Optimal. Leaf size=20 \[ \frac{x}{c}-\frac{\log (a x+1)}{a c} \]
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Rubi [A] time = 0.11509, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 43} \[ \frac{x}{c}-\frac{\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx\\ &=\frac{a \int \frac{e^{-2 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=\frac{a \int \frac{x}{1+a x} \, dx}{c}\\ &=\frac{a \int \left (\frac{1}{a}-\frac{1}{a (1+a x)}\right ) \, dx}{c}\\ &=\frac{x}{c}-\frac{\log (1+a x)}{a c}\\ \end{align*}
Mathematica [A] time = 0.0175695, size = 22, normalized size = 1.1 \[ \frac{a \left (\frac{x}{a}-\frac{\log (a x+1)}{a^2}\right )}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 21, normalized size = 1.1 \begin{align*}{\frac{x}{c}}-{\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 = 1.02752, size = 27, normalized size = 1.35 \begin{align*} \frac{x}{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 = 1.76688, size = 38, normalized size = 1.9 \begin{align*} \frac{a x - \log \left (a x + 1\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.269226, size = 17, normalized size = 0.85 \begin{align*} a \left (\frac{x}{a c} - \frac{\log{\left (a x + 1 \right )}}{a^{2} c}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13968, size = 28, normalized size = 1.4 \begin{align*} \frac{x}{c} - \frac{\log \left ({\left | a x + 1 \right |}\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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