Integrand size = 22, antiderivative size = 22 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\text {Int}\left (\frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3},x\right ) \]
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Not integrable
Time = 0.04 (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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx \\ \end{align*}
Not integrable
Time = 0.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx \]
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Not integrable
Time = 9.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )^{2}}{\arctan \left (a x \right )^{3}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 8.87 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=c^{2} \left (\int \frac {x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {2 a^{2} x^{2} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{4} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]
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Not integrable
Time = 1.15 (sec) , antiderivative size = 333, normalized size of antiderivative = 15.14 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 193.41 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2} x^{m}}{\arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^2}{\arctan (a x)^3} \, dx=\int \frac {x^m\,{\left (c\,a^2\,x^2+c\right )}^2}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]
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