$\begin{array}{l}\phantom{\phantom{;}3x^{2}+1;}{-81x^{8}\phantom{-;x^n}+27x^{6}\phantom{-;x^n}-9x^{4}\phantom{-;x^n}+3x^{2}\phantom{-;x^n}-1\phantom{;}\phantom{;}}\\\phantom{;}3x^{2}+1\overline{\smash{)}-243x^{10}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}3x^{2}+1;}\underline{\phantom{;}243x^{10}\phantom{-;x^n}+81x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}243x^{10}+81x^{8};}\phantom{;}81x^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}+1-;x^n;}\underline{-81x^{8}\phantom{-;x^n}-27x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-81x^{8}-27x^{6}-;x^n;}-27x^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}+1-;x^n-;x^n;}\underline{\phantom{;}27x^{6}\phantom{-;x^n}+9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;\phantom{;}27x^{6}+9x^{4}-;x^n-;x^n;}\phantom{;}9x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}+1-;x^n-;x^n-;x^n;}\underline{-9x^{4}\phantom{-;x^n}-3x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;-9x^{4}-3x^{2}-;x^n-;x^n-;x^n;}-3x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}3x^{2}+1-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}3x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{;;;;\phantom{;}3x^{2}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n;}\phantom{;}2\phantom{;}\phantom{;}\\\end{array}$

$-81x^{8}+27x^{6}-9x^{4}+3x^{2}-1+\frac{2}{1+3x^2}$