f1[x_, n_] :=
1/(1 + 10000000000000000000000000 (\[Pi]^8 (-2 Sqrt[(89 + x)^2] +
Sqrt[31363 + (n + Sqrt[-31363 + 4 (89 + x)^2])^2])^8 -
4/3 \[Pi]^10 (-2 Sqrt[(89 + x)^2] + Sqrt[
31363 + (n + Sqrt[-31363 + 4 (89 + x)^2])^2])^10 +
38/45 \[Pi]^12 (-2 Sqrt[(89 + x)^2] + Sqrt[
31363 + (n + Sqrt[-31363 + 4 (89 + x)^2])^2])^12))
NIntegrate[(E^(2 I Pi (n + Sqrt[321 + 712 x + 4 x^2])))^21 N[
f1[x, n], 25], {x, 0, 10^35}, {n, -1/100, 1/100},
MaxRecursion -> 500, Method -> "LevinRule"]
The above works flawlessly, but once I try to set the Working Precision of the numerical integral like below, the integration time increases exponentially.
NIntegrate[(E^(2 I Pi (n + Sqrt[321 + 712 x + 4 x^2])))^21 N[
f1[x, n], 25], {x, 0, 10^35}, {n, -1/100, 1/100},
MaxRecursion -> 500, Method -> "LevinRule", WorkingPrecision -> 25]
Is there any way to speed this up and get that level of accurate digits of computation?, I know this is because of the default machine precision on most systems, but is there a work around?
