\(\int \sqrt {2 x-x^2} \, dx\) [144]

Optimal result

Integrand size = 13, antiderivative size = 33 \[ \int \sqrt {2 x-x^2} \, dx=-\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{2} \arcsin (1-x) \]

[Out]

1/2*arcsin(-1+x)-1/2*(1-x)*(-x^2+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {626, 633, 222} \[ \int \sqrt {2 x-x^2} \, dx=-\frac {1}{2} \arcsin (1-x)-\frac {1}{2} \sqrt {2 x-x^2} (1-x) \]

[In]

Int[Sqrt[2*x - x^2],x]

[Out]

-1/2*((1 - x)*Sqrt[2*x - x^2]) - ArcSin[1 - x]/2

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}+\frac {1}{2} \int \frac {1}{\sqrt {2 x-x^2}} \, dx \\ & = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,2-2 x\right ) \\ & = -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{2} \arcsin (1-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \sqrt {-((-2+x) x)} \left (-1+x+\frac {2 \log \left (\sqrt {-2+x}-\sqrt {x}\right )}{\sqrt {-2+x} \sqrt {x}}\right ) \]

[In]

Integrate[Sqrt[2*x - x^2],x]

[Out]

(Sqrt[-((-2 + x)*x)]*(-1 + x + (2*Log[Sqrt[-2 + x] - Sqrt[x]])/(Sqrt[-2 + x]*Sqrt[x])))/2

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {\left (-1+x \right ) x \left (-2+x \right )}{2 \sqrt {-x \left (-2+x \right )}}+\frac {\arcsin \left (-1+x \right )}{2}\)	\(25\)
default	\(-\frac {\left (-2 x +2\right ) \sqrt {-x^{2}+2 x}}{4}+\frac {\arcsin \left (-1+x \right )}{2}\)	\(26\)
pseudoelliptic	\(-\arctan \left (\frac {\sqrt {-x \left (-2+x \right )}}{x}\right )+\frac {\left (-1+x \right ) \sqrt {-x \left (-2+x \right )}}{2}\)	\(30\)
meijerg	\(-\frac {2 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-3 x +3\right ) \sqrt {1-\frac {x}{2}}}{12}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )}{2}\right )}{\sqrt {\pi }}\)	\(47\)
trager	\(\left (-\frac {1}{2}+\frac {x}{2}\right ) \sqrt {-x^{2}+2 x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+2 x}+x -1\right )}{2}\)	\(49\)

[In]

int((-x^2+2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-1+x)*x*(-2+x)/(-x*(-2+x))^(1/2)+1/2*arcsin(-1+x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} {\left (x - 1\right )} - \arctan \left (\frac {\sqrt {-x^{2} + 2 \, x}}{x}\right ) \]

[In]

integrate((-x^2+2*x)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(-x^2 + 2*x)*(x - 1) - arctan(sqrt(-x^2 + 2*x)/x)

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \sqrt {2 x-x^2} \, dx=\left (\frac {x}{2} - \frac {1}{2}\right ) \sqrt {- x^{2} + 2 x} + \frac {\operatorname {asin}{\left (x - 1 \right )}}{2} \]

[In]

integrate((-x**2+2*x)**(1/2),x)

[Out]

(x/2 - 1/2)*sqrt(-x**2 + 2*x) + asin(x - 1)/2

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \arcsin \left (-x + 1\right ) \]

[In]

integrate((-x^2+2*x)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^2 + 2*x)*x - 1/2*sqrt(-x^2 + 2*x) - 1/2*arcsin(-x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \sqrt {2 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} {\left (x - 1\right )} + \frac {1}{2} \, \arcsin \left (x - 1\right ) \]

[In]

integrate((-x^2+2*x)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 + 2*x)*(x - 1) + 1/2*arcsin(x - 1)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \sqrt {2 x-x^2} \, dx=\frac {\mathrm {asin}\left (x-1\right )}{2}+\left (\frac {x}{2}-\frac {1}{2}\right )\,\sqrt {2\,x-x^2} \]

[In]

int((2*x - x^2)^(1/2),x)

[Out]

asin(x - 1)/2 + (x/2 - 1/2)*(2*x - x^2)^(1/2)