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

Optimal. Leaf size=36 \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{1}{2} \sqrt{2 x-x^2} \]

[Out]

-Sqrt[2*x - x^2]/2 + ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi [A]  time = 0.0565279, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{1}{2} \sqrt{2 x-x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2*x - x^2]/(2 - 2*x),x]

[Out]

-Sqrt[2*x - x^2]/2 + ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi in Sympy [A]  time = 9.82599, size = 24, normalized size = 0.67 \[ - \frac{\sqrt{- x^{2} + 2 x}}{2} + \frac{\operatorname{atanh}{\left (\sqrt{- x^{2} + 2 x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+2*x)**(1/2)/(2-2*x),x)

[Out]

-sqrt(-x**2 + 2*x)/2 + atanh(sqrt(-x**2 + 2*x))/2

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Mathematica [B]  time = 0.0513368, size = 73, normalized size = 2.03 \[ \frac{\sqrt{-(x-2) x} \left (-\sqrt{x-2} \sqrt{x}+\tan ^{-1}\left (\frac{\sqrt{x}-2}{\sqrt{x-2}}\right )+\tan ^{-1}\left (\frac{\sqrt{x}+2}{\sqrt{x-2}}\right )\right )}{2 \sqrt{x-2} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2*x - x^2]/(2 - 2*x),x]

[Out]

(Sqrt[-((-2 + x)*x)]*(-(Sqrt[-2 + x]*Sqrt[x]) + ArcTan[(-2 + Sqrt[x])/Sqrt[-2 +
x]] + ArcTan[(2 + Sqrt[x])/Sqrt[-2 + x]]))/(2*Sqrt[-2 + x]*Sqrt[x])

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Maple [A]  time = 0.007, size = 29, normalized size = 0.8 \[ -{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}+1}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+2*x)^(1/2)/(2-2*x),x)

[Out]

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

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Maxima [A]  time = 0.762376, size = 61, normalized size = 1.69 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-x^2 + 2*x) + 1/2*log(2*sqrt(-x^2 + 2*x)/abs(x - 1) + 2/abs(x - 1))

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Fricas [A]  time = 0.219054, size = 77, normalized size = 2.14 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int \frac{\sqrt{- x^{2} + 2 x}}{x - 1}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+2*x)**(1/2)/(2-2*x),x)

[Out]

-Integral(sqrt(-x**2 + 2*x)/(x - 1), x)/2

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GIAC/XCAS [A]  time = 0.22004, size = 54, normalized size = 1.5 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \,{\rm ln}\left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-x^2 + 2*x) - 1/2*ln(-2*(sqrt(-x^2 + 2*x) - 1)/abs(-2*x + 2))