Optimal. Leaf size=35 \[ \frac{1}{a c (a x+1)}+\frac{2 \log (a x+1)}{a c}-\frac{x}{c} \]
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Rubi [A] time = 0.132358, antiderivative size = 35, 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} \[ \frac{1}{a c (a x+1)}+\frac{2 \log (a x+1)}{a c}-\frac{x}{c} \]
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 = 0.0268276, size = 28, normalized size = 0.8 \[ \frac{\frac{1}{a^2 x+a}+\frac{2 \log (a x+1)}{a}-x}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, 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 = 0.963086, size = 45, normalized size = 1.29 \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 = 1.79469, size = 88, normalized size = 2.51 \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 = 0.733326, size = 37, normalized size = 1.06 \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 = 1.31822, size = 74, normalized size = 2.11 \begin{align*} -\frac{a x + 1}{a c} - \frac{2 \, \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \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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