Optimal. Leaf size=20 \[ \frac{\log (a x+1)}{a c}-\frac{x}{c} \]
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Rubi [A] time = 0.0875015, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6131, 6129, 43} \[ \frac{\log (a x+1)}{a c}-\frac{x}{c} \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6129
Rule 43
Rubi steps
\begin{align*} \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 = 0.0152997, size = 18, normalized size = 0.9 \[ \frac{\log (a x+1)-a x}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, 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 = 0.940878, 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 = 2.20964, size = 39, normalized size = 1.95 \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 = 0.30708, size = 19, normalized size = 0.95 \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 [B] time = 1.25834, size = 55, normalized size = 2.75 \begin{align*} -\frac{a x + 1}{a c} - \frac{\log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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