+\frac {\arctanh \left (a x \right )}{a}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 11, normalized size = 0.92 \begin {gather*} -\frac {\log \left (a x - 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 11, normalized size = 0.92 \begin {gather*} -\frac {\log \left (a x - 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 8, normalized size = 0.67 \begin {gather*} - \frac {\log {\left (a x - 1 \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 12, normalized size = 1.00 \begin {gather*} -\frac {\log \left ({\left | a x - 1 \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 11, normalized size = 0.92 \begin {gather*} -\frac {\ln \left (a\,x-1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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