Integrand size = 14, antiderivative size = 25 \[ \int \frac {x^2}{5+2 x+x^2} \, dx=x-\frac {3}{2} \arctan \left (\frac {1+x}{2}\right )-\log \left (5+2 x+x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {717, 648, 632, 210, 642} \[ \int \frac {x^2}{5+2 x+x^2} \, dx=-\frac {3}{2} \arctan \left (\frac {x+1}{2}\right )-\log \left (x^2+2 x+5\right )+x \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 717
Rubi steps \begin{align*} \text {integral}& = x+\int \frac {-5-2 x}{5+2 x+x^2} \, dx \\ & = x-3 \int \frac {1}{5+2 x+x^2} \, dx-\int \frac {2+2 x}{5+2 x+x^2} \, dx \\ & = x-\log \left (5+2 x+x^2\right )+6 \text {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,2+2 x\right ) \\ & = x-\frac {3}{2} \arctan \left (\frac {1+x}{2}\right )-\log \left (5+2 x+x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{5+2 x+x^2} \, dx=x-\frac {3}{2} \arctan \left (\frac {1+x}{2}\right )-\log \left (5+2 x+x^2\right ) \]
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Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
default | \(x -\frac {3 \arctan \left (\frac {1}{2}+\frac {x}{2}\right )}{2}-\ln \left (x^{2}+2 x +5\right )\) | \(22\) |
risch | \(x -\frac {3 \arctan \left (\frac {1}{2}+\frac {x}{2}\right )}{2}-\ln \left (x^{2}+2 x +5\right )\) | \(22\) |
parallelrisch | \(x -\ln \left (x +1-2 i\right )+\frac {3 i \ln \left (x +1-2 i\right )}{4}-\ln \left (x +1+2 i\right )-\frac {3 i \ln \left (x +1+2 i\right )}{4}\) | \(37\) |
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{5+2 x+x^2} \, dx=x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{5+2 x+x^2} \, dx=x - \log {\left (x^{2} + 2 x + 5 \right )} - \frac {3 \operatorname {atan}{\left (\frac {x}{2} + \frac {1}{2} \right )}}{2} \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{5+2 x+x^2} \, dx=x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{5+2 x+x^2} \, dx=x - \frac {3}{2} \, \arctan \left (\frac {1}{2} \, x + \frac {1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \]
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Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{5+2 x+x^2} \, dx=x-\ln \left (x^2+2\,x+5\right )-\frac {3\,\mathrm {atan}\left (\frac {x}{2}+\frac {1}{2}\right )}{2} \]
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