3.312 \(\int \frac{1}{(2-2 x) \sqrt{2 x-x^2}} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]

[Out]

ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi [A]  time = 0.0330142, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((2 - 2*x)*Sqrt[2*x - x^2]),x]

[Out]

ArcTanh[Sqrt[2*x - x^2]]/2

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Rubi in Sympy [A]  time = 6.14732, size = 12, normalized size = 0.67 \[ \frac{\operatorname{atanh}{\left (\sqrt{- x^{2} + 2 x} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(sqrt(-x**2 + 2*x))/2

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Mathematica [B]  time = 0.0325672, size = 59, normalized size = 3.28 \[ -\frac{\sqrt{x-2} \sqrt{x} \left (\tan ^{-1}\left (\frac{\sqrt{x}-2}{\sqrt{x-2}}\right )+\tan ^{-1}\left (\frac{\sqrt{x}+2}{\sqrt{x-2}}\right )\right )}{2 \sqrt{-(x-2) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 - 2*x)*Sqrt[2*x - x^2]),x]

[Out]

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

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Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.687985, size = 42, normalized size = 2.33 \[ \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \]

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

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Fricas [A]  time = 0.221766, size = 59, normalized size = 3.28 \[ \frac{1}{2} \, \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220043, size = 35, normalized size = 1.94 \[ -\frac{1}{2} \,{\rm ln}\left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]

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