Optimal. Leaf size=12 \[ -\frac{\tanh ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.0487918, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6129, 35, 206} \[ -\frac{\tanh ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 35
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=-\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 = 0.0085515, size = 12, normalized size = 1. \[ -\frac{\tanh ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 30, normalized size = 2.5 \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 = 1.016, size = 39, normalized size = 3.25 \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 [A] time = 1.581, size = 59, normalized size = 4.92 \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 [A] time = 0.428533, size = 20, normalized size = 1.67 \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 = 1.15308, size = 34, normalized size = 2.83 \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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