partial fractions when the fraction cannot be decomposed I am trying to find partial fractions of $\frac {1}{(x^2+1)^2}$. All the coefficients I get are zeros except the coefficient for the constant term which is 1, leaving me with the fraction I started with, so it seems like the fraction cannot be decomposed. How can I then go about writing this fraction as the sum $\frac {1}{2}\left [\frac {1}{x^2+1}-\frac{x^2-1}{(x^2+1)^2}\right]$? Is it just manipulation and trial and error?

In the real-valued partial fraction method, your term
$$
\frac{1}{(x^2+1)^2}
$$
is the best you can do.  Evaluation of integrals
$$
\int\frac{dx}{(ax^2+bx+c)^n}
$$
is done as follows:  For $n=1$, complete the square and it is an arctangent.  For $n>1$ there is a reduction formula from integration by parts:
\begin{align}
\int\frac{dx}{(ax^2+bx+c)^n} &= \frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}
\\ &\qquad+\frac{(2n-3)2a}{(n-1)(4ac-b^2)}
\int\frac{dx}{(ax^2+bx+c)^{n-1}}
\end{align}
