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

Optimal. Leaf size=32 \[ \frac{3}{2} \log \left (x^2-2 x+4\right )+\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

ArcTan[(1 - x)/Sqrt[3]]/Sqrt[3] + (3*Log[4 - 2*x + x^2])/2

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Rubi [A]  time = 0.0406884, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3}{2} \log \left (x^2-2 x+4\right )+\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(1 - x)/Sqrt[3]]/Sqrt[3] + (3*Log[4 - 2*x + x^2])/2

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Rubi in Sympy [A]  time = 5.83509, size = 32, normalized size = 1. \[ \frac{3 \log{\left (x^{2} - 2 x + 4 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0154952, size = 31, normalized size = 0.97 \[ \frac{3}{2} \log \left (x^2-2 x+4\right )-\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[(-1 + x)/Sqrt[3]]/Sqrt[3]) + (3*Log[4 - 2*x + x^2])/2

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Maple [A]  time = 0.007, size = 29, normalized size = 0.9 \[{\frac{3\,\ln \left ({x}^{2}-2\,x+4 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{3}}{6}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4+3*x)/(x^2-2*x+4),x)

[Out]

3/2*ln(x^2-2*x+4)-1/3*3^(1/2)*arctan(1/6*(2*x-2)*3^(1/2))

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Maxima [A]  time = 0.874097, size = 35, normalized size = 1.09 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.274774, size = 42, normalized size = 1.31 \[ \frac{1}{6} \, \sqrt{3}{\left (3 \, \sqrt{3} \log \left (x^{2} - 2 \, x + 4\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*(3*sqrt(3)*log(x^2 - 2*x + 4) - 2*arctan(1/3*sqrt(3)*(x - 1)))

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Sympy [A]  time = 0.220161, size = 36, normalized size = 1.12 \[ \frac{3 \log{\left (x^{2} - 2 x + 4 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.26077, size = 35, normalized size = 1.09 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \,{\rm ln}\left (x^{2} - 2 \, x + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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