Optimal. Leaf size=31 \[ \frac{1}{c (1-a x)}-\frac{\log (1-a x)}{c}+\frac{\log (x)}{c} \]
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Rubi [A] time = 0.0866635, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 44} \[ \frac{1}{c (1-a x)}-\frac{\log (1-a x)}{c}+\frac{\log (x)}{c} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )} \, dx &=\frac{\int \frac{1}{x (1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{1}{x}+\frac{a}{(-1+a x)^2}-\frac{a}{-1+a x}\right ) \, dx}{c}\\ &=\frac{1}{c (1-a x)}+\frac{\log (x)}{c}-\frac{\log (1-a x)}{c}\\ \end{align*}
Mathematica [A] time = 0.0198567, size = 24, normalized size = 0.77 \[ \frac{\frac{1}{1-a x}-\log (1-a x)+\log (x)}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 31, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{c}}-{\frac{1}{c \left ( ax-1 \right ) }}-{\frac{\ln \left ( ax-1 \right ) }{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958175, size = 41, normalized size = 1.32 \begin{align*} -\frac{\log \left (a x - 1\right )}{c} + \frac{\log \left (x\right )}{c} - \frac{1}{a c x - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03213, size = 84, normalized size = 2.71 \begin{align*} -\frac{{\left (a x - 1\right )} \log \left (a x - 1\right ) -{\left (a x - 1\right )} \log \left (x\right ) + 1}{a c x - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.364284, size = 19, normalized size = 0.61 \begin{align*} - \frac{1}{a c x - c} + \frac{\log{\left (x \right )} - \log{\left (x - \frac{1}{a} \right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20644, size = 43, normalized size = 1.39 \begin{align*} -\frac{\log \left ({\left | a x - 1 \right |}\right )}{c} + \frac{\log \left ({\left | x \right |}\right )}{c} - \frac{1}{{\left (a x - 1\right )} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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