Improper integrals involving reciprocal trigonometric, Anger and Weber functions with respect to their order and argument

In this paper, we consider improper integrals involving the Anger and Weber functions, the order and argument of which depend on the integration variable, as well as reciprocal trigonometric and rational functions. Closed-form expressions for integrals under discussion are derived using real and complex analysis. We also extend the Hardy's theorem on integral inversion for Cauchy's principal values to the case of an infinite number of singularities.