Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{x^m}{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2},x\right ) \]
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Rubi [A] time = 0.119258, 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 = {} \[ \int \frac{x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx &=\int \frac{x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.53358, size = 0, normalized size = 0. \[ \int \frac{x^m}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.208, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( \arctan \left ( ax \right ) \right ) ^{2}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, 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{x^{m}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{2}}\,{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} x^{m}}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{m}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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