One caveat: some combinations of tweaks might give a wrong result. The variant below claims pi
instead of 2pi
.
In[714]:= AbsoluteTiming[
Integrate[
Simplify@TrigToExp[Cos[Cos[Theta]] Cosh[Sin[Theta]]], {Theta, 0,
2 Pi}, GenerateConditions -> False]]
(* Out[714]= {0.527774, \[Pi]} *)
Also, some of the timing discrepancies can have to do caching of intermediate results.