$\begin{array}{l}\phantom{\phantom{;}x^{2}+1;}{\phantom{;}x^{6}\phantom{-;x^n}-x^{4}\phantom{-;x^n}+x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+1\overline{\smash{)}\phantom{;}x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}x^{2}+1;}\underline{-x^{8}\phantom{-;x^n}-x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{8}-x^{6};}-x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}+1-;x^n;}\underline{\phantom{;}x^{6}\phantom{-;x^n}+x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}x^{6}+x^{4}-;x^n;}\phantom{;}x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n;}\underline{-x^{4}\phantom{-;x^n}-x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-x^{4}-x^{2}-;x^n-;x^n;}-x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}x^{2}+1-;x^n-;x^n-;x^n;}\underline{\phantom{;}x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{;;;\phantom{;}x^{2}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n;}\phantom{;}1\phantom{;}\phantom{;}\\\end{array}$

$x^{6}-x^{4}+x^{2}-1+\frac{1}{x^2+1}$