General result from Integrate differs from result with special values

Question

Can someone explain why these outputs differ?

In[5]:= g1 = 
   Integrate[Exp[-I x (n + 1)]/( Exp[I x] + 2), {x, 0, 2 Pi}, 
   Assumptions -> Element[n, Integers]];

In[6]:= g1 /. n -> 3

During evaluation of In[6]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.

Out[6]= Indeterminate

In[7]:= g2 = Integrate[Exp[-I x (3 + 1)]/( Exp[I x] + 2), {x, 0, 2 Pi}]

Out[7]= π/16

The last is correct, the first seems to apply some incorrect assumptions? Note

In[8]:= g1

Out[8]= 2^(-1 - n) Beta[-(1/2), -1 - n, 0] Sin[n π]

See Background for context.

Answer 1

Here is a workaround

g1[n_?IntegerQ] := Integrate[Exp[-I x (n + 1)]/(Exp[I x] + 2), {x, 0, 2 Pi} ];    
{g1[1], g1[2], g1[3] }
{π/4, -(π/8), π/16}