Integrand size = 17, antiderivative size = 38 \[ \int (1-x) \sqrt {1-x^2} \, dx=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {\arcsin (x)}{2} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 38, 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 = {655, 201, 222} \[ \int (1-x) \sqrt {1-x^2} \, dx=\frac {\arcsin (x)}{2}+\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} x \sqrt {1-x^2} \]
[In]
[Out]
Rule 201
Rule 222
Rule 655
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (1-x^2\right )^{3/2}+\int \sqrt {1-x^2} \, dx \\ & = \frac {1}{2} x \sqrt {1-x^2}+\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = \frac {1}{2} x \sqrt {1-x^2}+\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} \sin ^{-1}(x) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int (1-x) \sqrt {1-x^2} \, dx=-\frac {1}{6} \sqrt {1-x^2} \left (-2-3 x+2 x^2\right )-\arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
[In]
[Out]
Time = 2.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{3}+\frac {\arcsin \left (x \right )}{2}+\frac {x \sqrt {-x^{2}+1}}{2}\) | \(29\) |
risch | \(\frac {\left (2 x^{2}-3 x -2\right ) \left (x^{2}-1\right )}{6 \sqrt {-x^{2}+1}}+\frac {\arcsin \left (x \right )}{2}\) | \(32\) |
trager | \(\left (-\frac {1}{3} x^{2}+\frac {1}{2} x +\frac {1}{3}\right ) \sqrt {-x^{2}+1}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{2}\) | \(49\) |
meijerg | \(\frac {i \left (-2 i \sqrt {\pi }\, x \sqrt {-x^{2}+1}-2 i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{4 \sqrt {\pi }}-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (-2 x^{2}+2\right ) \sqrt {-x^{2}+1}}{3}}{4 \sqrt {\pi }}\) | \(65\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05 \[ \int (1-x) \sqrt {1-x^2} \, dx=-\frac {1}{6} \, {\left (2 \, x^{2} - 3 \, x - 2\right )} \sqrt {-x^{2} + 1} - \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.03 \[ \int (1-x) \sqrt {1-x^2} \, dx=- \frac {x^{2} \sqrt {1 - x^{2}}}{3} + \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\sqrt {1 - x^{2}}}{3} + \frac {\operatorname {asin}{\left (x \right )}}{2} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int (1-x) \sqrt {1-x^2} \, dx=\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int (1-x) \sqrt {1-x^2} \, dx=-\frac {1}{6} \, {\left ({\left (2 \, x - 3\right )} x - 2\right )} \sqrt {-x^{2} + 1} + \frac {1}{2} \, \arcsin \left (x\right ) \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int (1-x) \sqrt {1-x^2} \, dx=\frac {\mathrm {asin}\left (x\right )}{2}+\sqrt {1-x^2}\,\left (-\frac {x^2}{3}+\frac {x}{2}+\frac {1}{3}\right ) \]
[In]
[Out]