Integrand size = 20, antiderivative size = 19 \[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=\arctan \left (\frac {3-2 x}{\sqrt {-4+2 x+x^2}}\right ) \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {738, 210} \[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=\arctan \left (\frac {3-2 x}{\sqrt {x^2+2 x-4}}\right ) \]
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Rule 210
Rule 738
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {6-4 x}{\sqrt {-4+2 x+x^2}}\right )\right ) \\ & = \tan ^{-1}\left (\frac {3-2 x}{\sqrt {-4+2 x+x^2}}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=-2 \arctan \left (1-x+\sqrt {-4+2 x+x^2}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21
method | result | size |
default | \(-\arctan \left (\frac {-6+4 x}{2 \sqrt {\left (-1+x \right )^{2}-5+4 x}}\right )\) | \(23\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{2}+2 x -4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{-1+x}\right )\) | \(44\) |
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Time = 0.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=-2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x - 4} + 1\right ) \]
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\[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=- \int \frac {1}{x \sqrt {x^{2} + 2 x - 4} - \sqrt {x^{2} + 2 x - 4}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=-\arcsin \left (\frac {2 \, \sqrt {5} x}{5 \, {\left | x - 1 \right |}} - \frac {3 \, \sqrt {5}}{5 \, {\left | x - 1 \right |}}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=-2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x - 4} + 1\right ) \]
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Timed out. \[ \int \frac {1}{(1-x) \sqrt {-4+2 x+x^2}} \, dx=-\int \frac {1}{\left (x-1\right )\,\sqrt {x^2+2\,x-4}} \,d x \]
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