3.2393 \(\int \frac{1}{(1-x) \sqrt{-4+2 x+x^2}} \, dx\)

Optimal. Leaf size=19 \[ \tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right ) \]

[Out]

ArcTan[(3 - 2*x)/Sqrt[-4 + 2*x + x^2]]

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Rubi [A]  time = 0.0351389, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)*Sqrt[-4 + 2*x + x^2]),x]

[Out]

ArcTan[(3 - 2*x)/Sqrt[-4 + 2*x + x^2]]

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Rubi in Sympy [A]  time = 6.13359, size = 19, normalized size = 1. \[ \operatorname{atan}{\left (\frac{- 4 x + 6}{2 \sqrt{x^{2} + 2 x - 4}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)/(x**2+2*x-4)**(1/2),x)

[Out]

atan((-4*x + 6)/(2*sqrt(x**2 + 2*x - 4)))

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Mathematica [A]  time = 0.026845, size = 31, normalized size = 1.63 \[ \tan ^{-1}\left (\frac{2 (1-x)+1}{\sqrt{(1-x)^2-4 (1-x)-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)*Sqrt[-4 + 2*x + x^2]),x]

[Out]

ArcTan[(1 + 2*(1 - x))/Sqrt[-1 - 4*(1 - x) + (1 - x)^2]]

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Maple [A]  time = 0.01, size = 23, normalized size = 1.2 \[ -\arctan \left ({\frac{-6+4\,x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+4\,x-5}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)/(x^2+2*x-4)^(1/2),x)

[Out]

-arctan(1/2*(-6+4*x)/((-1+x)^2+4*x-5)^(1/2))

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Maxima [A]  time = 0.744551, size = 36, normalized size = 1.89 \[ -\arcsin \left (\frac{2 \, \sqrt{5} x}{5 \,{\left | x - 1 \right |}} - \frac{3 \, \sqrt{5}}{5 \,{\left | x - 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(x^2 + 2*x - 4)*(x - 1)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21944, size = 24, normalized size = 1.26 \[ -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(x^2 + 2*x - 4)*(x - 1)),x, algorithm="fricas")

[Out]

-2*arctan(-x + sqrt(x^2 + 2*x - 4) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{x \sqrt{x^{2} + 2 x - 4} - \sqrt{x^{2} + 2 x - 4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-x)/(x**2+2*x-4)**(1/2),x)

[Out]

-Integral(1/(x*sqrt(x**2 + 2*x - 4) - sqrt(x**2 + 2*x - 4)), x)

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GIAC/XCAS [A]  time = 0.207849, size = 24, normalized size = 1.26 \[ -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(sqrt(x^2 + 2*x - 4)*(x - 1)),x, algorithm="giac")

[Out]

-2*arctan(-x + sqrt(x^2 + 2*x - 4) + 1)