Optimal. Leaf size=13 \[ x-\frac{2 \log (a x+1)}{a} \]
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Rubi [A] time = 0.013253, antiderivative size = 13, normalized size of antiderivative = 1., 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} \[ x-\frac{2 \log (a x+1)}{a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6125
Rule 43
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*}
Mathematica [A] time = 0.010677, size = 13, normalized size = 1. \[ x-\frac{2 \log (a x+1)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 14, normalized size = 1.1 \begin{align*} x-2\,{\frac{\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01314, size = 18, normalized size = 1.38 \begin{align*} x - \frac{2 \, \log \left (a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73219, size = 35, normalized size = 2.69 \begin{align*} \frac{a x - 2 \, \log \left (a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09364, size = 10, normalized size = 0.77 \begin{align*} x - \frac{2 \log{\left (a x + 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1491, size = 19, normalized size = 1.46 \begin{align*} x - \frac{2 \, \log \left ({\left | a x + 1 \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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