Integrand size = 11, antiderivative size = 33 \[ \int \sqrt {(4-x) x} \, dx=-\frac {1}{2} (2-x) \sqrt {4 x-x^2}-2 \arcsin \left (1-\frac {x}{2}\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1976, 626, 633, 222} \[ \int \sqrt {(4-x) x} \, dx=-2 \arcsin \left (1-\frac {x}{2}\right )-\frac {1}{2} \sqrt {4 x-x^2} (2-x) \]
[In]
[Out]
Rule 222
Rule 626
Rule 633
Rule 1976
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {4 x-x^2} \, dx \\ & = -\frac {1}{2} (2-x) \sqrt {4 x-x^2}+2 \int \frac {1}{\sqrt {4 x-x^2}} \, dx \\ & = -\frac {1}{2} (2-x) \sqrt {4 x-x^2}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{16}}} \, dx,x,4-2 x\right ) \\ & = -\frac {1}{2} (2-x) \sqrt {4 x-x^2}-2 \sin ^{-1}\left (1-\frac {x}{2}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \sqrt {(4-x) x} \, dx=\frac {1}{2} \sqrt {-((-4+x) x)} \left (-2+x-\frac {16 \text {arctanh}\left (\frac {\sqrt {-4+x}}{-2+\sqrt {x}}\right )}{\sqrt {-4+x} \sqrt {x}}\right ) \]
[In]
[Out]
Time = 1.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {\left (x -2\right ) x \left (x -4\right )}{2 \sqrt {-x \left (x -4\right )}}+2 \arcsin \left (-1+\frac {x}{2}\right )\) | \(27\) |
default | \(-\frac {\left (4-2 x \right ) \sqrt {-x^{2}+4 x}}{4}+2 \arcsin \left (-1+\frac {x}{2}\right )\) | \(28\) |
pseudoelliptic | \(-4 \arctan \left (\frac {\sqrt {-x \left (x -4\right )}}{x}\right )+\frac {\left (x -2\right ) \sqrt {-x \left (x -4\right )}}{2}\) | \(30\) |
meijerg | \(-\frac {8 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \left (-\frac {3 x}{2}+3\right ) \sqrt {-\frac {x}{4}+1}}{12}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {x}}{2}\right )}{2}\right )}{\sqrt {\pi }}\) | \(41\) |
trager | \(\left (-1+\frac {x}{2}\right ) \sqrt {-x^{2}+4 x}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}+4 x}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\) | \(57\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \sqrt {(4-x) x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 4 \, x} {\left (x - 2\right )} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 4 \, x}}{x}\right ) \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \sqrt {(4-x) x} \, dx=\left (\frac {x}{2} - 1\right ) \sqrt {- x^{2} + 4 x} + 2 \operatorname {asin}{\left (\frac {x}{2} - 1 \right )} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \sqrt {(4-x) x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 4 \, x} x - \sqrt {-x^{2} + 4 \, x} - 2 \, \arcsin \left (-\frac {1}{2} \, x + 1\right ) \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \sqrt {(4-x) x} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 4 \, x} {\left (x - 2\right )} + 2 \, \arcsin \left (\frac {1}{2} \, x - 1\right ) \]
[In]
[Out]
Time = 21.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \sqrt {(4-x) x} \, dx=2\,\mathrm {asin}\left (\frac {x}{2}-1\right )+\left (\frac {x}{2}-1\right )\,\sqrt {4\,x-x^2} \]
[In]
[Out]