Optimal. Leaf size=32 \[ -\frac{2}{a c (1-a x)}-\frac{\log (1-a x)}{a c} \]
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Rubi [A] time = 0.0637809, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ -\frac{2}{a c (1-a x)}-\frac{\log (1-a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{c-a c x} \, dx\\ &=-\frac{\int \frac{1+a x}{(1-a x)^2} \, dx}{c}\\ &=-\frac{\int \left (\frac{2}{(-1+a x)^2}+\frac{1}{-1+a x}\right ) \, dx}{c}\\ &=-\frac{2}{a c (1-a x)}-\frac{\log (1-a x)}{a c}\\ \end{align*}
Mathematica [A] time = 0.0149316, size = 30, normalized size = 0.94 \[ -\frac{\frac{2}{a (1-a x)}+\frac{\log (1-a x)}{a}}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 31, normalized size = 1. \begin{align*} 2\,{\frac{1}{ac \left ( ax-1 \right ) }}-{\frac{\ln \left ( ax-1 \right ) }{ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02256, size = 41, normalized size = 1.28 \begin{align*} \frac{2}{a^{2} c x - a c} - \frac{\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 = 1.45197, size = 63, normalized size = 1.97 \begin{align*} -\frac{{\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 = 0.352277, size = 20, normalized size = 0.62 \begin{align*} \frac{2}{a^{2} c x - a c} - \frac{\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 = 1.14623, size = 42, normalized size = 1.31 \begin{align*} -\frac{\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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