Integrand size = 6, antiderivative size = 22 \[ \int \arctan \left (\sqrt {x}\right ) \, dx=-\sqrt {x}+\arctan \left (\sqrt {x}\right )+x \arctan \left (\sqrt {x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4930, 52, 65, 209} \[ \int \arctan \left (\sqrt {x}\right ) \, dx=x \arctan \left (\sqrt {x}\right )+\arctan \left (\sqrt {x}\right )-\sqrt {x} \]
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Rule 52
Rule 65
Rule 209
Rule 4930
Rubi steps \begin{align*} \text {integral}& = x \arctan \left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{1+x} \, dx \\ & = -\sqrt {x}+x \arctan \left (\sqrt {x}\right )+\frac {1}{2} \int \frac {1}{\sqrt {x} (1+x)} \, dx \\ & = -\sqrt {x}+x \arctan \left (\sqrt {x}\right )+\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right ) \\ & = -\sqrt {x}+\arctan \left (\sqrt {x}\right )+x \arctan \left (\sqrt {x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \arctan \left (\sqrt {x}\right ) \, dx=-\sqrt {x}+(1+x) \arctan \left (\sqrt {x}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\arctan \left (\sqrt {x}\right )+x \arctan \left (\sqrt {x}\right )-\sqrt {x}\) | \(17\) |
default | \(\arctan \left (\sqrt {x}\right )+x \arctan \left (\sqrt {x}\right )-\sqrt {x}\) | \(17\) |
parts | \(\arctan \left (\sqrt {x}\right )+x \arctan \left (\sqrt {x}\right )-\sqrt {x}\) | \(17\) |
meijerg | \(-\sqrt {x}+\frac {\left (3 x +3\right ) \arctan \left (\sqrt {x}\right )}{3}\) | \(18\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \arctan \left (\sqrt {x}\right ) \, dx={\left (x + 1\right )} \arctan \left (\sqrt {x}\right ) - \sqrt {x} \]
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Time = 0.45 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \arctan \left (\sqrt {x}\right ) \, dx=- \sqrt {x} + x \operatorname {atan}{\left (\sqrt {x} \right )} + \operatorname {atan}{\left (\sqrt {x} \right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \arctan \left (\sqrt {x}\right ) \, dx=x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \arctan \left (\sqrt {x}\right ) \, dx=x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \arctan \left (\sqrt {x}\right ) \, dx=\mathrm {atan}\left (\sqrt {x}\right )+x\,\mathrm {atan}\left (\sqrt {x}\right )-\sqrt {x} \]
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