Optimal. Leaf size=15 \[ \frac {2 \log (a x+1)}{a}-x \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6125, 43} \[ \frac {2 \log (a x+1)}{a}-x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6125
Rubi steps
\begin {align*} \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 = 15, normalized size = 1.00 \[ \frac {2 \log (a x+1)}{a}-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 17, normalized size = 1.13 \[ -\frac {a x - 2 \, \log \left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 64, normalized size = 4.27 \[ -a^{2} {\left (\frac {a x + 1}{a^{3}} + \frac {2 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a^{3}} - \frac {1}{{\left (a x + 1\right )} a^{3}}\right )} - \frac {1}{{\left (a x + 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 1.07 \[ -x +\frac {2 \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 15, normalized size = 1.00 \[ -x + \frac {2 \, \log \left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 15, normalized size = 1.00 \[ \frac {2\,\ln \left (a\,x+1\right )}{a}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 10, normalized size = 0.67 \[ - x + \frac {2 \log {\left (a x + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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