Optimal. Leaf size=14 \[ \frac {2 \log (1-a x)}{a}+x \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6167, 6125, 43} \[ \frac {2 \log (1-a x)}{a}+x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6125
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \, dx\\ &=-\int \frac {1+a x}{1-a x} \, dx\\ &=-\int \left (-1-\frac {2}{-1+a x}\right ) \, dx\\ &=x+\frac {2 \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \frac {2 \log (1-a x)}{a}+x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 16, normalized size = 1.14 \[ \frac {a x + 2 \, \log \left (a x - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 14, normalized size = 1.00 \[ x + \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 1.00 \[ x +\frac {2 \ln \left (a x -1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 13, normalized size = 0.93 \[ x + \frac {2 \, \log \left (a x - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 13, normalized size = 0.93 \[ x+\frac {2\,\ln \left (a\,x-1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 10, normalized size = 0.71 \[ x + \frac {2 \log {\left (a x - 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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