This thread has been fun to follow. Why is it that the following produces Missing["NotAvailable"], and that only after over 3 minutes? I removed the "N" from NIntegrate[] in the second panel of Marco's first post.

AbsoluteTiming@WolframAlpha[ToString[Integrate[Cos[Cos[Theta]] Cosh[Sin[Theta]], {Theta, 0, 2 Pi}]], {{"PossibleClosedForm", 1}, "ComputableData"}]

Dear Daniel,

ok, I understand the problem, I think.

Thanks,

Marco

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.

And if we finally use all of the additional tweaks that Integrate offers:

AbsoluteTiming[Integrate[Simplify@TrigToExp[Cos[Cos[Theta]] Cosh[Sin[Theta]]], {Theta, 0, 2 Pi}, PrincipalValue -> True, GenerateConditions -> False]]

we get down to a quarter of a second - on my current machine that is nearly 700 times faster than the "naive integration".

Cheers,

M.

I don't understand why but even without GenerateConditions->False and just the PrincipalValue it gains considerably. Adding both has a positive effect on the calculation time on my machine.

Cheers,

M.

PS: Why is it that TrigToExp speeds the process up so much? I understand that the Exp function might speed it up, but I would have assumed that MMA does that automatically. Is there an algorithmic advantage of not doing this?

Still a bit faster:

AbsoluteTiming[Integrate[Simplify@TrigToExp[Cos[Cos[Theta]] Cosh[Sin[Theta]]], {Theta, 0, 2 Pi}, PrincipalValue -> True]]

takes about a third of a second.

Cheers,

M.

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.

How long did it take you to do it without Mathematica?