Optimal. Leaf size=18 \[ -\frac{e^{-2 \tan ^{-1}(a x)}}{2 a c} \]
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Rubi [A] time = 0.028431, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {5071} \[ -\frac{e^{-2 \tan ^{-1}(a x)}}{2 a c} \]
Antiderivative was successfully verified.
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Rule 5071
Rubi steps
\begin{align*} \int \frac{e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=-\frac{e^{-2 \tan ^{-1}(a x)}}{2 a c}\\ \end{align*}
Mathematica [C] time = 0.0081919, size = 34, normalized size = 1.89 \[ -\frac{(1-i a x)^{-i} (1+i a x)^i}{2 a c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.036, size = 18, normalized size = 1. \begin{align*} -{\frac{1}{2\,ac{{\rm e}^{2\,\arctan \left ( ax \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06396, size = 31, normalized size = 1.72 \begin{align*} -\frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{a^{3} c x^{2} + a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95275, size = 42, normalized size = 2.33 \begin{align*} -\frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11948, size = 20, normalized size = 1.11 \begin{align*} -\frac{e^{\left (-2 \, \arctan \left (a x\right )\right )}}{2 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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