3.342 \(\int \frac{1}{1+x^4+x^8} \, dx\)

Optimal. Leaf size=88 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

-ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(1 + 2*x)/Sqrt[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 = 0.0986079, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4 + x^8)^(-1),x]

[Out]

-ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(1 + 2*x)/Sqrt[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 = 21.1603, size = 83, normalized size = 0.94 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(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 + s
qrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/6 + sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/6

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Mathematica [A]  time = 0.0350676, size = 68, normalized size = 0.77 \[ \frac{-\log \left (-x^2+\sqrt{3} x-1\right )+\log \left (x^2+\sqrt{3} x+1\right )+2 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 67, normalized size = 0.8 \[{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^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)+1/6*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. \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{2} \, \int \frac{x^{2} - 1}{x^{4} - x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^8 + x^4 + 1),x, 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)) - 1/2*integrate((x^2 - 1)/(x^4 - x^2 + 1), x)

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Fricas [A]  time = 0.269096, size = 90, normalized size = 1.02 \[ \frac{1}{12} \, \sqrt{3}{\left (2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{3} + 2 \, x\right )}\right ) + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \log \left (\frac{6 \, x^{3} + \sqrt{3}{\left (x^{4} + 5 \, x^{2} + 1\right )} + 6 \, x}{x^{4} - x^{2} + 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.515138, size = 82, normalized size = 0.93 \[ \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{3}}{3} + \frac{2 \sqrt{3} x}{3} \right )}\right )}{12} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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

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GIAC/XCAS [A]  time = 0.300524, size = 97, normalized size = 1.1 \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{12} \, \sqrt{3}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} + \frac{2}{x} \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^8 + x^4 + 1),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))