The integral is given by:

To solve the integral, we will use a known integral representation involving Chebyshev polynomials of the second kind and a series expansion for the denominator term.

The integral is given by:

$$I = \int_{0}^{\infty} \frac{x U_n \left[ a (a^2 + x^2)^{-1/2} \right]}{(a^2 + x^2)^{\frac{1}{2}n+1} (e^{\pi x} + 1)} dx$$

Step 1: Rewrite the term $(e^{\pi x} + 1)^{-1}$ as a series.
We use the geometric series expansion:

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