Integrand size = 10, antiderivative size = 19 \[ \int \frac {1}{2-x+x^2} \, dx=-\frac {2 \arctan \left (\frac {1-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {632, 210} \[ \int \frac {1}{2-x+x^2} \, dx=-\frac {2 \arctan \left (\frac {1-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Rule 210
Rule 632
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,-1+2 x\right )\right ) \\ & = -\frac {2 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2-x+x^2} \, dx=\frac {2 \arctan \left (\frac {-1+2 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Time = 1.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {2 \sqrt {7}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {7}}{7}\right )}{7}\) | \(17\) |
risch | \(\frac {2 \sqrt {7}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {7}}{7}\right )}{7}\) | \(17\) |
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Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{2-x+x^2} \, dx=\frac {2}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x - 1\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {1}{2-x+x^2} \, dx=\frac {2 \sqrt {7} \operatorname {atan}{\left (\frac {2 \sqrt {7} x}{7} - \frac {\sqrt {7}}{7} \right )}}{7} \]
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none
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{2-x+x^2} \, dx=\frac {2}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x - 1\right )}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{2-x+x^2} \, dx=\frac {2}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x - 1\right )}\right ) \]
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Time = 8.79 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{2-x+x^2} \, dx=\frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {\sqrt {7}\,\left (2\,x-1\right )}{7}\right )}{7} \]
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