I would like to find a closed formula for the coefficients of the generating $ f(x)=-{\frac {{x}^{4}+6\,{x}^{3}-2\,{x}^{2}+6\,x+1}{ \left( x+1 \right) \left( {x}^{2}+1 \right) \left( x-1 \right) ^{3}}} $ .

I am trying to do the following. \begin{align} f(x)=\sum_{i,j,k_1,k_2,k_3=0}^{\infty} (-x)^i (-x^2)^j x^{k_1+k_2+k_3}(1+6x-2x^2+6x^3+x^4). \end{align}

The expression is very complicated. Are there some simpler method (or some software) to find a closed formula for the coefficients? Thank you very much.