Optimal. Leaf size=16 \[ \frac{2 \sqrt{\tan ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.0243045, 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 = {4884} \[ \frac{2 \sqrt{\tan ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}} \, dx &=\frac{2 \sqrt{\tan ^{-1}(a x)}}{a c}\\ \end{align*}
Mathematica [A] time = 0.0031818, size = 16, normalized size = 1. \[ \frac{2 \sqrt{\tan ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 15, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88247, size = 36, normalized size = 2.25 \begin{align*} \frac{2 \, \sqrt{\arctan \left (a x\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 = 1.1066, size = 19, normalized size = 1.19 \begin{align*} \frac{2 \, \sqrt{\arctan \left (a x\right )}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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