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

Optimal. Leaf size=275 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}} \]

[Out]

-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 - 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]) - 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])

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Rubi [A]  time = 0.590313, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^4 + x^8)^(-1),x]

[Out]

-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 - 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]) - 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])

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Rubi in Sympy [A]  time = 26.0321, size = 529, normalized size = 1.92 \[ \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + \frac{1}{2}\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (\frac{1}{2} + \frac{\sqrt{3}}{2}\right ) \log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{1}{2} + \frac{\sqrt{3}}{2}\right ) \log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{2} + \sqrt{3} \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{2} + \sqrt{3} \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2}}{2} + \sqrt{3} \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2}}{2} + \sqrt{3} \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [C]  time = 0.0170151, size = 42, normalized size = 0.15 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1^3 + 2*#1^7) & ]/4

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Maple [C]  time = 0.009, size = 30, normalized size = 0.1 \[{\frac{\sum _{{\it \_R}={\it RootOf} \left ( 9\,{{\it \_Z}}^{4}+1 \right ) }{\it \_R}\,\ln \left ( 3\,{{\it \_R}}^{2}+3\,{\it \_R}\,x+{x}^{2} \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^8-x^4+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^8 - x^4 + 1), x)

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Fricas [A]  time = 0.229715, size = 257, normalized size = 0.93 \[ -\frac{1}{6} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{\sqrt{3} \sqrt{2} x + 2}{\sqrt{3} \sqrt{2} x + 2 \, x^{2} + 2 \, \sqrt{x^{4} + \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1}}\right ) - \frac{1}{6} \, \sqrt{3} \sqrt{2} \arctan \left (-\frac{\sqrt{3} \sqrt{2} x - 2}{\sqrt{3} \sqrt{2} x - 2 \, x^{2} - 2 \, \sqrt{x^{4} - \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1}}\right ) + \frac{1}{24} \, \sqrt{3} \sqrt{2} \log \left (x^{4} + \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) - \frac{1}{24} \, \sqrt{3} \sqrt{2} \log \left (x^{4} - \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*sqrt(2)*arctan((sqrt(3)*sqrt(2)*x + 2)/(sqrt(3)*sqrt(2)*x + 2*x^2 +
2*sqrt(x^4 + sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1))) - 1/6*sqrt(3)*sqrt(2)*arc
tan(-(sqrt(3)*sqrt(2)*x - 2)/(sqrt(3)*sqrt(2)*x - 2*x^2 - 2*sqrt(x^4 - sqrt(3)*s
qrt(2)*(x^3 + x) + 3*x^2 + 1))) + 1/24*sqrt(3)*sqrt(2)*log(x^4 + sqrt(3)*sqrt(2)
*(x^3 + x) + 3*x^2 + 1) - 1/24*sqrt(3)*sqrt(2)*log(x^4 - sqrt(3)*sqrt(2)*(x^3 +
x) + 3*x^2 + 1)

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Sympy [A]  time = 0.289391, size = 165, normalized size = 0.6 \[ \frac{\sqrt{6} \left (2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{3} - \frac{1}{3} \right )} + 2 \operatorname{atan}{\left (\sqrt{6} x^{3} - 4 x^{2} + 2 \sqrt{6} x - 3 \right )}\right )}{24} + \frac{\sqrt{6} \left (2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{3} + \frac{1}{3} \right )} + 2 \operatorname{atan}{\left (\sqrt{6} x^{3} + 4 x^{2} + 2 \sqrt{6} x + 3 \right )}\right )}{24} - \frac{\sqrt{6} \log{\left (x^{4} - \sqrt{6} x^{3} + 3 x^{2} - \sqrt{6} x + 1 \right )}}{24} + \frac{\sqrt{6} \log{\left (x^{4} + \sqrt{6} x^{3} + 3 x^{2} + \sqrt{6} x + 1 \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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

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GIAC/XCAS [A]  time = 0.208233, size = 277, normalized size = 1.01 \[ \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{24} \, \sqrt{6}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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