Two definite integrals involving products of four Legendre functions

The definite integrals \( \int _{-1}^1xP_\nu (x)]^4\mathrm{d}x\) and \( \int _{0}^1xP_\nu (x)]^2\{P_\nu (x)]^2-P_\nu (-x)]^2\}\mathrm{d}x\) are evaluated in closed form, where \( P_\nu \) stands for the Legendre function of degree \( \nu \in \mathbb C\). Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green’s functions and Epstein zeta functions.