Optimal. Leaf size=37 \[ \frac{2}{a c (1-a x)}+\frac{3 \log (1-a x)}{a c}+\frac{x}{c} \]
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Rubi [A] time = 0.124074, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 77} \[ \frac{2}{a c (1-a x)}+\frac{3 \log (1-a x)}{a c}+\frac{x}{c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx\\ &=\frac{a \int \frac{e^{2 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=\frac{a \int \frac{x (1+a x)}{(1-a x)^2} \, dx}{c}\\ &=\frac{a \int \left (\frac{1}{a}+\frac{2}{a (-1+a x)^2}+\frac{3}{a (-1+a x)}\right ) \, dx}{c}\\ &=\frac{x}{c}+\frac{2}{a c (1-a x)}+\frac{3 \log (1-a x)}{a c}\\ \end{align*}
Mathematica [A] time = 0.0283811, size = 30, normalized size = 0.81 \[ \frac{a x+\frac{2}{1-a x}+3 \log (1-a x)}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 36, normalized size = 1. \begin{align*}{\frac{x}{c}}+3\,{\frac{\ln \left ( ax-1 \right ) }{ac}}-2\,{\frac{1}{ac \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02992, size = 47, normalized size = 1.27 \begin{align*} \frac{x}{c} - \frac{2}{a^{2} c x - a c} + \frac{3 \, \log \left (a x - 1\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54618, size = 86, normalized size = 2.32 \begin{align*} \frac{a^{2} x^{2} - a x + 3 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{a^{2} c x - a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.362151, size = 26, normalized size = 0.7 \begin{align*} - \frac{2}{a^{2} c x - a c} + \frac{x}{c} + \frac{3 \log{\left (a x - 1 \right )}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14996, size = 49, normalized size = 1.32 \begin{align*} \frac{x}{c} + \frac{3 \, \log \left ({\left | a x - 1 \right |}\right )}{a c} - \frac{2}{{\left (a x - 1\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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