Optimal. Leaf size=39 \[ \frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2} \]
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Rubi [A] time = 0.0413822, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6167, 6126, 77} \[ \frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} x \, dx &=\int e^{4 \tanh ^{-1}(a x)} x \, dx\\ &=\int \frac{x (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac{4}{a}+x+\frac{4}{a (-1+a x)^2}+\frac{8}{a (-1+a x)}\right ) \, dx\\ &=\frac{4 x}{a}+\frac{x^2}{2}+\frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0283569, size = 39, normalized size = 1. \[ \frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 36, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+4\,{\frac{x}{a}}-4\,{\frac{1}{{a}^{2} \left ( ax-1 \right ) }}+8\,{\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 = 1.08211, size = 55, normalized size = 1.41 \begin{align*} \frac{a x^{2} + 8 \, x}{2 \, a} - \frac{4}{a^{3} x - a^{2}} + \frac{8 \, \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 = 1.83994, size = 109, normalized size = 2.79 \begin{align*} \frac{a^{3} x^{3} + 7 \, a^{2} x^{2} - 8 \, a x + 16 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 8}{2 \,{\left (a^{3} x - a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.327189, size = 31, normalized size = 0.79 \begin{align*} \frac{x^{2}}{2} - \frac{4}{a^{3} x - a^{2}} + \frac{4 x}{a} + \frac{8 \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 [A] time = 1.12868, size = 86, normalized size = 2.21 \begin{align*} \frac{\frac{{\left (a x - 1\right )}^{2}{\left (\frac{10}{a x - 1} + 1\right )}}{a} - \frac{16 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{8}{{\left (a x - 1\right )} a}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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