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 = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 43
Rule 6129
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.46, size = 28, normalized size = 0.90 \[ \frac {{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 30, normalized size = 0.97 \[ \frac {\log \left ({\left | a x - 1 \right |}\right )}{a c} - \frac {2}{{\left (a x - 1\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 30, normalized size = 0.97 \[ \frac {\ln \left (a x -1\right )}{c a}-\frac {2}{c a \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 29, normalized size = 0.94 \[ -\frac {2}{a^{2} c x - a c} + \frac {\log \left (a x - 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 28, normalized size = 0.90 \[ \frac {2}{a\,\left (c-a\,c\,x\right )}+\frac {\ln \left (a\,x-1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.65 \[ - \frac {2}{a^{2} c x - a c} + \frac {\log {\left (a x - 1 \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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