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

Optimal. Leaf size=154 \[ -\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-1/(7*x^7) + 1/(3*x^3) + ArcTan[(1 - 2*x)/Sqrt[3]]/(4*Sqrt[3]) - ArcTan[Sqrt[3]
- 2*x]/4 - ArcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) + ArcTan[Sqrt[3] + 2*x]/4 - Log
[1 - x + x^2]/8 + Log[1 + x + x^2]/8 + Log[1 - Sqrt[3]*x + x^2]/(8*Sqrt[3]) - Lo
g[1 + Sqrt[3]*x + x^2]/(8*Sqrt[3])

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Rubi [A]  time = 0.284647, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714 \[ -\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(1 + x^4 + x^8)),x]

[Out]

-1/(7*x^7) + 1/(3*x^3) + ArcTan[(1 - 2*x)/Sqrt[3]]/(4*Sqrt[3]) - ArcTan[Sqrt[3]
- 2*x]/4 - ArcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) + ArcTan[Sqrt[3] + 2*x]/4 - Log
[1 - x + x^2]/8 + Log[1 + x + x^2]/8 + Log[1 - Sqrt[3]*x + x^2]/(8*Sqrt[3]) - Lo
g[1 + Sqrt[3]*x + x^2]/(8*Sqrt[3])

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Rubi in Sympy [A]  time = 62.7571, size = 141, normalized size = 0.92 \[ - \frac{\log{\left (x^{2} - x + 1 \right )}}{8} + \frac{\log{\left (x^{2} + x + 1 \right )}}{8} + \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{12} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{4} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{4} + \frac{1}{3 x^{3}} - \frac{1}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2 - x + 1)/8 + log(x**2 + x + 1)/8 + sqrt(3)*log(x**2 - sqrt(3)*x + 1)/2
4 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/24 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/1
2 - sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/12 + atan(2*x - sqrt(3))/4 + atan(2*x +
sqrt(3))/4 + 1/(3*x**3) - 1/(7*x**7)

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Mathematica [C]  time = 0.693319, size = 171, normalized size = 1.11 \[ -\frac{1}{7 x^7}+\frac{1}{3 x^3}-\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )}{2 \sqrt{-6+6 i \sqrt{3}}}+\frac{\left (\sqrt{3}-i\right ) \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )}{2 \sqrt{-6-6 i \sqrt{3}}}-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-1/(7*x^7) + 1/(3*x^3) + ((I + Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*x)/2])/(2*Sqrt[-
6 + (6*I)*Sqrt[3]]) + ((-I + Sqrt[3])*ArcTan[((1 + I*Sqrt[3])*x)/2])/(2*Sqrt[-6
- (6*I)*Sqrt[3]]) - ArcTan[(-1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) - ArcTan[(1 + 2*x)/Sq
rt[3]]/(4*Sqrt[3]) - Log[1 - x + x^2]/8 + Log[1 + x + x^2]/8

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Maple [A]  time = 0.012, size = 119, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}-{\frac{1}{7\,{x}^{7}}}+{\frac{1}{3\,{x}^{3}}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(x^8+x^4+1),x)

[Out]

1/8*ln(x^2+x+1)-1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/24*ln(1+x^2-x*3^(1/2)
)*3^(1/2)+1/4*arctan(2*x-3^(1/2))-1/24*ln(1+x^2+x*3^(1/2))*3^(1/2)+1/4*arctan(2*
x+3^(1/2))-1/7/x^7+1/3/x^3-1/8*ln(x^2-x+1)-1/12*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. \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{7 \, x^{4} - 3}{21 \, x^{7}} + \frac{1}{2} \, \int \frac{x^{2}}{x^{4} - x^{2} + 1}\,{d x} + \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2
*x - 1)) + 1/21*(7*x^4 - 3)/x^7 + 1/2*integrate(x^2/(x^4 - x^2 + 1), x) + 1/8*lo
g(x^2 + x + 1) - 1/8*log(x^2 - x + 1)

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Fricas [A]  time = 0.274457, size = 250, normalized size = 1.62 \[ -\frac{\sqrt{3}{\left (84 \, \sqrt{3} x^{7} \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) + 84 \, \sqrt{3} x^{7} \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) - 21 \, \sqrt{3} x^{7} \log \left (x^{2} + x + 1\right ) + 21 \, \sqrt{3} x^{7} \log \left (x^{2} - x + 1\right ) + 42 \, x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 42 \, x^{7} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 21 \, x^{7} \log \left (x^{2} + \sqrt{3} x + 1\right ) - 21 \, x^{7} \log \left (x^{2} - \sqrt{3} x + 1\right ) - 8 \, \sqrt{3}{\left (7 \, x^{4} - 3\right )}\right )}}{504 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/504*sqrt(3)*(84*sqrt(3)*x^7*arctan(1/(2*x + sqrt(3) + 2*sqrt(x^2 + sqrt(3)*x
+ 1))) + 84*sqrt(3)*x^7*arctan(1/(2*x - sqrt(3) + 2*sqrt(x^2 - sqrt(3)*x + 1)))
- 21*sqrt(3)*x^7*log(x^2 + x + 1) + 21*sqrt(3)*x^7*log(x^2 - x + 1) + 42*x^7*arc
tan(1/3*sqrt(3)*(2*x + 1)) + 42*x^7*arctan(1/3*sqrt(3)*(2*x - 1)) + 21*x^7*log(x
^2 + sqrt(3)*x + 1) - 21*x^7*log(x^2 - sqrt(3)*x + 1) - 8*sqrt(3)*(7*x^4 - 3))/x
^7

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Sympy [A]  time = 3.12639, size = 209, normalized size = 1.36 \[ \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x - \frac{1}{2} + \frac{\sqrt{3} i}{6} - 18432 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - \frac{1}{2} - 18432 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} - \frac{\sqrt{3} i}{6} \right )} + \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x + \frac{1}{2} + \frac{\sqrt{3} i}{6} - 18432 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x + \frac{1}{2} - 18432 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} - \frac{\sqrt{3} i}{6} \right )} + \operatorname{RootSum}{\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log{\left (- 18432 t^{5} - 4 t + x \right )} \right )\right )} + \frac{7 x^{4} - 3}{21 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/8 - sqrt(3)*I/24)*log(x - 1/2 + sqrt(3)*I/6 - 18432*(1/8 - sqrt(3)*I/24)**5)
+ (1/8 + sqrt(3)*I/24)*log(x - 1/2 - 18432*(1/8 + sqrt(3)*I/24)**5 - sqrt(3)*I/6
) + (-1/8 - sqrt(3)*I/24)*log(x + 1/2 + sqrt(3)*I/6 - 18432*(-1/8 - sqrt(3)*I/24
)**5) + (-1/8 + sqrt(3)*I/24)*log(x + 1/2 - 18432*(-1/8 + sqrt(3)*I/24)**5 - sqr
t(3)*I/6) + RootSum(2304*_t**4 + 48*_t**2 + 1, Lambda(_t, _t*log(-18432*_t**5 -
4*_t + x))) + (7*x**4 - 3)/(21*x**7)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{8} + x^{4} + 1\right )} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^8 + x^4 + 1)*x^8), x)