Optimal. Leaf size=16 \[ \frac{\tan ^{-1}(a x)^3}{3 a c} \]
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Rubi [A] time = 0.0235709, antiderivative size = 16, 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 = {4884} \[ \frac{\tan ^{-1}(a x)^3}{3 a c} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx &=\frac{\tan ^{-1}(a x)^3}{3 a c}\\ \end{align*}
Mathematica [A] time = 0.0033675, size = 16, normalized size = 1. \[ \frac{\tan ^{-1}(a x)^3}{3 a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 15, normalized size = 0.9 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{3}}{3\,ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52937, size = 19, normalized size = 1.19 \begin{align*} \frac{\arctan \left (a x\right )^{3}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12695, size = 34, normalized size = 2.12 \begin{align*} \frac{\arctan \left (a x\right )^{3}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14008, size = 19, normalized size = 1.19 \begin{align*} \frac{\arctan \left (a x\right )^{3}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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