Optimal. Leaf size=25 \[ -\frac{c \log (x)}{a}+\frac{4 c \log (a x+1)}{a}-c x \]
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Rubi [A] time = 0.0656118, antiderivative size = 25, 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, 72} \[ -\frac{c \log (x)}{a}+\frac{4 c \log (a x+1)}{a}-c x \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6129
Rule 72
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)^2}{x (1+a x)} \, dx}{a}\\ &=-\frac{c \int \left (a+\frac{1}{x}-\frac{4 a}{1+a x}\right ) \, dx}{a}\\ &=-c x-\frac{c \log (x)}{a}+\frac{4 c \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0364962, size = 25, normalized size = 1. \[ -\frac{c \log (x)}{a}+\frac{4 c \log (a x+1)}{a}-c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 26, normalized size = 1. \begin{align*} -cx-{\frac{c\ln \left ( x \right ) }{a}}+4\,{\frac{c\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960578, size = 34, normalized size = 1.36 \begin{align*} -c x + \frac{4 \, c \log \left (a x + 1\right )}{a} - \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 = 2.29126, size = 57, normalized size = 2.28 \begin{align*} -\frac{a c x - 4 \, c \log \left (a x + 1\right ) + 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 = 0.426521, size = 19, normalized size = 0.76 \begin{align*} - c x - \frac{c \left (\log{\left (x \right )} - 4 \log{\left (x + \frac{1}{a} \right )}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18673, size = 76, normalized size = 3.04 \begin{align*} -\frac{{\left (a x + 1\right )} c}{a} - \frac{3 \, c \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{c \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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