3.1358 \(\int \frac{1}{x^8 \left (1-x^6\right )} \, dx\)

Optimal. Leaf size=85 \[ -\frac{1}{7 x^7}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]

[Out]

-1/(7*x^7) - x^(-1) + ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x)/S
qrt[3]]/(2*Sqrt[3]) + ArcTanh[x]/3 - Log[1 - x + x^2]/12 + Log[1 + x + x^2]/12

_______________________________________________________________________________________

Rubi [A]  time = 0.295853, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{1}{7 x^7}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(1 - x^6)),x]

[Out]

-1/(7*x^7) - x^(-1) + ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x)/S
qrt[3]]/(2*Sqrt[3]) + ArcTanh[x]/3 - Log[1 - x + x^2]/12 + Log[1 + x + x^2]/12

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.9757, size = 78, normalized size = 0.92 \[ - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{\operatorname{atanh}{\left (x \right )}}{3} - \frac{1}{x} - \frac{1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(-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 - 1/(7*x**7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0551721, size = 87, normalized size = 1.02 \[ \frac{1}{84} \left (-\frac{12}{x^7}-7 \log \left (x^2-x+1\right )+7 \log \left (x^2+x+1\right )-\frac{84}{x}-14 \log (1-x)+14 \log (x+1)-14 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-14 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(1 - x^6)),x]

[Out]

(-12/x^7 - 84/x - 14*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 14*Sqrt[3]*ArcTan[(1 +
2*x)/Sqrt[3]] - 14*Log[1 - x] + 14*Log[1 + x] - 7*Log[1 - x + x^2] + 7*Log[1 +
x + x^2])/84

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 76, normalized size = 0.9 \[ -{\frac{1}{7\,{x}^{7}}}-{x}^{-1}+{\frac{\ln \left ({x}^{2}+x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( -1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(-x^6+1),x)

[Out]

-1/7/x^7-1/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))+1/6*ln(1+x)

_______________________________________________________________________________________

Maxima [A]  time = 1.59913, size = 104, normalized size = 1.22 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{7 \, x^{6} + 1}{7 \, x^{7}} + \frac{1}{12} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^8),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)) - 1/7*(7*x^6 + 1)/x^7 + 1/12*log(x^2 + x + 1) - 1/12*log(x^2 - x + 1) + 1
/6*log(x + 1) - 1/6*log(x - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.226576, size = 147, normalized size = 1.73 \[ \frac{\sqrt{3}{\left (7 \, \sqrt{3} x^{7} \log \left (x^{2} + x + 1\right ) - 7 \, \sqrt{3} x^{7} \log \left (x^{2} - x + 1\right ) + 14 \, \sqrt{3} x^{7} \log \left (x + 1\right ) - 14 \, \sqrt{3} x^{7} \log \left (x - 1\right ) - 42 \, x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 42 \, x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \, \sqrt{3}{\left (7 \, x^{6} + 1\right )}\right )}}{252 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^8),x, algorithm="fricas")

[Out]

1/252*sqrt(3)*(7*sqrt(3)*x^7*log(x^2 + x + 1) - 7*sqrt(3)*x^7*log(x^2 - x + 1) +
14*sqrt(3)*x^7*log(x + 1) - 14*sqrt(3)*x^7*log(x - 1) - 42*x^7*arctan(1/3*sqrt(
3)*(2*x + 1)) - 42*x^7*arctan(1/3*sqrt(3)*(2*x - 1)) - 12*sqrt(3)*(7*x^6 + 1))/x
^7

_______________________________________________________________________________________

Sympy [A]  time = 0.987873, size = 95, normalized size = 1.12 \[ - \frac{\log{\left (x - 1 \right )}}{6} + \frac{\log{\left (x + 1 \right )}}{6} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\log{\left (x^{2} + x + 1 \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{6} - \frac{7 x^{6} + 1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(-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 - (7*x**6 + 1)/(7*x**7)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222968, size = 107, normalized size = 1.26 \[ -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{7 \, x^{6} + 1}{7 \, x^{7}} + \frac{1}{12} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{1}{12} \,{\rm ln}\left (x^{2} - x + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((x^6 - 1)*x^8),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)) - 1/7*(7*x^6 + 1)/x^7 + 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))