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

Optimal. Leaf size=33 \[ -\frac {1}{2} (1-x) \sqrt {2 x-x^2}-\frac {1}{2} \sin ^{-1}(1-x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {626, 633, 222} \begin {gather*} -\frac {1}{2} \sqrt {2 x-x^2} (1-x)-\frac {1}{2} \sin ^{-1}(1-x) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \sqrt {2 x-x^2} \, dx &=-\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} \sin ^{-1}(1-x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 43, normalized size = 1.30 \begin {gather*} \frac {1}{2} \sqrt {-((-2+x) x)} \left (-1+x-\frac {2 \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-2+x}{x}}}\right )}{\sqrt {-2+x} \sqrt {x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2*x - x^2],x]

[Out]

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

________________________________________________________________________________________

Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[2*x - x^2],x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 26, normalized size = 0.79

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	\(-\frac {\left (-2 x +2\right ) \sqrt {-x^{2}+2 x}}{4}+\frac {\arcsin \left (-1+x \right )}{2}\)	\(26\)
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	\(\left (-\frac {1}{2}+\frac {x}{2}\right ) \sqrt {-x^{2}+2 x}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+2 x}+x -1\right )}{2}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-2*x+2)*(-x^2+2*x)^(1/2)+1/2*arcsin(-1+x)

________________________________________________________________________________________

Maxima [A]
time = 0.44, size = 36, normalized size = 1.09 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]
time = 0.32, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} {\left (x - 1\right )} - \arctan \left (\frac {\sqrt {-x^{2} + 2 \, x}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- x^{2} + 2 x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-x**2 + 2*x), x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 30, normalized size = 0.91 \begin {gather*} 2 \left (\frac {x}{4}-\frac 1{4}\right ) \sqrt {-x^{2}+2 x}+\frac {\arcsin \left (x-1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.17, size = 24, normalized size = 0.73 \begin {gather*} \frac {\mathrm {asin}\left (x-1\right )}{2}+\left (\frac {x}{2}-\frac {1}{2}\right )\,\sqrt {2\,x-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - x^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________