Optimal. Leaf size=25 \[ -\frac {2 \log (a x+1)}{a^2}+\frac {2 x}{a}-\frac {x^2}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6126, 77} \[ -\frac {2 \log (a x+1)}{a^2}+\frac {2 x}{a}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6126
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} x \, dx &=\int \frac {x (1-a x)}{1+a x} \, dx\\ &=\int \left (\frac {2}{a}-x-\frac {2}{a (1+a x)}\right ) \, dx\\ &=\frac {2 x}{a}-\frac {x^2}{2}-\frac {2 \log (1+a x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ -\frac {2 \log (a x+1)}{a^2}+\frac {2 x}{a}-\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 25, normalized size = 1.00 \[ -\frac {a^{2} x^{2} - 4 \, a x + 4 \, \log \left (a x + 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 52, normalized size = 2.08 \[ \frac {\frac {{\left (a x + 1\right )}^{2} {\left (\frac {6}{a x + 1} - 1\right )}}{a} + \frac {4 \, \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 24, normalized size = 0.96 \[ \frac {2 x}{a}-\frac {x^{2}}{2}-\frac {2 \ln \left (a x +1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 26, normalized size = 1.04 \[ -\frac {a x^{2} - 4 \, x}{2 \, a} - \frac {2 \, \log \left (a x + 1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 23, normalized size = 0.92 \[ \frac {2\,x}{a}-\frac {2\,\ln \left (a\,x+1\right )}{a^2}-\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 20, normalized size = 0.80 \[ - \frac {x^{2}}{2} + \frac {2 x}{a} - \frac {2 \log {\left (a x + 1 \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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