Optimal. Leaf size=14 \[ -\frac{\log (a x+1)}{a c} \]
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Rubi [A] time = 0.0546414, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 31} \[ -\frac{\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{c-a c x} \, dx &=-\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 = 0.0074188, size = 14, normalized size = 1. \[ -\frac{\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 15, 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 = 1.01747, size = 19, normalized size = 1.36 \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 = 1.43935, size = 28, normalized size = 2. \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 = 0.242826, size = 12, normalized size = 0.86 \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 = 1.1383, size = 20, normalized size = 1.43 \begin{align*} -\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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