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

Optimal. Leaf size=40 \[ \frac{5}{6} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x)+\sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - (2*Log[1 - x])/3 + (5*Log[1 + x + x^2])/6

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Rubi [A]  time = 0.0639313, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ \frac{5}{6} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x)+\sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-3 + x^2)/(-1 + x^3),x]

[Out]

Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - (2*Log[1 - x])/3 + (5*Log[1 + x + x^2])/6

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Rubi in Sympy [A]  time = 9.78696, size = 39, normalized size = 0.98 \[ - \frac{2 \log{\left (- x + 1 \right )}}{3} + \frac{5 \log{\left (x^{2} + x + 1 \right )}}{6} + \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0197222, size = 50, normalized size = 1.25 \[ \frac{1}{3} \log \left (1-x^3\right )+\frac{1}{2} \log \left (x^2+x+1\right )-\log (1-x)+\sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 32, normalized size = 0.8 \[{\frac{5\,\ln \left ({x}^{2}+x+1 \right ) }{6}}+\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) \sqrt{3}-{\frac{2\,\ln \left ( -1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-3)/(x^3-1),x)

[Out]

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

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Maxima [A]  time = 1.52211, size = 42, normalized size = 1.05 \[ \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{5}{6} \, \log \left (x^{2} + x + 1\right ) - \frac{2}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22236, size = 42, normalized size = 1.05 \[ \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{5}{6} \, \log \left (x^{2} + x + 1\right ) - \frac{2}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.152633, size = 42, normalized size = 1.05 \[ - \frac{2 \log{\left (x - 1 \right )}}{3} + \frac{5 \log{\left (x^{2} + x + 1 \right )}}{6} + \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-3)/(x**3-1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.239933, size = 43, normalized size = 1.08 \[ \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{5}{6} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{2}{3} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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