Optimal. Leaf size=31 \[ \frac{2}{a c (1-a x)}+\frac{\log (1-a x)}{a c} \]
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Rubi [A] time = 0.0381136, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {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 6129
Rule 43
Rubi steps
\begin{align*} \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 = 0.0162467, size = 25, normalized size = 0.81 \[ \frac{\frac{2}{1-a x}+\log (1-a x)}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 30, 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 = 0.934844, size = 39, normalized size = 1.26 \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 = 1.68026, size = 62, normalized size = 2. \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 = 0.333816, size = 20, normalized size = 0.65 \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 = 1.2026, size = 41, normalized size = 1.32 \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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