How do I use NMinimize to solve equations obtained using NIntegrate?

Question

I wish to minimize following equation with respect to variables $h,p,q,t$:

$\qquad f(h,p,q,t)=\int_0^1 \int_0^1 \frac{dx\,dy}{\sqrt{h^2+x^2+y^2-2\,x\,y\space cos(p)+2\,h\,y\,cos(q)-2\,h\,x\,cos(t)}}$.

If I try to integrate this double integral using Mathematica it reduces it to a single integral:

$\qquad f(h,p,q,t)=- \int_o^1 \ln\big[-x \, cos(p)+h \, cos(q)+ \sqrt{h^2 +x^2-2\,h\,x \,cos(t)}\,\big] dx \\ \qquad + \int_0^1 \ln \big [1-x\, cos(p)+h\,cos(q)+ \sqrt{1+h^2+x^2-2\,x\,cos(q) - 2\,h\,x\,cos(t)} \big] dx $

Any further operation using Integrate does not work. So, I tried to use NIntegrate instead. NIntegrate doesn't work if you have have variables in the integrand that are not a variable of integration. So, the way out is this:

I created the integral as a function of the variables that don't participate in the integral.

 FUNC[p_, q_, h_, t_] := 
   NIntegrate[Log[-x Cos[p] + h Cos[q] + Sqrt[h^2 + x^2 - 2 h x Cos[t]]], {x, 0, 1}]

Now FUNC[Pi/3, Pi/2, 6, Pi/6] gives me 1.66971, which is great. But I really need the integral to run a minimization on the variables h,p,q, and t.

So, I tried this next:

NMinimize[
  {FUNC[p, q, h, t], {-Pi <= p <= Pi, -Pi <= q <= Pi, h >= 0, -Pi <= t <= Pi}}, 
  {p, q, h, t}]

And I get errors.

Is my entire approach wrong? Or it is partially wrong? Any feedback is most welcome.

Solution

FUNC[(p_)?NumericQ, (q_)?NumericQ, (h_)?NumericQ, (t_)?NumericQ] := 
   NIntegrate[
    If[(-x)*Cos[p] + h*Cos[q] + Sqrt[h^2 + x^2 - 2*h*x*Cos[t]] > 0, 
      Log[(-x)*Cos[p] + h*Cos[q] + Sqrt[h^2 + x^2 - 2*h*x*Cos[t]]]], 
    {x, 0, 1}];
NMinimize[
  {Re[FUNC[p, q, h, t]], {-Pi <= p <= Pi, -Pi <= q <= Pi, h >= 0, -Pi <= t <= Pi}}, 
  {p, q, h, t}]

Mathematica result:

{-10.5339, {p -> -0.0248342, q -> 2.54783, h -> 0.0265091, t -> -2.523}}

This NMinimize makes sense to me. I'm minimizing energy here and I didn't incorporated an negative sign and this the minimized result is negative.