Optimal. Leaf size=36 \[ \frac{1}{a c (1-a x)}+\frac{2 \log (1-a x)}{a c}+\frac{x}{c} \]
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Rubi [A] time = 0.159508, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6157, 6150, 43} \[ \frac{1}{a c (1-a x)}+\frac{2 \log (1-a x)}{a c}+\frac{x}{c} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx\\ &=\frac{a^2 \int \frac{e^{2 \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=\frac{a^2 \int \frac{x^2}{(1-a x)^2} \, dx}{c}\\ &=\frac{a^2 \int \left (\frac{1}{a^2}+\frac{1}{a^2 (-1+a x)^2}+\frac{2}{a^2 (-1+a x)}\right ) \, dx}{c}\\ &=\frac{x}{c}+\frac{1}{a c (1-a x)}+\frac{2 \log (1-a x)}{a c}\\ \end{align*}
Mathematica [A] time = 0.0290702, size = 28, normalized size = 0.78 \[ \frac{\frac{1}{a-a^2 x}+\frac{2 \log (1-a x)}{a}+x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 36, normalized size = 1. \begin{align*}{\frac{x}{c}}+2\,{\frac{\ln \left ( ax-1 \right ) }{ac}}-{\frac{1}{ac \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05927, size = 47, normalized size = 1.31 \begin{align*} \frac{x}{c} - \frac{1}{a^{2} c x - a c} + \frac{2 \, \log \left (a x - 1\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52547, size = 86, normalized size = 2.39 \begin{align*} \frac{a^{2} x^{2} - a x + 2 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 1}{a^{2} c x - a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.31619, size = 36, normalized size = 1. \begin{align*} a^{2} \left (- \frac{1}{a^{4} c x - a^{3} c} + \frac{x}{a^{2} c} + \frac{2 \log{\left (a x - 1 \right )}}{a^{3} c}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08939, size = 49, normalized size = 1.36 \begin{align*} \frac{x}{c} + \frac{2 \, \log \left ({\left | a x - 1 \right |}\right )}{a c} - \frac{1}{{\left (a x - 1\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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