Integrand size = 17, antiderivative size = 26 \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=-\sqrt {4 x-x^2}-4 \arcsin \left (1-\frac {x}{2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {654, 633, 222} \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=-4 \arcsin \left (1-\frac {x}{2}\right )-\sqrt {4 x-x^2} \]
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Rule 222
Rule 633
Rule 654
Rubi steps \begin{align*} \text {integral}& = -\sqrt {4 x-x^2}+4 \int \frac {1}{\sqrt {4 x-x^2}} \, dx \\ & = -\sqrt {4 x-x^2}-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,4-2 x\right ) \\ & = -\sqrt {4 x-x^2}-4 \sin ^{-1}\left (1-\frac {x}{2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=\frac {(-4+x) x-8 \sqrt {-4+x} \sqrt {x} \log \left (\sqrt {-4+x}-\sqrt {x}\right )}{\sqrt {-((-4+x) x)}} \]
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Time = 1.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
default | \(4 \arcsin \left (-1+\frac {x}{2}\right )-\sqrt {-x^{2}+4 x}\) | \(23\) |
risch | \(\frac {x \left (x -4\right )}{\sqrt {-x \left (x -4\right )}}+4 \arcsin \left (-1+\frac {x}{2}\right )\) | \(23\) |
pseudoelliptic | \(-\sqrt {-x \left (x -4\right )}-8 \arctan \left (\frac {\sqrt {-x \left (x -4\right )}}{x}\right )\) | \(27\) |
meijerg | \(4 \arcsin \left (\frac {\sqrt {x}}{2}\right )+\frac {4 i \left (\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {-\frac {x}{4}+1}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )\right )}{\sqrt {\pi }}\) | \(45\) |
trager | \(-\sqrt {-x^{2}+4 x}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}+4 x}\right )\) | \(52\) |
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Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=-\sqrt {-x^{2} + 4 \, x} - 8 \, \arctan \left (\frac {\sqrt {-x^{2} + 4 \, x}}{x}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=- \sqrt {- x^{2} + 4 x} + 4 \operatorname {asin}{\left (\frac {x}{2} - 1 \right )} \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=-\sqrt {-x^{2} + 4 \, x} - 4 \, \arcsin \left (-\frac {1}{2} \, x + 1\right ) \]
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Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=-\sqrt {-x^{2} + 4 \, x} + 4 \, \arcsin \left (\frac {1}{2} \, x - 1\right ) \]
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Time = 19.76 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {2+x}{\sqrt {4 x-x^2}} \, dx=4\,\mathrm {asin}\left (\frac {x}{2}-1\right )-\sqrt {4\,x-x^2} \]
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