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

Optimal. Leaf size=98 \[ -\frac{1}{5 x^5}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{x}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

-1/(5*x^5) + x^(-1) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(1 + 2*x)/S
qrt[3]]/(2*Sqrt[3]) + Log[1 - Sqrt[3]*x + x^2]/(4*Sqrt[3]) - Log[1 + Sqrt[3]*x +
x^2]/(4*Sqrt[3])

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Rubi [A]  time = 0.166289, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643 \[ -\frac{1}{5 x^5}+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{1}{x}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 + x^4 + x^8)),x]

[Out]

-1/(5*x^5) + x^(-1) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(1 + 2*x)/S
qrt[3]]/(2*Sqrt[3]) + Log[1 - Sqrt[3]*x + x^2]/(4*Sqrt[3]) - Log[1 + Sqrt[3]*x +
x^2]/(4*Sqrt[3])

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Rubi in Sympy [A]  time = 34.8661, size = 94, normalized size = 0.96 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{1}{x} - \frac{1}{5 x^{5}} \]

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

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Mathematica [A]  time = 0.0637317, size = 95, normalized size = 0.97 \[ \frac{1}{60} \left (-\frac{12}{x^5}+5 \sqrt{3} \log \left (-x^2+\sqrt{3} x-1\right )-5 \sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )+\frac{60}{x}+10 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+10 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 75, normalized size = 0.8 \[{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{1}{5\,{x}^{5}}}+{x}^{-1}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^8+x^4+1),x)

[Out]

1/6*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/12*ln(1+x^2-x*3^(1/2))*3^(1/2)-1/12*ln
(1+x^2+x*3^(1/2))*3^(1/2)-1/5/x^5+1/x+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{5 \, x^{4} - 1}{5 \, x^{5}} + \frac{1}{2} \, \int \frac{x^{2} - 1}{x^{4} - x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x
- 1)) + 1/5*(5*x^4 - 1)/x^5 + 1/2*integrate((x^2 - 1)/(x^4 - x^2 + 1), x)

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Fricas [A]  time = 0.264504, size = 128, normalized size = 1.31 \[ \frac{\sqrt{3}{\left (10 \, x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{3} + 2 \, x\right )}\right ) + 10 \, x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + 5 \, x^{5} \log \left (-\frac{6 \, x^{3} - \sqrt{3}{\left (x^{4} + 5 \, x^{2} + 1\right )} + 6 \, x}{x^{4} - x^{2} + 1}\right ) + 4 \, \sqrt{3}{\left (5 \, x^{4} - 1\right )}\right )}}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*sqrt(3)*(10*x^5*arctan(1/3*sqrt(3)*(x^3 + 2*x)) + 10*x^5*arctan(1/3*sqrt(3)
*x) + 5*x^5*log(-(6*x^3 - sqrt(3)*(x^4 + 5*x^2 + 1) + 6*x)/(x^4 - x^2 + 1)) + 4*
sqrt(3)*(5*x^4 - 1))/x^5

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Sympy [A]  time = 0.696406, size = 94, normalized size = 0.96 \[ \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{3}}{3} + \frac{2 \sqrt{3} x}{3} \right )}\right )}{12} + \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{5 x^{4} - 1}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*(2*atan(sqrt(3)*x/3) + 2*atan(sqrt(3)*x**3/3 + 2*sqrt(3)*x/3))/12 + sqrt
(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12 + (5*x**
4 - 1)/(5*x**5)

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GIAC/XCAS [A]  time = 0.277816, size = 113, normalized size = 1.15 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{12} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} + \frac{2}{x} \right |}}\right ) + \frac{5 \, x^{4} - 1}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x
- 1)) + 1/12*sqrt(3)*ln(abs(2*x - 2*sqrt(3) + 2/x)/abs(2*x + 2*sqrt(3) + 2/x)) +
1/5*(5*x^4 - 1)/x^5