Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=-\frac {x^m}{a c \arctan (a x)}+\frac {m \text {Int}\left (\frac {x^{-1+m}}{\arctan (a x)},x\right )}{a c} \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {x^m}{a c \arctan (a x)}+\frac {m \int \frac {x^{-1+m}}{\arctan (a x)} \, dx}{a c} \\ \end{align*}
Not integrable
Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx \]
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Not integrable
Time = 3.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 3.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\frac {\int \frac {x^{m}}{a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c} \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 112.91 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int { \frac {x^{m}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx=\int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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