The result of integrate_1d is confusing

This is indeed weird. Looks like a bug. I just verified it also happens with recent CmdStan.

Could you please file it at https://github.com/stan-dev/stan/issues/ ? It would help if you tried if you can reproduce the issue with some even simpler integrand (say 1 / ( 1 + x^2) which should integrate to \pi over the whole real line)

Thanks for noticing this!