Optimal. Leaf size=13 \[ \frac{\log (a x+1)}{a c} \]
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Rubi [A] time = 0.0256917, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 31} \[ \frac{\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{c-a c x} \, dx &=\frac{\int \frac{1}{1+a x} \, dx}{c}\\ &=\frac{\log (1+a x)}{a c}\\ \end{align*}
Mathematica [A] time = 0.0059627, size = 13, normalized size = 1. \[ \frac{\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 14, normalized size = 1.1 \begin{align*}{\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 = 0.943681, size = 18, normalized size = 1.38 \begin{align*} \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 = 1.57094, size = 27, normalized size = 2.08 \begin{align*} \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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Sympy [A] time = 0.084929, size = 10, normalized size = 0.77 \begin{align*} \frac{\log{\left (a c x + c \right )}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23518, size = 36, normalized size = 2.77 \begin{align*} -\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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