Optimal. Leaf size=11 \[ \frac{\tanh ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.0268221, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6129, 35, 206} \[ \frac{\tanh ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 35
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac{\int \frac{1}{(1-a x) (1+a x)} \, dx}{c^2}\\ &=\frac{\int \frac{1}{1-a^2 x^2} \, dx}{c^2}\\ &=\frac{\tanh ^{-1}(a x)}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.007319, size = 11, normalized size = 1. \[ \frac{\tanh ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, size = 30, normalized size = 2.7 \begin{align*}{\frac{\ln \left ( ax+1 \right ) }{2\,a{c}^{2}}}-{\frac{\ln \left ( ax-1 \right ) }{2\,a{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.945176, size = 39, normalized size = 3.55 \begin{align*} \frac{\log \left (a x + 1\right )}{2 \, a c^{2}} - \frac{\log \left (a x - 1\right )}{2 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57252, size = 58, normalized size = 5.27 \begin{align*} \frac{\log \left (a x + 1\right ) - \log \left (a x - 1\right )}{2 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.158413, size = 22, normalized size = 2. \begin{align*} - \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{2} - \frac{\log{\left (x + \frac{1}{a} \right )}}{2}}{a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27475, size = 34, normalized size = 3.09 \begin{align*} \frac{\log \left ({\left | -\frac{2 \, c}{a c x - c} - 1 \right |}\right )}{2 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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