Optimal. Leaf size=16 \[ -x-\frac {2 \log (1-a x)}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, 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 = {6260, 45}
\begin {gather*} -\frac {2 \log (1-a x)}{a}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6260
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 = 16, normalized size = 1.00 \begin {gather*} -x-\frac {2 \log (1-a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 16, normalized size = 1.00
method | result | size |
default | \(-x -\frac {2 \ln \left (a x -1\right )}{a}\) | \(16\) |
norman | \(-x -\frac {2 \ln \left (a x -1\right )}{a}\) | \(16\) |
risch | \(-x -\frac {2 \ln \left (a x -1\right )}{a}\) | \(16\) |
meijerg | \(-\frac {-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}}{2 \sqrt {-a^{2}}}-\frac {\ln \left (-a^{2} x^{2}+1\right )}{a}+\frac {\arctanh \left (a x \right )}{a}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 15, normalized size = 0.94 \begin {gather*} -x - \frac {2 \, \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 = 0.35, size = 17, normalized size = 1.06 \begin {gather*} -\frac {a x + 2 \, \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 = 0.03, size = 12, normalized size = 0.75 \begin {gather*} - x - \frac {2 \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 = 16, normalized size = 1.00 \begin {gather*} -x - \frac {2 \, \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 = 0.03, size = 15, normalized size = 0.94 \begin {gather*} -x-\frac {2\,\ln \left (a\,x-1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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