3.331 \(\int \frac{x^5}{1+x^4+x^8} \, dx\)

Optimal. Leaf size=75 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{8} \log \left (x^4-x^2+1\right )-\frac{1}{8} \log \left (x^4+x^2+1\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.142103, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{8} \log \left (x^4-x^2+1\right )-\frac{1}{8} \log \left (x^4+x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(1 + x^4 + x^8),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.2466, size = 70, normalized size = 0.93 \[ \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{8} - \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} - \frac{1}{3}\right ) \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**8+x**4+1),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.209666, size = 94, normalized size = 1.25 \[ \frac{\sqrt{1-i \sqrt{3}} \left (\sqrt{3}-i\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x^2\right )+\sqrt{1+i \sqrt{3}} \left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x^2\right )}{4 \sqrt{6}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^5/(1 + x^4 + x^8),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 62, normalized size = 0.8 \[ -{\frac{\ln \left ({x}^{4}+{x}^{2}+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^8+x^4+1),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.818125, size = 82, normalized size = 1.09 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{8} \, \log \left (x^{4} + x^{2} + 1\right ) + \frac{1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 + x^4 + 1),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.261774, size = 88, normalized size = 1.17 \[ -\frac{1}{24} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{4} + x^{2} + 1\right ) - \sqrt{3} \log \left (x^{4} - x^{2} + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 + x^4 + 1),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.597719, size = 76, normalized size = 1.01 \[ \frac{\log{\left (x^{4} - x^{2} + 1 \right )}}{8} - \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**8+x**4+1),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.316075, size = 82, normalized size = 1.09 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{8} \,{\rm ln}\left (x^{4} + x^{2} + 1\right ) + \frac{1}{8} \,{\rm ln}\left (x^{4} - x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^8 + x^4 + 1),x, algorithm="giac")

[Out]

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