Multi Curve Modelling - Dealing with uncomfortable stochastic integrals

Today I'd like to put the focus on the multi curve modelling framework. The delevopement of financial markets after the credit crunch requires a new methodology for the valuation and modeling of financial instruments, since the basis spreads have increased and can no longer be neglected as it was done in the standard approach before.
These market adaptions require a multi curve modeling approach, where different curves for the calculation of discount and forward rates are used. From the UnRisk developer team, I was selected to analyze the latest innovations in multi curve modelling and to incorporate a new multi curve model to our UnRisk software product.
As the new model, we decided to extend the one factor Hull & White model to a multi curve interest rate model, having the form
where rd is the short interest rate process used for discounting, and rf  is the short rate interest rate process used for the calculation of the forward rates.
Using this model, one of the problems I encountered was the derivation of analytic formulas for the instruments used for the calibration process, i.e. zero coupon bonds, caps and swaptions. Therefore I had to solve some time consuming stochastic integrals. I don't want to go into details here, but to get an impression, here is a little out-take of the pricing formulas, which were computed by hand in flipchart format A1, since there was not enough space for a clear illustration of the formulas in an A4 format...