How to fix slow covergence and highly oscillatory integrand issues?

Question

I'm trying to numerically solve an integral in a specific region and then to visualize it as follows.

RegionPlot3D[
 NIntegrate[1/Sqrt[r] - 1/Sqrt[l + r Sin[t]], {r, l, t} ∈
  ImplicitRegion[r + l Sin[t] > 0 && l > 0 && r > 0, {r, l, t}]], {r, 
  0, 5}, {l, 0, 10}, {t, 0, pi/4}]

However, Mathematica complains that

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

I basically tried to get rid of potential singularities by taking that specific integration region into account. Yet, I have no idea about the slow convergence of highly oscillatory integrand.

Edit 1: There is also another warning saying

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 27 recursive bisections in l near {r,l,t} = {0.184661,1.641681647898248*10^690538901,0.184661}. NIntegrate obtained 8.935806122974667*10^22717757165+31581.2 I and 2.096578395728379`15.954589770191005*^22717757166 for the integral and error estimates.

How can I fix these issues?

Edit 2: What I am actually looking for is the 3D plot corresponding to the following integral function F(l,\theta) where r is a fixed number (say, 10, or whatever). I am particularly in trouble to get rid of singularities and divergent subsets of the variable's domains.

Answer 1

Maybe this will get you started:

ff[r_?NumericQ, t_?NumericQ, l_?NumericQ] := 
  NIntegrate[1/Sqrt[r] - 1/Sqrt[ll + r Sin[tt]], {tt, 0, t}, {ll, 0, l}];

Plot3D[ff[0.1, t, l], {t, 0, Pi}, {l, 0, 4}]

In the original code, there was a restriction, r + l Sin[t] > 0, but I'm not sure how you want to handle that. You might use Piecewise[] or multiply the integrand by Boole[r + ll Sin[tt] > 0], if appropriate.