Integrand size = 21, antiderivative size = 18 \[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{3/2}}{3 a c} \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5004} \[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{3/2}}{3 a c} \]
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Rule 5004
Rubi steps \begin{align*} \text {integral}& = \frac {2 \arctan (a x)^{3/2}}{3 a c} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {2 \arctan (a x)^{3/2}}{3 a c} \]
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Time = 3.86 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {2 \, \arctan \left (a x\right )^{\frac {3}{2}}}{3 \, a c} \]
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\[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \]
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Exception generated. \[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {2 \, \arctan \left (a x\right )^{\frac {3}{2}}}{3 \, a c} \]
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Time = 0.43 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {\arctan (a x)}}{c+a^2 c x^2} \, dx=\frac {2\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{3\,a\,c} \]
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