Optimal. Leaf size=25 \[ -\frac{c \log (x)}{a}+\frac{4 c \log (1-a x)}{a}+c x \]
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Rubi [A] time = 0.0816088, antiderivative size = 25, 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} \[ -\frac{c \log (x)}{a}+\frac{4 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 72
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right ) \, dx\\ &=-\frac{c \int \frac{e^{4 \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 = 0.0385445, size = 25, normalized size = 1. \[ -\frac{c \log (x)}{a}+\frac{4 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 1. \begin{align*} cx+4\,{\frac{c\ln \left ( ax-1 \right ) }{a}}-{\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 = 1.12228, size = 32, normalized size = 1.28 \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 = 1.54833, size = 55, normalized size = 2.2 \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 = 0.389204, size = 17, normalized size = 0.68 \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 = 1.15558, size = 74, normalized size = 2.96 \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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