Integrand size = 13, antiderivative size = 33 \[ \int \sqrt {2 x-x^2} \, dx=-\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{2} \arcsin (1-x) \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {626, 633, 222} \[ \int \sqrt {2 x-x^2} \, dx=-\frac {1}{2} \arcsin (1-x)-\frac {1}{2} \sqrt {2 x-x^2} (1-x) \]
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Rule 222
Rule 626
Rule 633
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}+\frac {1}{2} \int \frac {1}{\sqrt {2 x-x^2}} \, dx \\ & = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,2-2 x\right ) \\ & = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{2} \arcsin (1-x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \sqrt {-((-2+x) x)} \left (-1+x+\frac {2 \log \left (\sqrt {-2+x}-\sqrt {x}\right )}{\sqrt {-2+x} \sqrt {x}}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {\left (-1+x \right ) x \left (-2+x \right )}{2 \sqrt {-x \left (-2+x \right )}}+\frac {\arcsin \left (-1+x \right )}{2}\) | \(25\) |
default | \(-\frac {\left (-2 x +2\right ) \sqrt {-x^{2}+2 x}}{4}+\frac {\arcsin \left (-1+x \right )}{2}\) | \(26\) |
pseudoelliptic | \(-\arctan \left (\frac {\sqrt {-x \left (-2+x \right )}}{x}\right )+\frac {\left (-1+x \right ) \sqrt {-x \left (-2+x \right )}}{2}\) | \(30\) |
meijerg | \(-\frac {2 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-3 x +3\right ) \sqrt {1-\frac {x}{2}}}{12}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )}{2}\right )}{\sqrt {\pi }}\) | \(47\) |
trager | \(\left (-\frac {1}{2}+\frac {x}{2}\right ) \sqrt {-x^{2}+2 x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+2 x}+x -1\right )}{2}\) | \(49\) |
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Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} {\left (x - 1\right )} - \arctan \left (\frac {\sqrt {-x^{2} + 2 \, x}}{x}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \sqrt {2 x-x^2} \, dx=\left (\frac {x}{2} - \frac {1}{2}\right ) \sqrt {- x^{2} + 2 x} + \frac {\operatorname {asin}{\left (x - 1 \right )}}{2} \]
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Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \arcsin \left (-x + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} {\left (x - 1\right )} + \frac {1}{2} \, \arcsin \left (x - 1\right ) \]
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \sqrt {2 x-x^2} \, dx=\frac {\mathrm {asin}\left (x-1\right )}{2}+\left (\frac {x}{2}-\frac {1}{2}\right )\,\sqrt {2\,x-x^2} \]
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