NIntegrate with high error estimate

Question

Mathematica cannot get a sufficiently small error estimate (10^(-6) say) for the inegtals

NIntegrate[(R R1 (-R Cos[s] + R1 Cos[s1])^4)/((z - z1)^2 + (R Cos[s] -
       R1 Cos[s1])^2 + (R Sin[s] - R1 Sin[s1])^2)^(5/2), {s, 0, 
  2 Pi}, {z, -0.2705980500730985, 0.2705980500730985}, {R, 0, 
  1/2 Sqrt[1 - 16 z^2 + 32 z^4]}, {s1, 0, 
  2 Pi}, {z1, -0.7205904182426568, 0.7205904182426568}, {R1, 0, Sqrt[
  1 - 1.7313019390581716 z1^2 - 0.37467580052332317 z1^4]}, 
 AccuracyGoal -> 8, WorkingPrecision -> 32]

why?

Answer 1

You can Simplify a bit your integrand. Then, first, I have tried with MonteCarlo method to get a rough estimation of your integral. After a non-convergent result, the estimation is $\approx 0.3$. This gives me a clue about the result.

So following the suggestion of @MariuszIwaniuk:

NIntegrate[(R R1 (R Cos[s] - R1 Cos[s1])^4)/(R^2 + R1^2 + (z - z1)^2 - 
2 R R1 Cos[s - s1])^(5/2), {s, 0, 2 Pi}, {z, -0.2705980500730985,0.2705980500730985}, 
{R,0, 1/2 Sqrt[1 - 16 z^2 + 32 z^4]}, {s1, 0, 2 Pi}, 
{z1, -0.7205904182426568, 0.7205904182426568}, 
{R1, 0, Sqrt[1 - 1.7313019390581716 z1^2 - 0.37467580052332317 z1^4]}, 
Method -> "LocalAdaptive"]

(* 0.300715 *)

After 200 runs by using MonteCarlo, now the convergent result is:

Mean@Table[NIntegrate[(R R1 (R Cos[s] - R1 Cos[s1])^4)/
                      (R^2 + R1^2 + (z - z1)^2 - 2 R R1 Cos[s - s1])^(5/2),
{s, 0, \[Pi], 2 Pi}, {z, -0.2705980500730985, 0.2705980500730985}, 
{R, 0,1/2 Sqrt[1 - 16 z^2 + 32 z^4]}, {s1, 0, 2 Pi}, 
{z1, -0.7205904182426568, 0, 0.7205904182426568}, 
{R1, 0, Sqrt[1 - 1.7313019390581716 z1^2 - 0.37467580052332317 z1^4]}, 
 Method -> {"MonteCarlo", "MaxPoints" -> 1000000}], {200}]

(* 0.300865 *)

which is faster and quite approximated.