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

Optimal. Leaf size=8 \[ -\sin ^{-1}(1-2 x) \]

[Out]

-ArcSin[1 - 2*x]

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Rubi [A]  time = 0.00738297, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[x - x^2],x]

[Out]

-ArcSin[1 - 2*x]

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Rubi in Sympy [A]  time = 0.573428, size = 5, normalized size = 0.62 \[ \operatorname{asin}{\left (2 x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(2*x - 1)

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Mathematica [B]  time = 0.0142764, size = 38, normalized size = 4.75 \[ \frac{2 \sqrt{x-1} \sqrt{x} \log \left (\sqrt{x-1}+\sqrt{x}\right )}{\sqrt{-(x-1) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[x - x^2],x]

[Out]

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

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Maple [A]  time = 0.004, size = 7, normalized size = 0.9 \[ \arcsin \left ( 2\,x-1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+x)^(1/2),x)

[Out]

arcsin(2*x-1)

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Maxima [A]  time = 1.57631, size = 8, normalized size = 1. \[ \arcsin \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(2*x - 1)

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Fricas [A]  time = 0.214582, size = 22, normalized size = 2.75 \[ -2 \, \arctan \left (\frac{\sqrt{-x^{2} + x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*arctan(sqrt(-x^2 + x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+x)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 + x), x)

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GIAC/XCAS [A]  time = 0.208798, size = 8, normalized size = 1. \[ \arcsin \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(2*x - 1)