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

Optimal. Leaf size=15 \[ -\sqrt{2 x-x^2} \]

[Out]

-Sqrt[2*x - x^2]

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Rubi [A]  time = 0.00856658, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\sqrt{2 x-x^2} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x)/Sqrt[2*x - x^2],x]

[Out]

-Sqrt[2*x - x^2]

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Rubi in Sympy [A]  time = 1.32372, size = 10, normalized size = 0.67 \[ - \sqrt{- x^{2} + 2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 2*x)

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Mathematica [A]  time = 0.0138485, size = 12, normalized size = 0.8 \[ -\sqrt{-(x-2) x} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[-((-2 + x)*x)]

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Maple [A]  time = 0.004, size = 17, normalized size = 1.1 \[{x \left ( x-2 \right ){\frac{1}{\sqrt{-{x}^{2}+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x)/(-x^2+2*x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.695921, size = 18, normalized size = 1.2 \[ -\sqrt{-x^{2} + 2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 2*x)

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Fricas [A]  time = 0.262983, size = 18, normalized size = 1.2 \[ -\sqrt{-x^{2} + 2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 2*x)

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Sympy [A]  time = 0.30012, size = 10, normalized size = 0.67 \[ - \sqrt{- x^{2} + 2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 2*x)

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GIAC/XCAS [A]  time = 0.263443, size = 18, normalized size = 1.2 \[ -\sqrt{-x^{2} + 2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 2*x)