Integrand size = 17, antiderivative size = 17 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=\text {Int}\left (\frac {c+a^2 c x^2}{\arctan (a x)^2},x\right ) \]
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Not integrable
Time = 0.01 (sec) , antiderivative size = 17, 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 = {} \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=\int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx \\ \end{align*}
Not integrable
Time = 0.49 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=\int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx \]
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Not integrable
Time = 19.67 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00
\[\int \frac {a^{2} c \,x^{2}+c}{\arctan \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=\int { \frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=c \left (\int \frac {a^{2} x^{2}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx\right ) \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.53 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=\int { \frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 55.51 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.18 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=\int { \frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.46 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {c+a^2 c x^2}{\arctan (a x)^2} \, dx=\int \frac {c\,a^2\,x^2+c}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \]
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