Integrals of products of Hurwitz zeta functions and the Casimir effect in $phi^4$ field theories

We evaluate two integrals over $xin [0,1]$ involving products of the function $zeta_1(a,x)equiv zeta(a,x)-x^{-a}$ for $Re (a)>1$, where $zeta(a,x)$ is the Hurwitz zeta function. The evaluation of these integrals for the particular case of integer $ageq 2$ is also presented. As an application we calculate the $O(g)$ weak-coupling expansion coefficient $c_{1}(varepsilon)$ of the Casimir energy for a film with Dirichlet-Dirichlet boundary conditions, first stated by Symanzik [Schrodinger representation and Casimir effect in renormalizable quantum field theory, Nucl. Phys. B 190 (1981) 1-44] in the framework of $gphi^4_{4-varepsilon}$ theory.