3.2392 \(\int \frac{-1+x}{\sqrt{3-4 x+x^2}} \, dx\)

Optimal. Leaf size=34 \[ \sqrt{x^2-4 x+3}-\tanh ^{-1}\left (\frac{2-x}{\sqrt{x^2-4 x+3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0308259, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \sqrt{x^2-4 x+3}-\tanh ^{-1}\left (\frac{2-x}{\sqrt{x^2-4 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x)/Sqrt[3 - 4*x + x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.09982, size = 31, normalized size = 0.91 \[ \sqrt{x^{2} - 4 x + 3} + \operatorname{atanh}{\left (\frac{2 x - 4}{2 \sqrt{x^{2} - 4 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+x)/(x**2-4*x+3)**(1/2),x)

[Out]

sqrt(x**2 - 4*x + 3) + atanh((2*x - 4)/(2*sqrt(x**2 - 4*x + 3)))

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Mathematica [A]  time = 0.0238631, size = 33, normalized size = 0.97 \[ \sqrt{x^2-4 x+3}+\log \left (-\sqrt{x^2-4 x+3}-x+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x)/Sqrt[3 - 4*x + x^2],x]

[Out]

Sqrt[3 - 4*x + x^2] + Log[2 - x - Sqrt[3 - 4*x + x^2]]

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Maple [A]  time = 0.007, size = 26, normalized size = 0.8 \[ \sqrt{{x}^{2}-4\,x+3}+\ln \left ( x-2+\sqrt{{x}^{2}-4\,x+3} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x)/(x^2-4*x+3)^(1/2),x)

[Out]

(x^2-4*x+3)^(1/2)+ln(x-2+(x^2-4*x+3)^(1/2))

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Maxima [A]  time = 0.673139, size = 39, normalized size = 1.15 \[ \sqrt{x^{2} - 4 \, x + 3} + \log \left (2 \, x + 2 \, \sqrt{x^{2} - 4 \, x + 3} - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)/sqrt(x^2 - 4*x + 3),x, algorithm="maxima")

[Out]

sqrt(x^2 - 4*x + 3) + log(2*x + 2*sqrt(x^2 - 4*x + 3) - 4)

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Fricas [A]  time = 0.215033, size = 100, normalized size = 2.94 \[ -\frac{x^{2} +{\left (x - \sqrt{x^{2} - 4 \, x + 3} - 2\right )} \log \left (-x + \sqrt{x^{2} - 4 \, x + 3} + 2\right ) - \sqrt{x^{2} - 4 \, x + 3}{\left (x - 1\right )} - 3 \, x + 1}{x - \sqrt{x^{2} - 4 \, x + 3} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)/sqrt(x^2 - 4*x + 3),x, algorithm="fricas")

[Out]

-(x^2 + (x - sqrt(x^2 - 4*x + 3) - 2)*log(-x + sqrt(x^2 - 4*x + 3) + 2) - sqrt(x
^2 - 4*x + 3)*(x - 1) - 3*x + 1)/(x - sqrt(x^2 - 4*x + 3) - 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x - 1}{\sqrt{\left (x - 3\right ) \left (x - 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+x)/(x**2-4*x+3)**(1/2),x)

[Out]

Integral((x - 1)/sqrt((x - 3)*(x - 1)), x)

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GIAC/XCAS [A]  time = 0.211674, size = 41, normalized size = 1.21 \[ \sqrt{x^{2} - 4 \, x + 3} -{\rm ln}\left ({\left | -x + \sqrt{x^{2} - 4 \, x + 3} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)/sqrt(x^2 - 4*x + 3),x, algorithm="giac")

[Out]

sqrt(x^2 - 4*x + 3) - ln(abs(-x + sqrt(x^2 - 4*x + 3) + 2))