Eq 2 of this paper states this integral: \begin{align*} r^{-\beta} = \frac{1}{\Gamma(\beta)}\int_{-\infty}^{\infty} e^{-re^t + \beta t} dt \end{align*} Is there is name for this identity, or the class of identities of this form? And are there generalizations? I am dealing with a problem that requires computation of \begin{align*} \int_{\mathbb{R}^p}\exp\left(-\left\{\sum_{i=1}^{n}\lambda_i e^{\boldsymbol{\theta}_i^\intercal \mathbf{t}}\right\} + \boldsymbol{\alpha}^\intercal \mathbf{t} – \frac{1}{2}\mathbf{t}^\intercal \Gamma \mathbf{t}\right) d\mathbf{t} \end{align*} where $ \lambda_i \ge 0$ and $ \boldsymbol{\theta}_i, \boldsymbol{\alpha}, \mathbf{t} \in \mathbb{R}^p$ and $ \Gamma \in \mathbf{R}^{p\times p}$ . I’m willing to also settle for a very good approximation, if such exists.