Optimal. Leaf size=14 \[ \frac{1}{a c (a x+1)} \]
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Rubi [A] time = 0.0611401, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 32} \[ \frac{1}{a c (a x+1)} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\\ &=-\frac{\int \frac{1}{(1+a x)^2} \, dx}{c}\\ &=\frac{1}{a c (1+a x)}\\ \end{align*}
Mathematica [C] time = 0.0144369, size = 18, normalized size = 1.29 \[ -\frac{e^{-2 \coth ^{-1}(a x)}}{2 a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 15, normalized size = 1.1 \begin{align*}{\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 = 1.03499, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{a^{2} c x + a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49258, size = 26, normalized size = 1.86 \begin{align*} \frac{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 = 0.291627, size = 10, normalized size = 0.71 \begin{align*} \frac{1}{a^{2} c x + a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11098, size = 19, normalized size = 1.36 \begin{align*} \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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