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 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....

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....

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]

You can see it gets 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....

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 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....