<s>[INST] How to convert $\frac{1}{(1-x)(1-x^3)}$ into a sum of multiple fractions? What I mean is, that one can convert $\frac{1}{(1-x)^2(1-x^2)}$ into the following sum:
$\frac{1}{8}(\frac{1}{1+x}+\frac{1}{1-x} +\frac{2}{(1-x)^2} + \frac{4}{(1-x^3)})$
But I can't seem to do the same here, because when I try to simplify $1-x^3$ I get $(1-x)(1+x+x^2)$, and the second term can't be further simplified. 
For context, this is part of another problem in combinatorics, which I'm trying to solve using generating functions.
 [/INST] Since you know that$$\frac1{(x-1)(x^3-1)}=\frac1{(x-1)^2(x^2+x+1)},$$use this to deduce that$$\frac1{(x-1)(x^3-1)}=\frac{x+1}{3\left(x^2+x+1\right)}-\frac1{3(x-1)}+\frac1{3 (x-1)^2},$$by partial fraction decomposition.
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