Optimal. Leaf size=40 \[ \frac{m \text{Unintegrable}\left (\frac{x^{m-1}}{\tan ^{-1}(a x)},x\right )}{a c}-\frac{x^m}{a c \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.0846464, 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 = {} \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx \]
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 ) \tan ^{-1}(a x)^2} \, dx &=-\frac{x^m}{a c \tan ^{-1}(a x)}+\frac{m \int \frac{x^{-1+m}}{\tan ^{-1}(a x)} \, dx}{a c}\\ \end{align*}
Mathematica [A] time = 0.332036, size = 0, normalized size = 0. \[ \int \frac{x^m}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.437, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ({a}^{2}c{x}^{2}+c \right ) \left ( \arctan \left ( ax \right ) \right ) ^{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*} \frac{m \arctan \left (a x\right ) \int \frac{x^{m}}{x \arctan \left (a x\right )}\,{d x} - x^{m}}{a c \arctan \left (a x\right )} \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{x^{m}}{{\left (a^{2} c x^{2} + c\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{m}}{a^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )} + \operatorname{atan}^{2}{\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^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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