TimedIntegrate[exp, vars] is like Integrate, but stops after the number of seconds specified by the option Timing. Options of Integrate can be given and are passed on.
This should reach to be done
TimedIntegrate[Log[x^5], {x, 0, 1}, Timing -> 1]-5
This shouldn’t
TimedIntegrate[Log[Cos[x^5]], {x, 0, 1}, Timing -> 10, Integrate -> int]\text{int}\left(\log \left(\cos \left(x^5\right)\right),\{x,0,1\},\text{Assumptions}\to \varepsilon >0\right)