I assumed that it would be straightforward to find the partial fraction decomposition over the reals of the rational function $$f(x) = \frac{1}{(x^2 +1)^2}.$$ However, when I try what I thought would be the usual method of writing it as $$\frac{1}{(x^2+1)^2} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2+1)^2},$$ I find that the only choice of constants is $A = B = C = 0$, and $D = 1$, simply reproducing what I started with. Typically, one might think that this would be a logical approach to finding the indefinite integral of $f(x)$, but it seems to fail here. Could someone explain why this happens, or where I have made a mistake?
For the record, the integral is elementary using the substitution $x = \tan \theta$ and a trig identity in the result.