\begin{equation*}
  \begin{split}
  \tilde{M}_{\phi}(\alpha, \beta_p, \beta_v) & = \tfrac{1}{Z_{\tilde\phi}(\alpha, \beta_p, \beta_v) }
  \tfrac{\partial}{\partial \alpha }Z_{\tilde\phi}(\alpha, \beta_p, \beta_v) \\
 & =
   \tfrac{1}{Z_{\tilde\phi}(\alpha, \beta_p, \beta_v) } \biggl[ n
   \exp{(\alpha n)} + n \exp{(\alpha'')} \\
& \qquad\qquad\qquad\qquad + n(n-1)  \exp{\{\alpha (n-2) + \alpha''\}} 
   + C_{2, \phi}(\alpha, \beta_p, \beta_v) 
    + n\sqrt{\tfrac{n}{2\pi}}    \int_{2/n}^{1 - 2/n}
    \tfrac{\Delta_{\Phi_2}(ny) \exp\{ g_2(ny)\} }{
    (1-y)^{n(1-y) + \frac{1}{2}} y^{ny - \frac{1}{2} }}
    dy \biggr],
  \end{split}
\end{equation*}
