Why not evaluate the integrals formulaically using an implementation of Risch's algorithm (although Cherry's algorithmic extension to Rich's result seems more practical; but, as you know, that's not saying much

), and then evaluate the resulting formula numerically at the points you want?

If I interpret Cherry's paper correctly (which you gave me 4 months ago), he does implement his algorithm in Macsyma.

Is Mathematica able to compute this integral (I know, it's a dumb question; but it's worth a shot)?