Optimal. Leaf size=36 \[ \frac{2 x \sqrt{\tan ^{-1}(a x)}}{a c}-\frac{2 \text{Unintegrable}\left (\sqrt{\tan ^{-1}(a x)},x\right )}{a c} \]
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Rubi [A] time = 0.0490175, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}} \, dx &=\frac{2 x \sqrt{\tan ^{-1}(a x)}}{a c}-\frac{2 \int \sqrt{\tan ^{-1}(a x)} \, dx}{a c}\\ \end{align*}
Mathematica [A] time = 0.81425, size = 0, normalized size = 0. \[ \int \frac{x}{\left (c+a^2 c x^2\right ) \sqrt{\tan ^{-1}(a x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.148, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{{a}^{2}c{x}^{2}+c}{\frac{1}{\sqrt{\arctan \left ( ax \right ) }}}}\, dx \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x}{a^{2} x^{2} \sqrt{\operatorname{atan}{\left (a x \right )}} + \sqrt{\operatorname{atan}{\left (a x \right )}}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a^{2} c x^{2} + c\right )} \sqrt{\arctan \left (a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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