Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{\sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)},x\right ) \]
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Rubi [A] time = 0.038524, 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 = {} \[ \int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx &=\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx\\ \end{align*}
Mathematica [A] time = 0.261209, size = 0, normalized size = 0. \[ \int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.638, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\arctan \left ( ax \right ) }\sqrt{{a}^{2}c{x}^{2}+c}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} c x^{2} + c}}{\arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c}}{\arctan \left (a x\right )}, x\right ) \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*} \int \frac{\sqrt{c \left (a^{2} x^{2} + 1\right )}}{\operatorname{atan}{\left (a x \right )}}\, dx \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{a^{2} c x^{2} + c}}{\arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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