Hi,
I know that this is not nice and defeats the purpose, but because numerical integration is quite fast:
AbsoluteTiming[NIntegrate[Cos[Cos[Theta]] Cosh[Sin[Theta]], {Theta, 0, 2 Pi}]]
finishes in 0.006338 secs. The problem is that it does not give the symbolic solution. In this case, because the result is simple this dirty trick works:
AbsoluteTiming@WolframAlpha[ToString[NIntegrate[Cos[Cos[Theta]] Cosh[Sin[Theta]], {Theta, 0, 2 Pi}]], {{"PossibleClosedForm", 1}, "ComputableData"}]
runs for less than 5 seconds including the request to WolframAlpha. As I said, this does not give the proper symbolic solution and does not generalise to other problems.
Cheers,
M.
PS: Ups, I just noticed a better solution:
AbsoluteTiming[Integrate[TrigToExp[Cos[Cos[Theta]] Cosh[Sin[Theta]]], {Theta, 0, 2 Pi}]]
gives the symbolic result and runs slightly longer than a second.