This is fixed in Sage 7.6.beta2:

┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 7.6.beta2, Release Date: 2017-02-01               │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Warning: this is a prerelease version, and it may be unstable.     ┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
sage: integrate(x/(1+x^2),x,0,infinity)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
...
ValueError: Integral is divergent.



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