Issues with integral

[pinkman]

Nice explanations and tricks, thanks.

Bernard,
Could you please explain the first warning in the message from the console?

Simplification assuming taylorx0 near π/4
Simplification assuming taylorx0 near π/4
Warning, choosing root of [1,0,%%%{-2,[1,1]%%%}+%%%{-2,[1,0]%%%},0,%%%{1,[2,2]%%%}+%%%{-2,[2,1]%%%}+%%%{1,[2,0]%%%}] at parameters values [-72,58]
Warning, 1/4*π not checked