3.1352 \(\int \frac{1}{x^2 \left (1-x^6\right )} \, dx\)

Optimal. Leaf size=52 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} x}{1-x^2}\right )}{2 \sqrt{3}}+\frac{1}{6} \tanh ^{-1}\left (\frac{x}{x^2+1}\right )-\frac{1}{x}+\frac{1}{3} \tanh ^{-1}(x) \]

[Out]

-x^(-1) - 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 = 0.296585, antiderivative size = 78, normalized size of antiderivative = 1.5, number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(1 - x^6)),x]

[Out]

-x^(-1) + ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x)/Sqrt[3]]/(2*S
qrt[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 = 49.879, size = 71, normalized size = 1.37 \[ - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{\operatorname{atanh}{\left (x \right )}}{3} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(-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/x

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Mathematica [A]  time = 0.0413572, size = 86, normalized size = 1.65 \[ -\frac{x \log \left (x^2-x+1\right )-x \log \left (x^2+x+1\right )+2 x \log (1-x)-2 x \log (x+1)+2 \sqrt{3} x \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \sqrt{3} x \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )+12}{12 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(1 - x^6)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 71, normalized size = 1.4 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{x}^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(-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/x

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Maxima [A]  time = 1.58067, size = 95, normalized size = 1.83 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{x} + \frac{1}{12} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^2),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/x + 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 = 0.227857, size = 120, normalized size = 2.31 \[ \frac{\sqrt{3}{\left (\sqrt{3} x \log \left (x^{2} + x + 1\right ) - \sqrt{3} x \log \left (x^{2} - x + 1\right ) + 2 \, \sqrt{3} x \log \left (x + 1\right ) - 2 \, \sqrt{3} x \log \left (x - 1\right ) - 6 \, x \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 6 \, x \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \, \sqrt{3}\right )}}{36 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^2),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(sqrt(3)*x*log(x^2 + x + 1) - sqrt(3)*x*log(x^2 - x + 1) + 2*sqrt(3
)*x*log(x + 1) - 2*sqrt(3)*x*log(x - 1) - 6*x*arctan(1/3*sqrt(3)*(2*x + 1)) - 6*
x*arctan(1/3*sqrt(3)*(2*x - 1)) - 12*sqrt(3))/x

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Sympy [A]  time = 0.825597, size = 87, normalized size = 1.67 \[ - \frac{\log{\left (x - 1 \right )}}{6} + \frac{\log{\left (x + 1 \right )}}{6} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{6} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(-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/x

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GIAC/XCAS [A]  time = 0.224792, size = 97, normalized size = 1.87 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{x} + \frac{1}{12} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{12} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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