3.365 \(\int \frac{1}{x^6 \left (1-x^4+x^8\right )} \, dx\)

Optimal. Leaf size=287 \[ -\frac{1}{5 x^5}-\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}} \]

[Out]

-1/(5*x^5) - x^(-1) + ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(2*Sqr
t[6]) + ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[6]) - ArcTan
[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(2*Sqrt[6]) - ArcTan[(Sqrt[2 + Sqr
t[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[6]) - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]
/(4*Sqrt[6]) + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(4*Sqrt[6]) - Log[1 - Sqrt[2 +
Sqrt[3]]*x + x^2]/(4*Sqrt[6]) + Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2]/(4*Sqrt[6])

_______________________________________________________________________________________

Rubi [A]  time = 0.572112, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562 \[ -\frac{1}{5 x^5}-\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 - x^4 + x^8)),x]

[Out]

-1/(5*x^5) - x^(-1) + ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(2*Sqr
t[6]) + ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[6]) - ArcTan
[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(2*Sqrt[6]) - ArcTan[(Sqrt[2 + Sqr
t[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[6]) - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]
/(4*Sqrt[6]) + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(4*Sqrt[6]) - Log[1 - Sqrt[2 +
Sqrt[3]]*x + x^2]/(4*Sqrt[6]) + Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2]/(4*Sqrt[6])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 86.3535, size = 518, normalized size = 1.8 \[ \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (\frac{1}{2} + \frac{\sqrt{3}}{2}\right ) \log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{1}{2} + \frac{\sqrt{3}}{2}\right ) \log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \sqrt{\sqrt{3} + 2} + \frac{\left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \sqrt{\sqrt{3} + 2} + \frac{\left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2}}{2} + \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2}}{2} + \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} - \frac{1}{x} - \frac{1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(x**8-x**4+1),x)

[Out]

sqrt(3)*(-sqrt(3)/2 + 1/2)*log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(12*sqrt(-sqrt(3
) + 2)) - sqrt(3)*(-sqrt(3)/2 + 1/2)*log(x**2 + x*sqrt(-sqrt(3) + 2) + 1)/(12*sq
rt(-sqrt(3) + 2)) - sqrt(3)*(1/2 + sqrt(3)/2)*log(x**2 - x*sqrt(sqrt(3) + 2) + 1
)/(12*sqrt(sqrt(3) + 2)) + sqrt(3)*(1/2 + sqrt(3)/2)*log(x**2 + x*sqrt(sqrt(3) +
2) + 1)/(12*sqrt(sqrt(3) + 2)) - sqrt(3)*(-sqrt(sqrt(3) + 2) + (1 + sqrt(3))*sq
rt(sqrt(3) + 2)/2)*atan((2*x - sqrt(sqrt(3) + 2))/sqrt(-sqrt(3) + 2))/(6*sqrt(-s
qrt(3) + 2)*sqrt(sqrt(3) + 2)) - sqrt(3)*(-sqrt(sqrt(3) + 2) + (1 + sqrt(3))*sqr
t(sqrt(3) + 2)/2)*atan((2*x + sqrt(sqrt(3) + 2))/sqrt(-sqrt(3) + 2))/(6*sqrt(-sq
rt(3) + 2)*sqrt(sqrt(3) + 2)) - sqrt(3)*(-(-sqrt(3) + 1)*sqrt(-sqrt(3) + 2)/2 +
sqrt(-sqrt(3) + 2))*atan((2*x - sqrt(-sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(6*sqrt(-
sqrt(3) + 2)*sqrt(sqrt(3) + 2)) - sqrt(3)*(-(-sqrt(3) + 1)*sqrt(-sqrt(3) + 2)/2
+ sqrt(-sqrt(3) + 2))*atan((2*x + sqrt(-sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(6*sqrt
(-sqrt(3) + 2)*sqrt(sqrt(3) + 2)) - 1/x - 1/(5*x**5)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0239242, size = 54, normalized size = 0.19 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right ]-\frac{1}{5 x^5}-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 - x^4 + x^8)),x]

[Out]

-1/(5*x^5) - x^(-1) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^3)/(-1 + 2*#1^
4) & ]/4

_______________________________________________________________________________________

Maple [C]  time = 0.014, size = 43, normalized size = 0.2 \[ -{\frac{1}{5\,{x}^{5}}}-{x}^{-1}-{\frac{\sum _{{\it \_R}={\it RootOf} \left ( 9\,{{\it \_Z}}^{4}+1 \right ) }{\it \_R}\,\ln \left ( 9\,{{\it \_R}}^{3}x-3\,{{\it \_R}}^{2}+{x}^{2} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^8-x^4+1),x)

[Out]

-1/5/x^5-1/x-1/4*sum(_R*ln(9*_R^3*x-3*_R^2+x^2),_R=RootOf(9*_Z^4+1))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{5 \, x^{4} + 1}{5 \, x^{5}} - \int \frac{x^{6}}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - x^4 + 1)*x^6),x, algorithm="maxima")

[Out]

-1/5*(5*x^4 + 1)/x^5 - integrate(x^6/(x^8 - x^4 + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.270996, size = 302, normalized size = 1.05 \[ \frac{20 \, \sqrt{3} \sqrt{2} x^{5} \arctan \left (\frac{\sqrt{3} \sqrt{2} + 3 \, x}{\sqrt{3} \sqrt{2} x^{2} + \sqrt{3} \sqrt{2} \sqrt{x^{4} + \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1} + 3 \, x}\right ) - 20 \, \sqrt{3} \sqrt{2} x^{5} \arctan \left (\frac{\sqrt{3} \sqrt{2} - 3 \, x}{\sqrt{3} \sqrt{2} x^{2} + \sqrt{3} \sqrt{2} \sqrt{x^{4} - \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1} - 3 \, x}\right ) + 5 \, \sqrt{3} \sqrt{2} x^{5} \log \left (x^{4} + \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) - 5 \, \sqrt{3} \sqrt{2} x^{5} \log \left (x^{4} - \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) - 120 \, x^{4} - 24}{120 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - x^4 + 1)*x^6),x, algorithm="fricas")

[Out]

1/120*(20*sqrt(3)*sqrt(2)*x^5*arctan((sqrt(3)*sqrt(2) + 3*x)/(sqrt(3)*sqrt(2)*x^
2 + sqrt(3)*sqrt(2)*sqrt(x^4 + sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1) + 3*x)) -
20*sqrt(3)*sqrt(2)*x^5*arctan((sqrt(3)*sqrt(2) - 3*x)/(sqrt(3)*sqrt(2)*x^2 + sqr
t(3)*sqrt(2)*sqrt(x^4 - sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1) - 3*x)) + 5*sqrt(
3)*sqrt(2)*x^5*log(x^4 + sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1) - 5*sqrt(3)*sqrt
(2)*x^5*log(x^4 - sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1) - 120*x^4 - 24)/x^5

_______________________________________________________________________________________

Sympy [A]  time = 0.866249, size = 180, normalized size = 0.63 \[ \frac{\sqrt{6} \left (- 2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{3} - \frac{1}{3} \right )} - 2 \operatorname{atan}{\left (\sqrt{6} x^{3} - 4 x^{2} + 2 \sqrt{6} x - 3 \right )}\right )}{24} + \frac{\sqrt{6} \left (- 2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{3} + \frac{1}{3} \right )} - 2 \operatorname{atan}{\left (\sqrt{6} x^{3} + 4 x^{2} + 2 \sqrt{6} x + 3 \right )}\right )}{24} - \frac{\sqrt{6} \log{\left (x^{4} - \sqrt{6} x^{3} + 3 x^{2} - \sqrt{6} x + 1 \right )}}{24} + \frac{\sqrt{6} \log{\left (x^{4} + \sqrt{6} x^{3} + 3 x^{2} + \sqrt{6} x + 1 \right )}}{24} - \frac{5 x^{4} + 1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(x**8-x**4+1),x)

[Out]

sqrt(6)*(-2*atan(sqrt(6)*x/3 - 1/3) - 2*atan(sqrt(6)*x**3 - 4*x**2 + 2*sqrt(6)*x
- 3))/24 + sqrt(6)*(-2*atan(sqrt(6)*x/3 + 1/3) - 2*atan(sqrt(6)*x**3 + 4*x**2 +
2*sqrt(6)*x + 3))/24 - sqrt(6)*log(x**4 - sqrt(6)*x**3 + 3*x**2 - sqrt(6)*x + 1
)/24 + sqrt(6)*log(x**4 + sqrt(6)*x**3 + 3*x**2 + sqrt(6)*x + 1)/24 - (5*x**4 +
1)/(5*x**5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285729, size = 293, normalized size = 1.02 \[ -\frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{5 \, x^{4} + 1}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - x^4 + 1)*x^6),x, algorithm="giac")

[Out]

-1/12*sqrt(6)*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) - 1/12*sqrt(
6)*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) - 1/12*sqrt(6)*arctan((
4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) - 1/12*sqrt(6)*arctan((4*x - sqrt(
6) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/24*sqrt(6)*ln(x^2 + 1/2*x*(sqrt(6) + sqrt
(2)) + 1) - 1/24*sqrt(6)*ln(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/24*sqrt(6)*
ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/24*sqrt(6)*ln(x^2 - 1/2*x*(sqrt(6) -
sqrt(2)) + 1) - 1/5*(5*x^4 + 1)/x^5