Integral of integral — it takes too much time

Question

When I evaluate the following expression in Mathematica, it takes so much time that I don't want to wait for the evaluation to complete. So I think that there must be a better approach.

wewint[lx_?NumericQ, ly_?NumericQ] := NIntegrate[
    Cos[lx]*Cos[qx] + Sin[lx]*Sin[qx], {qx, -Pi, Pi}, {qy, -Pi, Pi}, 
    Method -> Automatic];
NIntegrate[wewint[lx, ly], {lx, -Pi, Pi}, {ly, -Pi, Pi}, Method -> Automatic]

If I use Cos[lx-qx] instead of (Cos[lx]*Cos[qx] + Sin[lx]*Sin[qx]), the effect is the same. In reality, the expression I am working with is much more complicated and it couldn't be calculated analytically.

Answer 1

You may have some luck playing with various methods. It is also important to decrease the number of integrations - one 4-dimensional integral can often compute much faster than 2 two-dimensional ones (meaning 2d integral of 2d integrals). So, we have:

NIntegrate[Cos[lx]*Cos[qx]+Sin[lx]*Sin[qx],{qx,-Pi,Pi},{qy,-Pi,Pi},
  {lx,-Pi,Pi},{ly,-Pi,Pi},Method->"LocalAdaptive"]//AbsoluteTiming

(* {0.619141,0.}  *)

while for example

NIntegrate[Cos[lx]^2*Cos[qx]^2+Sin[lx]*Sin[qx],{qx,-Pi,Pi},{qy,-Pi,Pi},
   {lx,-Pi,Pi},{ly,-Pi,Pi},Method->"LocalAdaptive"]//AbsoluteTiming

(* {1.188476,389.636}  *)

and

{#,N[#]}&@Integrate[Cos[lx]^2*Cos[qx]^2+Sin[lx]*Sin[qx],{qx,-Pi,Pi},{qy,-Pi,Pi},
  {lx,-Pi,Pi},{ly,-Pi,Pi}]

(*  {4 \[Pi]^4,389.636}  *)

just to confirm.