Optimal. Leaf size=23 \[ \frac{c \log (x)}{a}-\frac{4 c \log (a x+1)}{a}+c x \]
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Rubi [A] time = 0.0822374, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6167, 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 6167
Rule 6131
Rule 6129
Rule 72
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx &=-\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 = 0.0388513, size = 23, 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 = 0.045, size = 24, 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 = 1.02825, size = 31, normalized size = 1.35 \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 = 1.85841, size = 55, normalized size = 2.39 \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 = 0.381285, size = 17, normalized size = 0.74 \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 [A] time = 1.13548, size = 34, normalized size = 1.48 \begin{align*} c x - \frac{4 \, c \log \left ({\left | a x + 1 \right |}\right )}{a} + \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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