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

Optimal. Leaf size=41 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right )+\log (x) \]

[Out]

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

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Rubi [A]  time = 0.0718375, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - x^4 + x^8)),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.4691, size = 41, normalized size = 1. \[ \frac{\log{\left (x^{4} \right )}}{4} - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} - \frac{1}{3}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0209464, size = 55, normalized size = 1.34 \[ \log (x)-\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 35, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ({x}^{8}-{x}^{4}+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^8-x^4+1),x)

[Out]

ln(x)-1/8*ln(x^8-x^4+1)+1/12*3^(1/2)*arctan(1/3*(2*x^4-1)*3^(1/2))

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Maxima [A]  time = 0.827508, size = 51, normalized size = 1.24 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/8*log(x^8 - x^4 + 1) + 1/4*log(
x^4)

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Fricas [A]  time = 0.256825, size = 58, normalized size = 1.41 \[ -\frac{1}{24} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{8} - x^{4} + 1\right ) - 8 \, \sqrt{3} \log \left (x\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*sqrt(3)*(sqrt(3)*log(x^8 - x^4 + 1) - 8*sqrt(3)*log(x) - 2*arctan(1/3*sqrt
(3)*(2*x^4 - 1)))

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Sympy [A]  time = 0.383117, size = 41, normalized size = 1. \[ \log{\left (x \right )} - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**8-x**4+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.275213, size = 51, normalized size = 1.24 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \,{\rm ln}\left (x^{8} - x^{4} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/8*ln(x^8 - x^4 + 1) + 1/4*ln(x^
4)