Integrand size = 19, antiderivative size = 19 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\text {Int}\left (\frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}},x\right ) \]
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Not integrable
Time = 0.01 (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 = {} \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx \\ \end{align*}
Not integrable
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx \]
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Not integrable
Time = 1.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89
\[\int \frac {a^{2} c \,x^{2}+c}{\sqrt {\arctan \left (a x \right )}}d x\]
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Exception generated. \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.76 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=c \left (\int \frac {a^{2} x^{2}}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx + \int \frac {1}{\sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx\right ) \]
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Exception generated. \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 75.74 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.16 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int { \frac {a^{2} c x^{2} + c}{\sqrt {\arctan \left (a x\right )}} \,d x } \]
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Not integrable
Time = 0.52 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {c+a^2 c x^2}{\sqrt {\arctan (a x)}} \, dx=\int \frac {c\,a^2\,x^2+c}{\sqrt {\mathrm {atan}\left (a\,x\right )}} \,d x \]
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