I won’t bring the subject of numerical computation into this discussion, but I spent several years learning to compute the digits of \int_0^\infty{e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right)}dx.
You can read about my adventure at
How to calculate the digits of the MKB constant .
If you like numeric computations, of much interest is the story of how I came across this integral, by investigating \sum _{x=0}^{\infty } e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) since the 1990s, at Try to beat these MRB constant records!