3.343 \(\int \frac{1}{x^2 \left (1+x^4+x^8\right )} \, dx\)

Optimal. Leaf size=145 \[ -\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.226359, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571 \[ -\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(1 + x^4 + x^8)),x]

[Out]

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

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Rubi in Sympy [A]  time = 55.7349, size = 131, normalized size = 0.9 \[ - \frac{\log{\left (x^{2} - x + 1 \right )}}{8} + \frac{\log{\left (x^{2} + x + 1 \right )}}{8} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{12} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{4} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(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/x

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Mathematica [C]  time = 0.386753, size = 140, normalized size = 0.97 \[ \frac{1}{24} \left (-3 \log \left (x^2-x+1\right )+3 \log \left (x^2+x+1\right )-\frac{24}{x}+2 i \sqrt{-6+6 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )-2 i \sqrt{-6-6 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(-24/x + (2*I)*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - (2*I)*Sq
rt[-6 - (6*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2] - 2*Sqrt[3]*ArcTan[(-1 + 2*
x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 3*Log[1 - x + x^2] + 3*Log[1
+ x + x^2])/24

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Maple [A]  time = 0.016, size = 114, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{24}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{24}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}-{x}^{-1}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [A]  time = 0.280712, size = 235, normalized size = 1.62 \[ \frac{\sqrt{3}{\left (4 \, \sqrt{3} x \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} + \sqrt{3} x + 1} + 3}\right ) + 4 \, \sqrt{3} x \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} - \sqrt{3} x + 1} - 3}\right ) + \sqrt{3} x \log \left (x^{2} + x + 1\right ) - \sqrt{3} x \log \left (x^{2} - x + 1\right ) - 2 \, x \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 2 \, x \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x \log \left (x^{2} + \sqrt{3} x + 1\right ) - x \log \left (x^{2} - \sqrt{3} x + 1\right ) - 8 \, \sqrt{3}\right )}}{24 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.9611, size = 218, normalized size = 1.5 \[ \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 442368 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{7} - 384 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{3} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 384 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{3} - 442368 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{7} \right )} + \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 442368 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{7} - 384 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{3} \right )} + \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 384 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{3} - 442368 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{7} \right )} + \operatorname{RootSum}{\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log{\left (- 442368 t^{7} - 384 t^{3} + x \right )} \right )\right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1/8 - sqrt(3)*I/24)*log(x - 442368*(-1/8 - sqrt(3)*I/24)**7 - 384*(-1/8 - sqrt
(3)*I/24)**3) + (-1/8 + sqrt(3)*I/24)*log(x - 384*(-1/8 + sqrt(3)*I/24)**3 - 442
368*(-1/8 + sqrt(3)*I/24)**7) + (1/8 - sqrt(3)*I/24)*log(x - 442368*(1/8 - sqrt(
3)*I/24)**7 - 384*(1/8 - sqrt(3)*I/24)**3) + (1/8 + sqrt(3)*I/24)*log(x - 384*(1
/8 + sqrt(3)*I/24)**3 - 442368*(1/8 + sqrt(3)*I/24)**7) + RootSum(2304*_t**4 + 4
8*_t**2 + 1, Lambda(_t, _t*log(-442368*_t**7 - 384*_t**3 + x))) - 1/x

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{8} + x^{4} + 1\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^2),x, algorithm="giac")

[Out]

integrate(1/((x^8 + x^4 + 1)*x^2), x)