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

Optimal. Leaf size=41 \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0395499, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857 \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^3)^(-1),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.50005, size = 37, normalized size = 0.9 \[ \frac{\log{\left (- x + 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-1),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00903536, size = 41, normalized size = 1. \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^3)^(-1),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 33, normalized size = 0.8 \[{\frac{\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.51389, size = 43, normalized size = 1.05 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{3} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.203758, size = 53, normalized size = 1.29 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) - 2 \, \sqrt{3} \log \left (x - 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*log(x^2 + x + 1) - 2*sqrt(3)*log(x - 1) + 6*arctan(1/3*sq
rt(3)*(2*x + 1)))

_______________________________________________________________________________________

Sympy [A]  time = 0.171504, size = 41, normalized size = 1. \[ \frac{\log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-1),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205249, size = 45, normalized size = 1.1 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \,{\rm ln}\left (x^{2} + x + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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