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 = 0.0239714, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 6126
Rule 77
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*}
Mathematica [A] time = 0.0108768, size = 25, normalized size = 1. \[ -\frac{2 \log (a x+1)}{a^2}+\frac{2 x}{a}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 24, normalized size = 1. \begin{align*} 2\,{\frac{x}{a}}-{\frac{{x}^{2}}{2}}-2\,{\frac{\ln \left ( ax+1 \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947156, size = 35, normalized size = 1.4 \begin{align*} -\frac{a x^{2} - 4 \, x}{2 \, a} - \frac{2 \, \log \left (a x + 1\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84173, size = 61, normalized size = 2.44 \begin{align*} -\frac{a^{2} x^{2} - 4 \, a x + 4 \, \log \left (a x + 1\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.286735, size = 20, normalized size = 0.8 \begin{align*} - \frac{x^{2}}{2} + \frac{2 x}{a} - \frac{2 \log{\left (a x + 1 \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16934, size = 70, normalized size = 2.8 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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