Optimal. Leaf size=13 \[ -\frac{c \log (x)}{a}-c x \]
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Rubi [A] time = 0.0541777, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6131, 6129, 43} \[ -\frac{c \log (x)}{a}-c x \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx &=-\frac{c \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac{c \int \frac{1+a x}{x} \, dx}{a}\\ &=-\frac{c \int \left (a+\frac{1}{x}\right ) \, dx}{a}\\ &=-c x-\frac{c \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.034086, size = 13, normalized size = 1. \[ -\frac{c \log (x)}{a}-c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 14, normalized size = 1.1 \begin{align*} -cx-{\frac{c\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936903, size = 18, normalized size = 1.38 \begin{align*} -c x - \frac{c \log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00883, size = 31, normalized size = 2.38 \begin{align*} -\frac{a c x + c \log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.095783, size = 12, normalized size = 0.92 \begin{align*} \frac{- a c x - c \log{\left (x \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13038, size = 19, normalized size = 1.46 \begin{align*} -c x - \frac{c \log \left ({\left | x \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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