3.144 \(\int \sqrt{2 x-x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0076645, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {612, 619, 216} \[ -\frac{1}{2} \sqrt{2 x-x^2} (1-x)-\frac{1}{2} \sin ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2*x - x^2],x]

[Out]

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

Rule 612

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 619

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]

Rule 216

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]

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} \operatorname{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 = 0.0391544, size = 32, normalized size = 0.97 \[ \frac{1}{2} (x-1) \sqrt{-(x-2) x}-\sin ^{-1}\left (\sqrt{1-\frac{x}{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2*x - x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 26, normalized size = 0.8 \begin{align*} -{\frac{-2\,x+2}{4}\sqrt{-{x}^{2}+2\,x}}+{\frac{\arcsin \left ( -1+x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.39716, size = 49, normalized size = 1.48 \begin{align*} \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{align*}

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)

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Fricas [A]  time = 2.16882, size = 80, normalized size = 2.42 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x - 1\right )} - \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{x}\right ) \end{align*}

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)

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

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)

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Giac [A]  time = 1.057, size = 31, normalized size = 0.94 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x - 1\right )} + \frac{1}{2} \, \arcsin \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)