3.313 \(\int \frac{1}{(2-2 x) \left (2 x-x^2\right )^{3/2}} \, dx\)

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]

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

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Rubi [A]  time = 0.0571368, 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[1/((2 - 2*x)*(2*x - x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.93971, size = 26, normalized size = 0.72 \[ \frac{\operatorname{atanh}{\left (\sqrt{- x^{2} + 2 x} \right )}}{2} - \frac{1}{2 \sqrt{- x^{2} + 2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2-2*x)/(-x**2+2*x)**(3/2),x)

[Out]

atanh(sqrt(-x**2 + 2*x))/2 - 1/(2*sqrt(-x**2 + 2*x))

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Mathematica [B]  time = 0.058522, size = 74, normalized size = 2.06 \[ -\frac{\sqrt{x-2} \sqrt{x} \tan ^{-1}\left (\frac{\sqrt{x}-2}{\sqrt{x-2}}\right )+\sqrt{x-2} \sqrt{x} \tan ^{-1}\left (\frac{\sqrt{x}+2}{\sqrt{x-2}}\right )+1}{2 \sqrt{-(x-2) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((2 - 2*x)*(2*x - x^2)^(3/2)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2-2*x)/(-x^2+2*x)^(3/2),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 = 0.692147, 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/((-x^2 + 2*x)^(3/2)*(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 = 0.219914, size = 107, normalized size = 2.97 \[ \frac{\sqrt{-x^{2} + 2 \, x} \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \sqrt{-x^{2} + 2 \, x} \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) - 1}{2 \, \sqrt{-x^{2} + 2 \, x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2/((-x^2 + 2*x)^(3/2)*(x - 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int \frac{1}{- x^{3} \sqrt{- x^{2} + 2 x} + 3 x^{2} \sqrt{- x^{2} + 2 x} - 2 x \sqrt{- x^{2} + 2 x}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2-2*x)/(-x**2+2*x)**(3/2),x)

[Out]

-Integral(1/(-x**3*sqrt(-x**2 + 2*x) + 3*x**2*sqrt(-x**2 + 2*x) - 2*x*sqrt(-x**2
+ 2*x)), x)/2

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GIAC/XCAS [A]  time = 0.222027, size = 66, normalized size = 1.83 \[ \frac{\sqrt{-x^{2} + 2 \, x}}{2 \,{\left (x^{2} - 2 \, x\right )}} - \frac{1}{2} \,{\rm ln}\left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2/((-x^2 + 2*x)^(3/2)*(x - 1)),x, algorithm="giac")

[Out]

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