I thought it was interesting that $\frac{u^2+1}{(u^2-2u-1)^2}$ has the very simple integral $-\frac{u}{u^2-2u-1}$ but both of $\frac{u^2}{(u^2-2u-1)^2}$ and $\frac{1}{(u^2-2u-1)^2}$ are very complicated (the transcendental parts cancel each other though).
So my question is how do I check by looking at a rational function whether or not it's a derivative of a rational function?
For example $\frac{1}{(x^2+1)^2}$ isn't but $\frac{x}{(x^2+1)^2}$ is. How can we tell in general?
On
For your last example, it is easy to see that a) you have a simple factor of $x$ in the numerator, and b) your denominator is a simple power of $1+x^2$. The integral is easily transformed into the form $\int du/(1+u)^2$, which is a rational function. This works for any power of $1+x^2$ greater than 1.
Examine the poles of your function (in the complex plane). If all residues are zero, you are in good shape.