3.27 \(\int \frac{1}{\sqrt{6 x-x^2}} \, dx\)

Optimal. Leaf size=10 \[ -\sin ^{-1}\left (1-\frac{x}{3}\right ) \]

[Out]

-ArcSin[1 - x/3]

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Rubi [A]  time = 0.0152482, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\sin ^{-1}\left (1-\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[6*x - x^2],x]

[Out]

-ArcSin[1 - x/3]

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Rubi in Sympy [A]  time = 1.34171, size = 5, normalized size = 0.5 \[ \operatorname{asin}{\left (\frac{x}{3} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2+6*x)**(1/2),x)

[Out]

asin(x/3 - 1)

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Mathematica [B]  time = 0.0142168, size = 38, normalized size = 3.8 \[ \frac{2 \sqrt{x-6} \sqrt{x} \log \left (\sqrt{x-6}+\sqrt{x}\right )}{\sqrt{-(x-6) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[6*x - x^2],x]

[Out]

(2*Sqrt[-6 + x]*Sqrt[x]*Log[Sqrt[-6 + x] + Sqrt[x]])/Sqrt[-((-6 + x)*x)]

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Maple [A]  time = 0.005, size = 7, normalized size = 0.7 \[ \arcsin \left ( -1+{\frac{x}{3}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+6*x)^(1/2),x)

[Out]

arcsin(-1+1/3*x)

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Maxima [A]  time = 0.781981, size = 11, normalized size = 1.1 \[ -\arcsin \left (-\frac{1}{3} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + 6*x),x, algorithm="maxima")

[Out]

-arcsin(-1/3*x + 1)

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Fricas [A]  time = 0.219675, size = 24, normalized size = 2.4 \[ -2 \, \arctan \left (\frac{\sqrt{-x^{2} + 6 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + 6*x),x, algorithm="fricas")

[Out]

-2*arctan(sqrt(-x^2 + 6*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x^{2} + 6 x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+6*x)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 + 6*x), x)

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GIAC/XCAS [A]  time = 0.212672, size = 8, normalized size = 0.8 \[ \arcsin \left (\frac{1}{3} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + 6*x),x, algorithm="giac")

[Out]

arcsin(1/3*x - 1)