3.2359 \(\int \frac{\sqrt{-1-x+x^2}}{1-x} \, dx\)

Optimal. Leaf size=65 \[ -\sqrt{x^2-x-1}-\tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right ) \]

[Out]

-Sqrt[-1 - x + x^2] - ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 - 2*x)
/(2*Sqrt[-1 - x + x^2])]/2

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Rubi [A]  time = 0.123848, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\sqrt{x^2-x-1}-\tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )+\frac{1}{2} \tanh ^{-1}\left (\frac{1-2 x}{2 \sqrt{x^2-x-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[-1 - x + x^2] - ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])] + ArcTanh[(1 - 2*x)
/(2*Sqrt[-1 - x + x^2])]/2

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Rubi in Sympy [A]  time = 30.5283, size = 48, normalized size = 0.74 \[ - \sqrt{x^{2} - x - 1} - \operatorname{atan}{\left (\frac{- x + 3}{2 \sqrt{x^{2} - x - 1}} \right )} - \frac{\operatorname{atanh}{\left (\frac{2 x - 1}{2 \sqrt{x^{2} - x - 1}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x**2 - x - 1) - atan((-x + 3)/(2*sqrt(x**2 - x - 1))) - atanh((2*x - 1)/(2
*sqrt(x**2 - x - 1)))/2

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Mathematica [A]  time = 0.0404164, size = 63, normalized size = 0.97 \[ \frac{1}{2} \left (-2 \sqrt{x^2-x-1}+\log \left (-2 \sqrt{x^2-x-1}+2 x-1\right )-2 \tan ^{-1}\left (\frac{3-x}{2 \sqrt{x^2-x-1}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-1 - x + x^2] - 2*ArcTan[(3 - x)/(2*Sqrt[-1 - x + x^2])] + Log[-1 + 2*x
- 2*Sqrt[-1 - x + x^2]])/2

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Maple [A]  time = 0.012, size = 46, normalized size = 0.7 \[ -\sqrt{ \left ( -1+x \right ) ^{2}+x-2}-{\frac{1}{2}\ln \left ( x-{\frac{1}{2}}+\sqrt{ \left ( -1+x \right ) ^{2}+x-2} \right ) }+\arctan \left ({\frac{x-3}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+x-2}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-x-1)^(1/2)/(1-x),x)

[Out]

-((-1+x)^2+x-2)^(1/2)-1/2*ln(x-1/2+((-1+x)^2+x-2)^(1/2))+arctan(1/2*(x-3)/((-1+x
)^2+x-2)^(1/2))

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Maxima [A]  time = 0.751294, size = 78, normalized size = 1.2 \[ -\sqrt{x^{2} - x - 1} + \arcsin \left (\frac{\sqrt{5} x}{5 \,{\left | x - 1 \right |}} - \frac{3 \, \sqrt{5}}{5 \,{\left | x - 1 \right |}}\right ) - \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - x - 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x^2 - x - 1) + arcsin(1/5*sqrt(5)*x/abs(x - 1) - 3/5*sqrt(5)/abs(x - 1)) -
1/2*log(2*x + 2*sqrt(x^2 - x - 1) - 1)

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Fricas [A]  time = 0.236291, size = 162, normalized size = 2.49 \[ \frac{8 \, x^{2} + 8 \,{\left (2 \, x - 2 \, \sqrt{x^{2} - x - 1} - 1\right )} \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) + 2 \,{\left (2 \, x - 2 \, \sqrt{x^{2} - x - 1} - 1\right )} \log \left (-2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1\right ) - 2 \, \sqrt{x^{2} - x - 1}{\left (4 \, x - 1\right )} - 6 \, x - 9}{4 \,{\left (2 \, x - 2 \, \sqrt{x^{2} - x - 1} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(8*x^2 + 8*(2*x - 2*sqrt(x^2 - x - 1) - 1)*arctan(-x + sqrt(x^2 - x - 1) + 1
) + 2*(2*x - 2*sqrt(x^2 - x - 1) - 1)*log(-2*x + 2*sqrt(x^2 - x - 1) + 1) - 2*sq
rt(x^2 - x - 1)*(4*x - 1) - 6*x - 9)/(2*x - 2*sqrt(x^2 - x - 1) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208015, size = 70, normalized size = 1.08 \[ -\sqrt{x^{2} - x - 1} + 2 \, \arctan \left (-x + \sqrt{x^{2} - x - 1} + 1\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} - x - 1} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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