Optimal. Leaf size=12 \[ \frac{\log \left (\tan ^{-1}(a x)\right )}{a c} \]
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Rubi [A] time = 0.0264302, antiderivative size = 12, 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 = {4882} \[ \frac{\log \left (\tan ^{-1}(a x)\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 4882
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx &=\frac{\log \left (\tan ^{-1}(a x)\right )}{a c}\\ \end{align*}
Mathematica [A] time = 0.0138428, size = 12, normalized size = 1. \[ \frac{\log \left (\tan ^{-1}(a x)\right )}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 13, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \arctan \left ( ax \right ) \right ) }{ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05453, size = 20, normalized size = 1.67 \begin{align*} \frac{\log \left (2 \,{\left | \arctan \left (a x\right ) \right |}\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85804, size = 32, normalized size = 2.67 \begin{align*} \frac{\log \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 = 1.15777, size = 18, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | \arctan \left (a x\right ) \right |}\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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