Integration of Rational function contain even power of variable $\int\frac{1}{1+x^6}dx$

Question

Evaluation of $\displaystyle \int\frac{1}{1+x^6}dx$

$\bf{My\; Try::}$Let $$I = \int\frac{1}{1+x^6}dx = \int\frac{1}{(1+x^2)(x^4-x^2+1)}dx$$

Using Partial fraction , above Integral is very lengthy, Can we solve it without using

partial fraction or any other way, Help me

Thanks

Answer 1

I would also recommend partial fractions but instead with

$$\dfrac{Ax+b}{x^2+\sqrt{3}x+1}+\dfrac{Cx+D}{x^2-\sqrt{3}x+1}+\dfrac{Ex+F}{x^2+1}$$