Optimal. Leaf size=11 \[ \frac{c \log (x)}{a}+c x \]
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Rubi [A] time = 0.0746432, antiderivative size = 11, 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, 43} \[ \frac{c \log (x)}{a}+c x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 43
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}{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 = 0.036415, size = 11, normalized size = 1. \[ \frac{c \log (x)}{a}+c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 12, 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 = 1.04582, size = 15, normalized size = 1.36 \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 = 1.40232, size = 30, normalized size = 2.73 \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 = 0.091203, size = 10, normalized size = 0.91 \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 = 1.11006, size = 16, normalized size = 1.45 \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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