Integrand size = 18, antiderivative size = 13 \[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=\arctan \left (\sqrt {3-4 x+x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {702, 209} \[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=\arctan \left (\sqrt {x^2-4 x+3}\right ) \]
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Rule 209
Rule 702
Rubi steps \begin{align*} \text {integral}& = 4 \text {Subst}\left (\int \frac {1}{4+4 x^2} \, dx,x,\sqrt {3-4 x+x^2}\right ) \\ & = \tan ^{-1}\left (\sqrt {3-4 x+x^2}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=-2 \arctan \left (\frac {\sqrt {3-4 x+x^2}}{-3+x}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\arctan \left (\sqrt {x^{2}-4 x +3}\right )\) | \(12\) |
default | \(-\arctan \left (\frac {1}{\sqrt {\left (-2+x \right )^{2}-1}}\right )\) | \(13\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{2}-4 x +3}}{-2+x}\right )\) | \(33\) |
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Time = 0.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=2 \, \arctan \left (-x + \sqrt {x^{2} - 4 \, x + 3} + 2\right ) \]
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\[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=\int \frac {1}{\sqrt {\left (x - 3\right ) \left (x - 1\right )} \left (x - 2\right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=-\arcsin \left (\frac {1}{{\left | x - 2 \right |}}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=2 \, \arctan \left (-x + \sqrt {x^{2} - 4 \, x + 3} + 2\right ) \]
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Timed out. \[ \int \frac {1}{(-2+x) \sqrt {3-4 x+x^2}} \, dx=\int \frac {1}{\left (x-2\right )\,\sqrt {x^2-4\,x+3}} \,d x \]
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