Clear["`*"]
b = Exp[I (a - I \[Eta]) (\[Epsilon] -
C1 (1/(1 - I C2 \[Epsilon]) + 1/(1 + I C2 \[Epsilon])))];
Integrate[b, {\[Epsilon], -Infinity, \[Epsilon]F},
Assumptions -> {a \[Element] Reals, \[Eta] \[Element] Reals,
C1 \[Element] Reals, C2 \[Element] Reals}]

请参看精品区《如果符号解算不出来且符号解不是必须的，试试数值解》