General polynomial-form expression for ∫ tanh(x)/x · sech(x)^L · e^{-Tx} dx

Establish, for every integer L ≥ 1, the existence of polynomials P_{L}^{0}(T), …, P_{L}^{L}(T), with degrees indicated by their superscripts, such that the integral ∫_{0}^{∞} (tanh(x)/x) · sech(x)^{L} · exp(−T x) dx equals Σ_{k=1}^{L} P_{L}^{L−k}(T) [∂ζ/∂s (−k, (T + 1 + (1/2)(1 + (−1)^{L}))/4) − ∂ζ/∂s (−k, (T + 3 + (1/2)(1 + (−1)^{L}))/4)] + P_{L}^{L}(T) [log Γ((T + 1 + (1/2)(1 + (−1)^{L}))/4) − log Γ((T + 3 + (1/2)(1 + (−1)^{L}))/4)].

Background

The paper explicitly evaluates ∫_{0}^{∞} (tanh(x)/x) * sech(x)^{L} * exp(−T x) dx for L = 1, 2, 3, 4 in terms of derivatives of the Hurwitz zeta function and logarithms of Gamma functions (equations (int_tanh_over_x_L1)–(int_tanh_over_x_L4)).

Motivated by these explicit cases, the author formulates a conjecture asserting that, for general L, the integral admits a structured expansion involving polynomial coefficients multiplying differences of Hurwitz zeta s-derivatives at negative integers and a log-Gamma difference, with the arguments depending on the parity of L. This general form would unify the observed patterns for small L.