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

Optimal. Leaf size=19 \[ \frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

(2*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3]

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Rubi [A]  time = 0.0290193, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)/(1 - x^3),x]

[Out]

(2*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3]

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Rubi in Sympy [A]  time = 4.68088, size = 22, normalized size = 1.16 \[ \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/3

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Mathematica [A]  time = 0.00801525, size = 19, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)/(1 - x^3),x]

[Out]

(2*ArcTan[(1 + 2*x)/Sqrt[3]])/Sqrt[3]

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Maple [A]  time = 0.003, size = 17, normalized size = 0.9 \[{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/(-x^3+1),x)

[Out]

2/3*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 1.60304, size = 22, normalized size = 1.16 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1))

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Fricas [A]  time = 0.211303, size = 22, normalized size = 1.16 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1))

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Sympy [A]  time = 0.101319, size = 26, normalized size = 1.37 \[ \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)/(-x**3+1),x)

[Out]

2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3

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GIAC/XCAS [A]  time = 0.206852, size = 22, normalized size = 1.16 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1))