Slow integration of trigonometric expression speeds up after TrigReduce

Question

I'm trying to integrate a trigonometric expression as

Integrate[Cos[2 t] Cos[3 t] Cos[4 t] Cos[5 t] Sin[2 t] Sin[3 t], {t, 0, 2 Pi}]

The final results is correct (which is 0), but it takes a very long time (about 1100 seconds), and it returns some output messages such as:

N::meprec :  Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -\[Pi]+10 ArcTan[Sqrt[10-2 Power[<<2>>]]/(5+Sqrt[5])].
N::meprec :  Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -\[Pi]+10 ArcTan[Sqrt[10-2 Power[<<2>>]]/(5+Sqrt[5])].
LessEqual::meprec :  Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -(1/5) Abs[-\[Pi]+10 ArcTan[Sqrt[Plus[<<2>>]]/Plus[<<2>>]]].
PossibleZeroQ::ztest1 :  Unable to decide whether numeric quantity -(\[Pi]/5)+I (Log[1-(I Sqrt[Plus[<<2>>]])/Plus[<<2>>]]-Log[1+I Power[<<2>>] Power[<<2>>]]) is equal to zero. Assuming it is.

In the end, I used TrigReduce to process this expression first then Integrate it (and got the result very fast).

But I am curious why the original version takes so long.

Answer 1

Simplification of the integrand leads to (removable) singularities:

Cos[2 t] Cos[3 t] Cos[4 t] Cos[5 t] Sin[2 t] Sin[3 t] // Simplify
(*  1/16 Csc[5 t] Sin[6 t] Sin[8 t] Sin[10 t]  *)

That leads to checking convergence, which I guess takes a long time.

You can turn off some of the checking, and Integrate[] takes somewhere between 0.2 and 4 seconds, depending on what's been loaded and computed already.

Integrate[
 Cos[2 t] Cos[3 t] Cos[4 t] Cos[5 t] Sin[2 t] Sin[3 t],
 {t, 0, 2 Pi}, 
 GenerateConditions -> False]
(*  0  *)

You can see that the original Integrate[] obtains the antiderivative fairly quickly and then output stops, from which I inferred it was dealing with the singularities (see How much time should one give Mathematica for an integral evaluation? for some debugging techniques):

Block[{Integrate`QuickLookUpDump`dbgPrintQT = Print},
 Integrate[
  Cos[2 t] Cos[3 t] Cos[4 t] Cos[5 t] Sin[2 t] Sin[3 t],
   {t, 0, 2 Pi}]
 ]
(* output contains Csc[5 t] Sin[6 t] Sin[8 t] Sin[10 t] *)

I aborted the computation, so I don't know what happens if you wait 1100 seconds....