Limit[Integrate[(1/(Cosh[x] + I Sinh[x] - 2)) (Sinh[x] + I Cosh[x]),{x,-a,a}],a->Infinity]Limit[Integrate[(1/(Cosh[x] + I Sinh[x] - 2)) (Sinh[x] + I Cosh[x]),{x,-a,a}],a->Infinity]

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