Integrand size = 18, antiderivative size = 27 \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=-\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {\arcsin (x)}{2} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {799, 794, 222} \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=-\frac {\arcsin (x)}{2}-\frac {1}{2} \sqrt {1-x^2} (2-x) \]
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Rule 222
Rule 794
Rule 799
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-x) x}{\sqrt {1-x^2}} \, dx \\ & = -\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {1}{2} (2-x) \sqrt {1-x^2}-\frac {1}{2} \sin ^{-1}(x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=\frac {1}{2} (-2+x) \sqrt {1-x^2}+\arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
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Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
risch | \(-\frac {\left (-2+x \right ) \left (x^{2}-1\right )}{2 \sqrt {-x^{2}+1}}-\frac {\arcsin \left (x \right )}{2}\) | \(25\) |
default | \(\frac {x \sqrt {-x^{2}+1}}{2}-\frac {\arcsin \left (x \right )}{2}-\sqrt {-\left (1+x \right )^{2}+2+2 x}\) | \(34\) |
trager | \(\left (-1+\frac {x}{2}\right ) \sqrt {-x^{2}+1}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{2}\) | \(45\) |
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} + \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 1.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=\begin {cases} \frac {x \sqrt {1 - x^{2}}}{2} - \sqrt {1 - x^{2}} - \frac {\operatorname {asin}{\left (x \right )}}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1} - \frac {1}{2} \, \arcsin \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x - 2\right )} - \frac {1}{2} \, \arcsin \left (x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {x \sqrt {1-x^2}}{1+x} \, dx=\left (\frac {x}{2}-1\right )\,\sqrt {1-x^2}-\frac {\mathrm {asin}\left (x\right )}{2} \]
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