Integrand size = 15, antiderivative size = 12 \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=-\arctan \left (\sqrt {3}-x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {632, 210} \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=-\arctan \left (\sqrt {3}-x\right ) \]
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Rule 210
Rule 632
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,-2 \sqrt {3}+2 x\right )\right ) \\ & = -\tan ^{-1}\left (\sqrt {3}-x\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=-\arctan \left (\sqrt {3}-x\right ) \]
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Time = 3.56 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75
method | result | size |
default | \(\arctan \left (x -\sqrt {3}\right )\) | \(9\) |
risch | \(\arctan \left (x -\sqrt {3}\right )\) | \(9\) |
parallelrisch | \(\frac {i \ln \left (x +i-\sqrt {3}\right )}{2}-\frac {i \ln \left (x -\sqrt {3}-i\right )}{2}\) | \(28\) |
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none
Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=-\arctan \left (-x + \sqrt {3}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=\operatorname {atan}{\left (x - \sqrt {3} \right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=\arctan \left (x - \sqrt {3}\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=\arctan \left (x - \sqrt {3}\right ) \]
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Time = 9.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx=\mathrm {atan}\left (x-\sqrt {3}\right ) \]
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