Integrand size = 20, antiderivative size = 20 \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\text {Int}\left (\frac {a+b \arctan (c x)}{\sqrt {d+e x^2}},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx \\ \end{align*}
Not integrable
Time = 2.88 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx \]
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Not integrable
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {a +b \arctan \left (c x \right )}{\sqrt {e \,x^{2}+d}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 2.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{\sqrt {d + e x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 52.41 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.15 \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{\sqrt {e x^{2} + d}} \,d x } \]
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Not integrable
Time = 0.70 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \arctan (c x)}{\sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{\sqrt {e\,x^2+d}} \,d x \]
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