Optimal. Leaf size=13 \[ \frac{e^{\tan ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.0258877, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, 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 = 0.0072651, size = 35, normalized size = 2.69 \[ \frac{(1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}}}{a c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.036, size = 13, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{\arctan \left ( ax \right ) }}}{ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49114, size = 16, normalized size = 1.23 \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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Fricas [A] time = 2.04406, size = 31, normalized size = 2.38 \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 = 1.10974, size = 16, normalized size = 1.23 \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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