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

Optimal. Leaf size=51 \[ -\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{x}{2}-\frac{1}{2} \log (1-x) \]

[Out]

-x/2 - Sqrt[2*x - x^2]/2 + ArcTanh[Sqrt[2*x - x^2]]/2 - Log[1 - x]/2

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Rubi [A]  time = 0.198276, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{x}{2}-\frac{1}{2} \log (1-x) \]

Antiderivative was successfully verified.

[In]  Int[x/(-x + Sqrt[2*x - x^2]),x]

[Out]

-x/2 - Sqrt[2*x - x^2]/2 + ArcTanh[Sqrt[2*x - x^2]]/2 - Log[1 - x]/2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0386552, size = 41, normalized size = 0.8 \[ \frac{1}{2} \left (-x-\sqrt{-(x-2) x}-2 \log (1-x)+\log \left (\sqrt{-(x-2) x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 38, normalized size = 0.8 \[ -{\frac{x}{2}}-{\frac{\ln \left ( -1+x \right ) }{2}}-{\frac{1}{2}\sqrt{- \left ( -1+x \right ) ^{2}+1}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x+(-x^2+2*x)^(1/2)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272809, size = 89, normalized size = 1.75 \[ -\frac{1}{2} \, x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \log \left (x - 1\right ) + \frac{1}{2} \, \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x - 1/2*sqrt(-x^2 + 2*x) - 1/2*log(x - 1) + 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. \[ - \int \frac{x}{x - \sqrt{- x^{2} + 2 x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.271647, size = 68, normalized size = 1.33 \[ -\frac{1}{2} \, x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \,{\rm ln}\left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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