NIntegrate[ Cos[2 1 Pi (2 Sqrt[(n + 8854800957672623)^2 - 78407500000000000000000000000000] + 0) - Pi]/(2 1 Pi)^2 (1000000/(10^40 Pi^8 (Sqrt[ 2 (Sqrt[(n + 8854800957672623)^2 - 78407500000000000000000000000000] + 0.001)^2 + 313630000000000000000000000000000] - Sqrt[2 (Sqrt[(n + 8854800957672623)^2 - 78407500000000000000000000000000])^2 + 313630000000000000000000000000000])^8 + 1)) , {n, 10^8, 10^16}, MaxRecursion -> 500]
The first part of the function is an highly oscillating function and the second part is a polynomial function, I've tried using various methods; "Global Adaptive", "LevinRule", but they are not working, the Global Adaptive seems to be the best option so far but its not monotonically decreasing after a number of integrand evaluations and hence not converging.
Additional Input It is accurately solving the problem from 0 to Infinity, which is weird.