Simplistically: If you want an accurate integration (step size and sampling apart) do it digitally. If you want a simple integrator do it in analogue.
As you say, problems of bias and leakage currents, and also component drifts and non-linearities, make analogue integrators a bit of a nightmare. But practically an integrator will always form part of a larger system, so you tend to choose the technology to match the rest of the system: it is hard to image a digital control system that would exit into analogue just to implement an integrator (although it may make sense the other way around).
The last time I used an analogue integrator* was nearly 20 years ago, to correct the DC offset in an audio system (to avoid the use of coupling capacitors), however this was inside a feedback loop so the exact parameters and stability were not critical.
*Of course a low-pass filter is actually an integrator, but I think we know what we mean by treating them seperately...
Andy Millar CEng MIET CMgr MCMIhttp://www.linkedin.com/in/millarandy
"The aim of argument, or of discussion, should not be victory, but progress." Joseph Joubert