Integrand size = 19, antiderivative size = 19 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=\text {Int}\left (\frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx \\ \end{align*}
Not integrable
Time = 0.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=\int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx \]
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Not integrable
Time = 28.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00
\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{2}}{\arctan \left (a x \right )}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 0.98 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=c^{2} \left (\int \frac {2 a^{2} x^{2}}{\operatorname {atan}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{4}}{\operatorname {atan}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}{\left (a x \right )}}\, dx\right ) \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 33.65 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.16 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{2}}{\arctan \left (a x\right )} \,d x } \]
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Not integrable
Time = 0.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\arctan (a x)} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^2}{\mathrm {atan}\left (a\,x\right )} \,d x \]
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