-Result in mathematica 11.3:
In[1]:= F[n_] :=
Integrate[Exp[I n x] Sin[x] x^-1 Log[x] , {x, 0, 2 Pi}]
F[1.] // Timing
Out[2]= {25.6563, -1.07069 + 0.649257 I}
-Result in mathematica 12:
In[1]:= F[n_] :=
Integrate[Exp[I n x] Sin[x] x^-1 Log[x] , {x, 0, 2 Pi}]
F[1.] // Timing
During evaluation of In[1]:= Infinity::indet: Indeterminate expression (0. +0. I) (-[Infinity]) encountered.
During evaluation of In[1]:= Infinity::indet: Indeterminate expression (0. +0. I) (-[Infinity]) encountered.
During evaluation of In[1]:= Power::infy: Infinite expression 1/0. encountered.
During evaluation of In[1]:= Power::infy: Infinite expression 1/0. encountered.
During evaluation of In[1]:= Infinity::indet: Indeterminate expression ComplexInfinity+ComplexInfinity encountered.
During evaluation of In[1]:= Infinity::indet: Indeterminate expression -[Infinity]+[Infinity] encountered.
During evaluation of In[1]:= Infinity::indet: Indeterminate expression -[Infinity]+[Infinity]-1. ExpIntegralEi[(0. -2. I) (3.14159 +ArcSin[x])]+1. ExpIntegralEi[(0. +2. I) (3.14159 +ArcSin[x])] encountered.
During evaluation of In[1]:= General::stop: Further output of Infinity::indet will be suppressed during this calculation.
During evaluation of In[1]:= Power::infy: Infinite expression 1/0. encountered.
During evaluation of In[1]:= General::stop: Further output of Power::infy will be suppressed during this calculation.
Out[2]= {222.344, -1.07069 + 0.649257 I}