Integrand size = 21, antiderivative size = 31 \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sqrt {1+x^2} \arctan \left (\sqrt {1+x^2}\right )-\frac {1}{2} \log \left (2+x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {267, 5315, 266} \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sqrt {x^2+1} \arctan \left (\sqrt {x^2+1}\right )-\frac {1}{2} \log \left (x^2+2\right ) \]
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Rule 266
Rule 267
Rule 5315
Rubi steps \begin{align*} \text {integral}& = \sqrt {1+x^2} \arctan \left (\sqrt {1+x^2}\right )-\int \frac {x}{2+x^2} \, dx \\ & = \sqrt {1+x^2} \arctan \left (\sqrt {1+x^2}\right )-\frac {1}{2} \log \left (2+x^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sqrt {1+x^2} \arctan \left (\sqrt {1+x^2}\right )-\frac {1}{2} \log \left (2+x^2\right ) \]
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Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(-\frac {\ln \left (x^{2}+2\right )}{2}+\arctan \left (\sqrt {x^{2}+1}\right ) \sqrt {x^{2}+1}\) | \(26\) |
default | \(-\frac {\ln \left (x^{2}+2\right )}{2}+\arctan \left (\sqrt {x^{2}+1}\right ) \sqrt {x^{2}+1}\) | \(26\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sqrt {x^{2} + 1} \arctan \left (\sqrt {x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
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Time = 0.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sqrt {x^{2} + 1} \operatorname {atan}{\left (\sqrt {x^{2} + 1} \right )} - \frac {\log {\left (x^{2} + 2 \right )}}{2} \]
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none
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sqrt {x^{2} + 1} \arctan \left (\sqrt {x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\sqrt {x^{2} + 1} \arctan \left (\sqrt {x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
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Time = 0.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {x \arctan \left (\sqrt {1+x^2}\right )}{\sqrt {1+x^2}} \, dx=\mathrm {atan}\left (\sqrt {x^2+1}\right )\,\sqrt {x^2+1}-\frac {\ln \left (x^2+2\right )}{2} \]
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