3.50 \(\int \frac{1}{-1+x^6} \, dx\)

Optimal. Leaf size=47 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} x}{1-x^2}\right )}{2 \sqrt{3}}-\frac{1}{6} \tanh ^{-1}\left (\frac{x}{x^2+1}\right )-\frac{1}{3} \tanh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.188862, antiderivative size = 73, normalized size of antiderivative = 1.55, number of steps used = 10, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857 \[ \frac{1}{12} \log \left (x^2-x+1\right )-\frac{1}{12} \log \left (x^2+x+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{3} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^6)^(-1),x]

[Out]

ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x)/Sqrt[3]]/(2*Sqrt[3]) -
ArcTanh[x]/3 + Log[1 - x + x^2]/12 - Log[1 + x + x^2]/12

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Rubi in Sympy [A]  time = 16.9956, size = 68, normalized size = 1.45 \[ \frac{\log{\left (x^{2} - x + 1 \right )}}{12} - \frac{\log{\left (x^{2} + x + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} - \frac{\operatorname{atanh}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**6-1),x)

[Out]

log(x**2 - x + 1)/12 - log(x**2 + x + 1)/12 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3)
)/6 - sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/6 - atanh(x)/3

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Mathematica [A]  time = 0.0207816, size = 75, normalized size = 1.6 \[ \frac{1}{12} \left (\log \left (x^2-x+1\right )-\log \left (x^2+x+1\right )+2 \log (1-x)-2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^6)^(-1),x]

[Out]

(-2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 2
*Log[1 - x] - 2*Log[1 + x] + Log[1 - x + x^2] - Log[1 + x + x^2])/12

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Maple [A]  time = 0., size = 66, normalized size = 1.4 \[{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^6-1),x)

[Out]

1/6*ln(-1+x)-1/12*ln(x^2+x+1)-1/6*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/6*ln(1+x
)+1/12*ln(x^2-x+1)-1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

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Maxima [A]  time = 1.527, size = 88, normalized size = 1.87 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{12} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{6} \, \log \left (x + 1\right ) + \frac{1}{6} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - 1),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x
- 1)) - 1/12*log(x^2 + x + 1) + 1/12*log(x^2 - x + 1) - 1/6*log(x + 1) + 1/6*lo
g(x - 1)

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Fricas [A]  time = 0.203179, size = 101, normalized size = 2.15 \[ -\frac{1}{36} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) - \sqrt{3} \log \left (x^{2} - x + 1\right ) + 2 \, \sqrt{3} \log \left (x + 1\right ) - 2 \, \sqrt{3} \log \left (x - 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - 1),x, algorithm="fricas")

[Out]

-1/36*sqrt(3)*(sqrt(3)*log(x^2 + x + 1) - sqrt(3)*log(x^2 - x + 1) + 2*sqrt(3)*l
og(x + 1) - 2*sqrt(3)*log(x - 1) + 6*arctan(1/3*sqrt(3)*(2*x + 1)) + 6*arctan(1/
3*sqrt(3)*(2*x - 1)))

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Sympy [A]  time = 0.358155, size = 83, normalized size = 1.77 \[ \frac{\log{\left (x - 1 \right )}}{6} - \frac{\log{\left (x + 1 \right )}}{6} + \frac{\log{\left (x^{2} - x + 1 \right )}}{12} - \frac{\log{\left (x^{2} + x + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**6-1),x)

[Out]

log(x - 1)/6 - log(x + 1)/6 + log(x**2 - x + 1)/12 - log(x**2 + x + 1)/12 - sqrt
(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/6 - sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/
6

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GIAC/XCAS [A]  time = 0.225219, size = 90, normalized size = 1.91 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{12} \,{\rm ln}\left (x^{2} + x + 1\right ) + \frac{1}{12} \,{\rm ln}\left (x^{2} - x + 1\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - 1),x, algorithm="giac")

[Out]

-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x
- 1)) - 1/12*ln(x^2 + x + 1) + 1/12*ln(x^2 - x + 1) - 1/6*ln(abs(x + 1)) + 1/6*
ln(abs(x - 1))