Some people don't like working over the complex plane. We can
avoid factoring over C and factor only to a real form; this
gives factors like , which significantly complicate
the actual integration part of the algorithm (instead of simply using
the rule (note this gets the correct
logarithmic form in the limit as )), we have to use
integration by parts and arctans. It's all do-able, of course, just
It's not obvious here if we can avoid factoring over the real
numbers. But some problems (e.g. integrating the derivative of
) will not require a final answer to be
expressed using any algebraic extensions of the rational numbers.
It would be nice if there was an algorithm which avoided these
extensions when it could get away with it.
What do we do when can't be factored? If the roots are
all simple then there isn't much problem: we write the answer
and for many purposes this form answers admirably, if we can compute
(which we can---it's just a residue, and in the general
case it's just ---for simple roots).