3.14 \(\int \sqrt{-8 x+x^2} \, dx\)

Optimal. Leaf size=37 \[ -\frac{1}{2} \sqrt{x^2-8 x} (4-x)-16 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-8 x}}\right ) \]

[Out]

-((4 - x)*Sqrt[-8*x + x^2])/2 - 16*ArcTanh[x/Sqrt[-8*x + x^2]]

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Rubi [A]  time = 0.0197001, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{1}{2} \sqrt{x^2-8 x} (4-x)-16 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-8 x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-8*x + x^2],x]

[Out]

-((4 - x)*Sqrt[-8*x + x^2])/2 - 16*ArcTanh[x/Sqrt[-8*x + x^2]]

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Rubi in Sympy [A]  time = 1.48839, size = 32, normalized size = 0.86 \[ - \frac{\left (- 2 x + 8\right ) \sqrt{x^{2} - 8 x}}{4} - 16 \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} - 8 x}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-8*x)**(1/2),x)

[Out]

-(-2*x + 8)*sqrt(x**2 - 8*x)/4 - 16*atanh(x/sqrt(x**2 - 8*x))

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Mathematica [A]  time = 0.0440485, size = 44, normalized size = 1.19 \[ \frac{1}{2} \sqrt{(x-8) x} \left (x-\frac{32 \log \left (\sqrt{x-8}+\sqrt{x}\right )}{\sqrt{x-8} \sqrt{x}}-4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-8*x + x^2],x]

[Out]

(Sqrt[(-8 + x)*x]*(-4 + x - (32*Log[Sqrt[-8 + x] + Sqrt[x]])/(Sqrt[-8 + x]*Sqrt[
x])))/2

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Maple [A]  time = 0.005, size = 33, normalized size = 0.9 \[{\frac{2\,x-8}{4}\sqrt{{x}^{2}-8\,x}}-8\,\ln \left ( x-4+\sqrt{{x}^{2}-8\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-8*x)^(1/2),x)

[Out]

1/4*(2*x-8)*(x^2-8*x)^(1/2)-8*ln(x-4+(x^2-8*x)^(1/2))

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Maxima [A]  time = 0.726829, size = 58, normalized size = 1.57 \[ \frac{1}{2} \, \sqrt{x^{2} - 8 \, x} x - 2 \, \sqrt{x^{2} - 8 \, x} - 8 \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 8 \, x} - 8\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 8*x),x, algorithm="maxima")

[Out]

1/2*sqrt(x^2 - 8*x)*x - 2*sqrt(x^2 - 8*x) - 8*log(2*x + 2*sqrt(x^2 - 8*x) - 8)

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Fricas [A]  time = 0.214963, size = 144, normalized size = 3.89 \[ -\frac{x^{4} - 16 \, x^{3} + 76 \, x^{2} - 16 \,{\left (x^{2} - \sqrt{x^{2} - 8 \, x}{\left (x - 4\right )} - 8 \, x + 8\right )} \log \left (-x + \sqrt{x^{2} - 8 \, x} + 4\right ) -{\left (x^{3} - 12 \, x^{2} + 36 \, x - 16\right )} \sqrt{x^{2} - 8 \, x} - 96 \, x - 32}{2 \,{\left (x^{2} - \sqrt{x^{2} - 8 \, x}{\left (x - 4\right )} - 8 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 8*x),x, algorithm="fricas")

[Out]

-1/2*(x^4 - 16*x^3 + 76*x^2 - 16*(x^2 - sqrt(x^2 - 8*x)*(x - 4) - 8*x + 8)*log(-
x + sqrt(x^2 - 8*x) + 4) - (x^3 - 12*x^2 + 36*x - 16)*sqrt(x^2 - 8*x) - 96*x - 3
2)/(x^2 - sqrt(x^2 - 8*x)*(x - 4) - 8*x + 8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{2} - 8 x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-8*x)**(1/2),x)

[Out]

Integral(sqrt(x**2 - 8*x), x)

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GIAC/XCAS [A]  time = 0.210566, size = 45, normalized size = 1.22 \[ \frac{1}{2} \, \sqrt{x^{2} - 8 \, x}{\left (x - 4\right )} + 8 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 8 \, x} + 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 - 8*x),x, algorithm="giac")

[Out]

1/2*sqrt(x^2 - 8*x)*(x - 4) + 8*ln(abs(-x + sqrt(x^2 - 8*x) + 4))