For hard to calculate integrals, limits or system of equations I wonder if you could use a vector instead of single variable as input. What I mean is that the algorithm would handle and make use of 3 or more values at a time.

For instance I wondered if maybe one can use future value prediction from a series of previous results. You can calculate a predicted future value x(n) and x(n+1) with a linear model. Where x(n) = sum(a(i)*x(n-i),{i=1..p}).

This guessed value could perhaps be used somehow.