3.17 \(\int \frac{3-x}{1-x^3} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x)+\frac{4 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0567307, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x)+\frac{4 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(3 - x)/(1 - x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.73538, size = 41, normalized size = 1. \[ - \frac{2 \log{\left (- x + 1 \right )}}{3} + \frac{\log{\left (x^{2} + x + 1 \right )}}{3} + \frac{4 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0143004, size = 41, normalized size = 1. \[ \frac{1}{3} \log \left (x^2+x+1\right )-\frac{2}{3} \log (1-x)+\frac{4 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 - x)/(1 - x^3),x]

[Out]

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

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Maple [A]  time = 0.007, size = 33, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}}+{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{2\,\ln \left ( -1+x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3-x)/(-x^3+1),x)

[Out]

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

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Maxima [A]  time = 1.52214, size = 43, normalized size = 1.05 \[ \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) - \frac{2}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219667, size = 53, normalized size = 1.29 \[ \frac{1}{9} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) - 2 \, \sqrt{3} \log \left (x - 1\right ) + 12 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(3)*(sqrt(3)*log(x^2 + x + 1) - 2*sqrt(3)*log(x - 1) + 12*arctan(1/3*sqr
t(3)*(2*x + 1)))

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Sympy [A]  time = 0.154906, size = 44, normalized size = 1.07 \[ - \frac{2 \log{\left (x - 1 \right )}}{3} + \frac{\log{\left (x^{2} + x + 1 \right )}}{3} + \frac{4 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3-x)/(-x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209048, size = 45, normalized size = 1.1 \[ \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{3} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{2}{3} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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