Optimal. Leaf size=18 \[ \frac{2 \tan ^{-1}(a x)^{3/2}}{3 a c} \]
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Rubi [A] time = 0.0242699, antiderivative size = 18, 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} \[ \frac{2 \tan ^{-1}(a x)^{3/2}}{3 a c} \]
Antiderivative was successfully verified.
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Rule 4884
Rubi steps
\begin{align*} \int \frac{\sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx &=\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a c}\\ \end{align*}
Mathematica [A] time = 0.0034562, size = 18, normalized size = 1. \[ \frac{2 \tan ^{-1}(a x)^{3/2}}{3 a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 15, normalized size = 0.8 \begin{align*}{\frac{2}{3\,ac} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}} \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 = 1.71705, size = 39, normalized size = 2.17 \begin{align*} \frac{2 \, \arctan \left (a x\right )^{\frac{3}{2}}}{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{\sqrt{\operatorname{atan}{\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 = 1.08891, size = 19, normalized size = 1.06 \begin{align*} \frac{2 \, \arctan \left (a x\right )^{\frac{3}{2}}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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