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 = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 44
Rule 6150
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.65, size = 35, normalized size = 1.13 \[ -\frac {{\left (a x - 1\right )} \log \left (a x - 1\right ) - {\left (a x - 1\right )} \log \relax (x) + 1}{a c x - c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 32, normalized size = 1.03 \[ -\frac {\log \left ({\left | a x - 1 \right |}\right )}{c} + \frac {\log \left ({\left | x \right |}\right )}{c} - \frac {1}{{\left (a x - 1\right )} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 1.00 \[ \frac {\ln \relax (x )}{c}-\frac {1}{c \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 30, normalized size = 0.97 \[ -\frac {\log \left (a x - 1\right )}{c} + \frac {\log \relax (x)}{c} - \frac {1}{a c x - c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 22, normalized size = 0.71 \[ \frac {2\,\mathrm {atanh}\left (2\,a\,x-1\right )}{c}+\frac {1}{c-a\,c\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 19, normalized size = 0.61 \[ - \frac {1}{a c x - c} + \frac {\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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