Optimal. Leaf size=19 \[ -\frac {x}{a}-\frac {\log (1-a x)}{a^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6285, 45}
\begin {gather*} -\frac {\log (1-a x)}{a^2}-\frac {x}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{\sqrt {1-a^2 x^2}} \, dx &=\int \frac {x}{1-a x} \, dx\\ &=\int \left (-\frac {1}{a}-\frac {1}{a (-1+a x)}\right ) \, dx\\ &=-\frac {x}{a}-\frac {\log (1-a x)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -\frac {x}{a}-\frac {\log (1-a x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 19, normalized size = 1.00
method | result | size |
default | \(-\frac {x}{a}-\frac {\ln \left (a x -1\right )}{a^{2}}\) | \(19\) |
norman | \(-\frac {x}{a}-\frac {\ln \left (a x -1\right )}{a^{2}}\) | \(19\) |
risch | \(-\frac {x}{a}-\frac {\ln \left (a x -1\right )}{a^{2}}\) | \(19\) |
meijerg | \(-\frac {-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}}{2 a \sqrt {-a^{2}}}-\frac {\ln \left (-a^{2} x^{2}+1\right )}{2 a^{2}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 18, normalized size = 0.95 \begin {gather*} -\frac {x}{a} - \frac {\log \left (a x - 1\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 15, normalized size = 0.79 \begin {gather*} -\frac {a x + \log \left (a x - 1\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 14, normalized size = 0.74 \begin {gather*} - \frac {x}{a} - \frac {\log {\left (a x - 1 \right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 19, normalized size = 1.00 \begin {gather*} -\frac {x}{a} - \frac {\log \left ({\left | a x - 1 \right |}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.86, size = 15, normalized size = 0.79 \begin {gather*} -\frac {\ln \left (a\,x-1\right )+a\,x}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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