Simplify[Integrate[1/((x^2 - a^2)^2 + b^4), {x, 0, Infinity}], a > 0 && b > 0]

ConditionalExpression[-((I [Pi])/(2 Sqrt[a^4 + b^4] (Sqrt[a^2 - I b^2] + Sqrt[a^2 + I b^2]))), Im[Sqrt[a^2 + I b^2]] > 0 && Im[Sqrt[a^2 - I b^2]] > 0]