Optimal. Leaf size=38 \[ -\frac{1}{a c (1-a x)}-\frac{2 \log (1-a x)}{a c}-\frac{x}{c} \]
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Rubi [A] time = 0.124496, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {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 6157
Rule 6150
Rule 43
Rubi steps
\begin{align*} \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 = 0.0203159, size = 38, normalized size = 1. \[ -\frac{1}{a c (1-a x)}-\frac{2 \log (1-a x)}{a c}-\frac{x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 36, normalized size = 1. \begin{align*} -{\frac{x}{c}}+{\frac{1}{ac \left ( ax-1 \right ) }}-2\,{\frac{\ln \left ( ax-1 \right ) }{ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962236, size = 46, normalized size = 1.21 \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 = 2.01568, size = 88, normalized size = 2.32 \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 = 0.874792, size = 37, normalized size = 0.97 \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 = 1.17655, size = 49, normalized size = 1.29 \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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