Optimal. Leaf size=33 \[ \frac{c}{a^2 x}-\frac{4 c \log (x)}{a}+\frac{8 c \log (1-a x)}{a}+c x \]
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Rubi [A] time = 0.0933374, antiderivative size = 33, 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, 6157, 6150, 88} \[ \frac{c}{a^2 x}-\frac{4 c \log (x)}{a}+\frac{8 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right ) \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right ) \, dx\\ &=-\frac{c \int \frac{e^{4 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac{c \int \frac{(1+a x)^3}{x^2 (1-a x)} \, dx}{a^2}\\ &=-\frac{c \int \left (-a^2+\frac{1}{x^2}+\frac{4 a}{x}-\frac{8 a^2}{-1+a x}\right ) \, dx}{a^2}\\ &=\frac{c}{a^2 x}+c x-\frac{4 c \log (x)}{a}+\frac{8 c \log (1-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0199034, size = 33, normalized size = 1. \[ \frac{c}{a^2 x}-\frac{4 c \log (x)}{a}+\frac{8 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 33, normalized size = 1. \begin{align*} cx+{\frac{c}{{a}^{2}x}}-4\,{\frac{c\ln \left ( x \right ) }{a}}+8\,{\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 = 1.02735, size = 43, normalized size = 1.3 \begin{align*} c x + \frac{8 \, c \log \left (a x - 1\right )}{a} - \frac{4 \, c \log \left (x\right )}{a} + \frac{c}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57583, size = 88, normalized size = 2.67 \begin{align*} \frac{a^{2} c x^{2} + 8 \, a c x \log \left (a x - 1\right ) - 4 \, a c x \log \left (x\right ) + c}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.457601, size = 26, normalized size = 0.79 \begin{align*} c x + \frac{4 c \left (- \log{\left (x \right )} + 2 \log{\left (x - \frac{1}{a} \right )}\right )}{a} + \frac{c}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11255, size = 89, normalized size = 2.7 \begin{align*} -\frac{4 \, c \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{4 \, c \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (a x - 1\right )} c}{a{\left (\frac{1}{a x - 1} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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