Optimal. Leaf size=15 \[ -\frac{1}{a c (a x+1)} \]
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Rubi [A] time = 0.0325568, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 32} \[ -\frac{1}{a c (a x+1)} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 32
Rubi steps
\begin{align*} \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 = 0.0164715, size = 18, normalized size = 1.2 \[ -\frac{e^{-2 \tanh ^{-1}(a x)}}{2 a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 16, 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 = 0.958754, size = 19, normalized size = 1.27 \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 = 2.06551, size = 27, normalized size = 1.8 \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 = 0.340862, size = 12, normalized size = 0.8 \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 = 1.14865, size = 20, normalized size = 1.33 \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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