  We consider the problem of symbolic-numeric integration of symbolic
functions, focusing on rational functions. Using a hybrid method allows the
stable yet efficient computation of symbolic antiderivatives while avoiding
issues of ill-conditioning to which numerical methods are susceptible. We
propose two alternative methods for exact input that compute the rational part
of the integral using Hermite reduction and then compute the transcendental
part two different ways using a combination of exact integration and efficient
numerical computation of roots. The symbolic computation is done within BPAS,
or Basic Polynomial Algebra Subprograms, which is a highly optimized
environment for polynomial computation on parallel architectures, while the
numerical computation is done using the highly optimized multiprecision
rootfinding package MPSolve. We show that both methods are forward and backward
stable in a structured sense and away from singularities tolerance
proportionality is achieved by adjusting the precision of the rootfinding
tasks.
