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

Optimal. Leaf size=37 \[ \frac{1}{6} \log \left (3 x^2-4 x+3\right )+\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{5}}\right )}{3 \sqrt{5}} \]

[Out]

ArcTan[(2 - 3*x)/Sqrt[5]]/(3*Sqrt[5]) + Log[3 - 4*x + 3*x^2]/6

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Rubi [A]  time = 0.0497561, antiderivative size = 37, 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 \[ \frac{1}{6} \log \left (3 x^2-4 x+3\right )+\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{5}}\right )}{3 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[(2 - 3*x)/Sqrt[5]]/(3*Sqrt[5]) + Log[3 - 4*x + 3*x^2]/6

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Rubi in Sympy [A]  time = 6.37102, size = 34, normalized size = 0.92 \[ \frac{\log{\left (3 x^{2} - 4 x + 3 \right )}}{6} - \frac{\sqrt{5} \operatorname{atan}{\left (\sqrt{5} \left (\frac{3 x}{5} - \frac{2}{5}\right ) \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(3*x**2 - 4*x + 3)/6 - sqrt(5)*atan(sqrt(5)*(3*x/5 - 2/5))/15

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Mathematica [A]  time = 0.0194873, size = 37, normalized size = 1. \[ \frac{1}{6} \log \left (3 x^2-4 x+3\right )-\frac{\tan ^{-1}\left (\frac{3 x-2}{\sqrt{5}}\right )}{3 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(-2 + 3*x)/Sqrt[5]]/(3*Sqrt[5]) + Log[3 - 4*x + 3*x^2]/6

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Maple [A]  time = 0.006, size = 31, normalized size = 0.8 \[{\frac{\ln \left ( 3\,{x}^{2}-4\,x+3 \right ) }{6}}-{\frac{\sqrt{5}}{15}\arctan \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{5}}{10}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x)/(3*x^2-4*x+3),x)

[Out]

1/6*ln(3*x^2-4*x+3)-1/15*5^(1/2)*arctan(1/10*(6*x-4)*5^(1/2))

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Maxima [A]  time = 0.888227, size = 41, normalized size = 1.11 \[ -\frac{1}{15} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (3 \, x - 2\right )}\right ) + \frac{1}{6} \, \log \left (3 \, x^{2} - 4 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(5)*arctan(1/5*sqrt(5)*(3*x - 2)) + 1/6*log(3*x^2 - 4*x + 3)

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Fricas [A]  time = 0.253113, size = 46, normalized size = 1.24 \[ \frac{1}{30} \, \sqrt{5}{\left (\sqrt{5} \log \left (3 \, x^{2} - 4 \, x + 3\right ) - 2 \, \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (3 \, x - 2\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*sqrt(5)*(sqrt(5)*log(3*x^2 - 4*x + 3) - 2*arctan(1/5*sqrt(5)*(3*x - 2)))

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Sympy [A]  time = 0.232691, size = 39, normalized size = 1.05 \[ \frac{\log{\left (x^{2} - \frac{4 x}{3} + 1 \right )}}{6} - \frac{\sqrt{5} \operatorname{atan}{\left (\frac{3 \sqrt{5} x}{5} - \frac{2 \sqrt{5}}{5} \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 - 4*x/3 + 1)/6 - sqrt(5)*atan(3*sqrt(5)*x/5 - 2*sqrt(5)/5)/15

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GIAC/XCAS [A]  time = 0.259884, size = 41, normalized size = 1.11 \[ -\frac{1}{15} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (3 \, x - 2\right )}\right ) + \frac{1}{6} \,{\rm ln}\left (3 \, x^{2} - 4 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(5)*arctan(1/5*sqrt(5)*(3*x - 2)) + 1/6*ln(3*x^2 - 4*x + 3)