Integrand size = 11, antiderivative size = 23 \[ \int \sqrt {1-x^2} \, dx=\frac {1}{2} x \sqrt {1-x^2}+\frac {\arcsin (x)}{2} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {201, 222} \[ \int \sqrt {1-x^2} \, dx=\frac {\arcsin (x)}{2}+\frac {1}{2} \sqrt {1-x^2} x \]
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Rule 201
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {1}{2} x \sqrt {1-x^2}+\frac {\arcsin (x)}{2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \sqrt {1-x^2} \, dx=\frac {1}{2} x \sqrt {1-x^2}-\arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {\arcsin \left (x \right )}{2}+\frac {x \sqrt {-x^{2}+1}}{2}\) | \(18\) |
risch | \(-\frac {x \left (x^{2}-1\right )}{2 \sqrt {-x^{2}+1}}+\frac {\arcsin \left (x \right )}{2}\) | \(23\) |
pseudoelliptic | \(\frac {x \sqrt {-x^{2}+1}}{2}-\frac {\arctan \left (\frac {\sqrt {-x^{2}+1}}{x}\right )}{2}\) | \(30\) |
meijerg | \(\frac {i \left (-2 i \sqrt {\pi }\, x \sqrt {-x^{2}+1}-2 i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{4 \sqrt {\pi }}\) | \(32\) |
trager | \(\frac {x \sqrt {-x^{2}+1}}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{2}\) | \(41\) |
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none
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \sqrt {1-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x - \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \sqrt {1-x^2} \, dx=\frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\left (x \right )}}{2} \]
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none
Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \sqrt {1-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \sqrt {1-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \sqrt {1-x^2} \, dx=\frac {\mathrm {asin}\left (x\right )}{2}+\frac {x\,\sqrt {1-x^2}}{2} \]
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