Optimal. Leaf size=16 \[ -\frac{e^{-\tan ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.0267153, antiderivative size = 16, 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} \[ -\frac{e^{-\tan ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Rule 5071
Rubi steps
\begin{align*} \int \frac{e^{-\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=-\frac{e^{-\tan ^{-1}(a x)}}{a c}\\ \end{align*}
Mathematica [C] time = 0.0084977, size = 36, normalized size = 2.25 \[ -\frac{(1-i a x)^{-\frac{i}{2}} (1+i a x)^{\frac{i}{2}}}{a c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, size = 16, normalized size = 1. \begin{align*} -{\frac{1}{ac{{\rm e}^{\arctan \left ( ax \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04651, size = 31, normalized size = 1.94 \begin{align*} -\frac{2 \, e^{\left (-\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 = 1.96871, size = 34, normalized size = 2.12 \begin{align*} -\frac{e^{\left (-\arctan \left (a x\right )\right )}}{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 = 1.09327, size = 20, normalized size = 1.25 \begin{align*} -\frac{e^{\left (-\arctan \left (a x\right )\right )}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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