3.1356 \(\int \frac{1}{x^6 \left (1-x^6\right )} \, dx\)

Optimal. Leaf size=80 \[ -\frac{1}{5 x^5}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]

[Out]

-1/(5*x^5) - 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 [A]  time = 0.205085, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{1}{5 x^5}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 - x^6)),x]

[Out]

-1/(5*x^5) - 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 = 36.4134, size = 75, normalized size = 0.94 \[ - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{\operatorname{atanh}{\left (x \right )}}{3} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(-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 - 1/(5*x**5)

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Mathematica [A]  time = 0.0473911, size = 82, normalized size = 1.02 \[ \frac{1}{60} \left (-\frac{12}{x^5}-5 \log \left (x^2-x+1\right )+5 \log \left (x^2+x+1\right )-10 \log (1-x)+10 \log (x+1)+10 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+10 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 - x^6)),x]

[Out]

(-12/x^5 + 10*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 10*Sqrt[3]*ArcTan[(1 + 2*x)/S
qrt[3]] - 10*Log[1 - x] + 10*Log[1 + x] - 5*Log[1 - x + x^2] + 5*Log[1 + x + x^2
])/60

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Maple [A]  time = 0.013, size = 71, normalized size = 0.9 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{1}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-x^6+1),x)

[Out]

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

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Maxima [A]  time = 1.57971, size = 95, normalized size = 1.19 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{5 \, x^{5}} + \frac{1}{12} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^6),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)) - 1/5/x^5 + 1/12*log(x^2 + x + 1) - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)
- 1/6*log(x - 1)

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Fricas [A]  time = 0.227786, size = 138, normalized size = 1.72 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3} x^{5} \log \left (x^{2} + x + 1\right ) - 5 \, \sqrt{3} x^{5} \log \left (x^{2} - x + 1\right ) + 10 \, \sqrt{3} x^{5} \log \left (x + 1\right ) - 10 \, \sqrt{3} x^{5} \log \left (x - 1\right ) + 30 \, x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 30 \, x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \, \sqrt{3}\right )}}{180 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/180*sqrt(3)*(5*sqrt(3)*x^5*log(x^2 + x + 1) - 5*sqrt(3)*x^5*log(x^2 - x + 1) +
10*sqrt(3)*x^5*log(x + 1) - 10*sqrt(3)*x^5*log(x - 1) + 30*x^5*arctan(1/3*sqrt(
3)*(2*x + 1)) + 30*x^5*arctan(1/3*sqrt(3)*(2*x - 1)) - 12*sqrt(3))/x^5

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Sympy [A]  time = 0.921895, size = 90, normalized size = 1.12 \[ - \frac{\log{\left (x - 1 \right )}}{6} + \frac{\log{\left (x + 1 \right )}}{6} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{6} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(-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 + sqr
t(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/6 + sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
/6 - 1/(5*x**5)

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GIAC/XCAS [A]  time = 0.222093, size = 97, normalized size = 1.21 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{5 \, x^{5}} + \frac{1}{12} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{12} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^6),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)) - 1/5/x^5 + 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))