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

Optimal. Leaf size=147 \[ -\frac{1}{3 x^3}+\frac{1}{8} \log \left (x^2-x+1\right )-\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.188172, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{3 x^3}+\frac{1}{8} \log \left (x^2-x+1\right )-\frac{1}{8} \log \left (x^2+x+1\right )+\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(1 + x^4 + x^8)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.9141, size = 134, normalized size = 0.91 \[ \frac{\log{\left (x^{2} - x + 1 \right )}}{8} - \frac{\log{\left (x^{2} + x + 1 \right )}}{8} + \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{12} - \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{4} - \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{4} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(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)/24
- sqrt(3)*log(x**2 + sqrt(3)*x + 1)/24 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/12
- sqrt(3)*atan(sqrt(3)*(2*x/3 + 1/3))/12 - atan(2*x - sqrt(3))/4 - atan(2*x + s
qrt(3))/4 - 1/(3*x**3)

_______________________________________________________________________________________

Mathematica [C]  time = 0.554714, size = 148, normalized size = 1.01 \[ \frac{1}{24} \left (-\frac{8}{x^3}+3 \log \left (x^2-x+1\right )-3 \log \left (x^2+x+1\right )-\frac{4 i \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )}{\sqrt{\frac{1}{6} i \left (\sqrt{3}+i\right )}}+\frac{4 i \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )}{\sqrt{-\frac{1}{6} i \left (\sqrt{3}-i\right )}}-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 114, normalized size = 0.8 \[ -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{24}}-{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{24}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}-{\frac{1}{3\,{x}^{3}}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{3 \, x^{3}} - \frac{1}{2} \, \int \frac{1}{x^{4} - x^{2} + 1}\,{d x} - \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 + x^4 + 1)*x^4),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)) - 1/3/x^3 - 1/2*integrate(1/(x^4 - x^2 + 1), x) - 1/8*log(x^2 + x + 1)
+ 1/8*log(x^2 - x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.275027, size = 259, normalized size = 1.76 \[ \frac{\sqrt{3}{\left (12 \, \sqrt{3} x^{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} + \sqrt{3} x + 1} + 3}\right ) + 12 \, \sqrt{3} x^{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} - \sqrt{3} x + 1} - 3}\right ) - 3 \, \sqrt{3} x^{3} \log \left (x^{2} + x + 1\right ) + 3 \, \sqrt{3} x^{3} \log \left (x^{2} - x + 1\right ) - 6 \, x^{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 6 \, x^{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 3 \, x^{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + 3 \, x^{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) - 8 \, \sqrt{3}\right )}}{72 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/72*sqrt(3)*(12*sqrt(3)*x^3*arctan(sqrt(3)/(2*sqrt(3)*x + 2*sqrt(3)*sqrt(x^2 +
sqrt(3)*x + 1) + 3)) + 12*sqrt(3)*x^3*arctan(sqrt(3)/(2*sqrt(3)*x + 2*sqrt(3)*sq
rt(x^2 - sqrt(3)*x + 1) - 3)) - 3*sqrt(3)*x^3*log(x^2 + x + 1) + 3*sqrt(3)*x^3*l
og(x^2 - x + 1) - 6*x^3*arctan(1/3*sqrt(3)*(2*x + 1)) - 6*x^3*arctan(1/3*sqrt(3)
*(2*x - 1)) - 3*x^3*log(x^2 + sqrt(3)*x + 1) + 3*x^3*log(x^2 - sqrt(3)*x + 1) -
8*sqrt(3))/x^3

_______________________________________________________________________________________

Sympy [A]  time = 3.00226, size = 197, normalized size = 1.34 \[ \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 1 - \frac{\sqrt{3} i}{3} - 9216 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 1 - 9216 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} + \frac{\sqrt{3} i}{3} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x + 1 - \frac{\sqrt{3} i}{3} - 9216 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x + 1 - 9216 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} + \frac{\sqrt{3} i}{3} \right )} + \operatorname{RootSum}{\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log{\left (- 9216 t^{5} - 8 t + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/8 + sqrt(3)*I/24)*log(x - 1 - sqrt(3)*I/3 - 9216*(1/8 + sqrt(3)*I/24)**5) + (
1/8 - sqrt(3)*I/24)*log(x - 1 - 9216*(1/8 - sqrt(3)*I/24)**5 + sqrt(3)*I/3) + (-
1/8 + sqrt(3)*I/24)*log(x + 1 - sqrt(3)*I/3 - 9216*(-1/8 + sqrt(3)*I/24)**5) + (
-1/8 - sqrt(3)*I/24)*log(x + 1 - 9216*(-1/8 - sqrt(3)*I/24)**5 + sqrt(3)*I/3) +
RootSum(2304*_t**4 + 48*_t**2 + 1, Lambda(_t, _t*log(-9216*_t**5 - 8*_t + x))) -
1/(3*x**3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^8 + x^4 + 1)*x^4), x)