Consider the following function:
$$ f(x)=\frac{x^{2}+x+2}{(1-x^{2})^{3}} $$
I'm trying to calculate the coefficient of $x^{12}$. I usully use the Binomial theorem to solve this problem.
First thing that comes to mind, is to do the following:
$$\frac{x^{2}+x+2}{(1-x^{2})^{3}} = (x^2+x+2)\cdot \sum_{r=0}^{\infty}{r+2 \choose r}x^{2r} $$
But what can I do with $x^2+x+2$?
First express $f(x)$ as the sum of six partial fractions. Next find their 12-th derivative, it is not difficult. Finally calculate $f^{(12)}(0)/12!$ which is the coefficent of $x^{12}$.