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 = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 77
Rule 6126
Rule 6167
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*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.00 \[ \frac {4}{a^2 (1-a x)}+\frac {8 \log (1-a x)}{a^2}+\frac {4 x}{a}+\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 49, normalized size = 1.26 \[ \frac {a^{3} x^{3} + 7 \, a^{2} x^{2} - 8 \, a x + 16 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 8}{2 \, {\left (a^{3} x - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 64, normalized size = 1.64 \[ \frac {\frac {{\left (a x - 1\right )}^{2} {\left (\frac {10}{a x - 1} + 1\right )}}{a} - \frac {16 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {8}{{\left (a x - 1\right )} a}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 0.92 \[ \frac {x^{2}}{2}+\frac {4 x}{a}+\frac {8 \ln \left (a x -1\right )}{a^{2}}-\frac {4}{a^{2} \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 41, normalized size = 1.05 \[ \frac {a x^{2} + 8 \, x}{2 \, a} - \frac {4}{a^{3} x - a^{2}} + \frac {8 \, \log \left (a x - 1\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 38, normalized size = 0.97 \[ \frac {8\,\ln \left (a\,x-1\right )}{a^2}+\frac {4\,x}{a}+\frac {x^2}{2}+\frac {4}{a\,\left (a-a^2\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 31, normalized size = 0.79 \[ \frac {x^{2}}{2} - \frac {4}{a^{3} x - a^{2}} + \frac {4 x}{a} + \frac {8 \log {\left (a x - 1 \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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