Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{x \left (a^2 c x^2+c\right )^2},x\right ) \]
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Rubi [A] time = 0.0595246, 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 = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx &=\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 1.68618, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.54, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{2}}\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{\sqrt{\operatorname{atan}{\left (a x \right )}}}{a^{4} x^{5} + 2 a^{2} x^{3} + x}\, dx}{c^{2}} \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{\sqrt{\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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