In[]:=
Assuming{π>ϕ>0,1>α>0},Integrate
Sqrt[1+Exp[2θ]-2Exp[θ]Cos[ϕ]]
1+αExp[θ]
,{θ,-ϕ,ϕ}
Out[]=
ϕ
∫
-ϕ
1+
2θ

-2
θ

Cos[ϕ]
1+
θ

α
θ
​
​