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

Optimal. Leaf size=140 \[ \frac{1}{8} \log \left (x^2-x+1\right )-\frac{1}{8} \log \left (x^2+x+1\right )-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.164832, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{1}{8} \log \left (x^2-x+1\right )-\frac{1}{8} \log \left (x^2+x+1\right )-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 + x^4 + x^8),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.8471, size = 128, normalized size = 0.91 \[ \frac{\log{\left (x^{2} - x + 1 \right )}}{8} - \frac{\log{\left (x^{2} + x + 1 \right )}}{8} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{24} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{12} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{4} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(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)/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

_______________________________________________________________________________________

Mathematica [C]  time = 0.296594, size = 135, normalized size = 0.96 \[ \frac{1}{48} \left (6 \log \left (x^2-x+1\right )-6 \log \left (x^2+x+1\right )+4 i \sqrt{-6-6 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )-4 i \sqrt{-6+6 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((4*I)*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - (4*I)*Sqrt[-6 +
(6*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3])*x)/2] - 4*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[
3]] - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] + 6*Log[1 - x + x^2] - 6*Log[1 + x + x
^2])/48

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 109, normalized size = 0.8 \[ -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}-{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^8 + x^4 + 1),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)) + 1/2*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 = 0.270786, size = 212, normalized size = 1.51 \[ -\frac{1}{24} \, \sqrt{3}{\left (4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} + \sqrt{3} x + 1} + 3}\right ) + 4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} - \sqrt{3} x + 1} - 3}\right ) + \sqrt{3} \log \left (x^{2} + x + 1\right ) - \sqrt{3} \log \left (x^{2} - x + 1\right ) + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \log \left (x^{2} + \sqrt{3} x + 1\right ) + \log \left (x^{2} - \sqrt{3} x + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.89033, size = 214, normalized size = 1.53 \[ \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x + 442368 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{7} - 192 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{3} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 192 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{3} + 442368 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{7} \right )} + \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x + 442368 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{7} - 192 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{3} \right )} + \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 192 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{3} + 442368 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{7} \right )} + \operatorname{RootSum}{\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log{\left (442368 t^{7} - 192 t^{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(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 - 192*(-1/8 - sqrt
(3)*I/24)**3) + (-1/8 + sqrt(3)*I/24)*log(x - 192*(-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 - 192*(1/8 - sqrt(3)*I/24)**3) + (1/8 + sqrt(3)*I/24)*log(x - 192*(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 - 192*_t**3 + x)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(x^8 + x^4 + 1), x)