Be $A_{ij}$ a symmetric matrix. Then I can easily write $$ \int \exp\left(-\frac{1}{2}\sum_{i,j}x_i A_{ij} x_j+\sum_{i} B_i x_i\right)\; d^nx= \sqrt{(2\pi)^n}\exp\left\{-\frac{1}{2}\mathrm{Tr}\log A\right\}\exp\left\{\frac{1}{2}\vec{B}^{T}A^{-1}\vec{B}\right\}. $$ and this formula holds both in the finite and inifinite dimensional case.

If I consider a functional integration, and a differential operator $\hat A$, the result is usually expressed as $$ \int \exp\left( - \frac 1 2 \varphi \hat A \varphi +J \varphi \right) D\varphi \; \propto \; \exp \left( {1\over 2} \int d^4x \; d^4y J\left ( x \right ) A^{-1}\left ( x - y \right ) J\left( y \right ) \right) $$ or in some cases as $$ \int \exp\left( - \frac 1 2 \varphi \hat A \varphi +J \varphi \right) D\varphi \; =\\\mathcal{N}\,\exp\left\{-\frac{1}{2}\mathrm{Tr}\log \hat A \right\}\, \exp \left( {1\over 2} \int dx \; dx^\prime J\left ( x \right ) A^{-1}\left ( x - x^\prime \right ) J\left( x^\prime \right ) \right) $$ where $\varphi \hat A \varphi = \int dx dx^\prime \varphi(x) A(x,x^\prime) \varphi(x^\prime) $ and $J\varphi = \int dx \varphi(x) J(x) $.

A more complicated situation I have to face now would be $$ \int D\psi D\varphi \exp\left\{ - \frac 1 2 \left[ \varphi \hat A \varphi + \psi \hat B \psi + \varphi \psi \hat C \psi \varphi \right] \right\} $$ where $\varphi \psi \hat C \psi \varphi=\int dx dx^\prime \varphi(x) \psi(x) C(x,x^\prime) \psi(x^\prime) \varphi(x^\prime)$ where $\psi$ is a Grassman-odd field. Anyway, the problem should be the same in the case the $\psi$ was a simple Grassman-even field.

My idea in this case would be rewriting it as $$ \int D\psi \exp\left\{ - \frac 1 2 \psi \hat B \psi \right\} \int D\varphi \exp\left\{ - \frac 1 2 \varphi \left[ \hat A + \psi \hat C \psi \right]\varphi \right\} $$ in such a way to obtain something similar to $$ \int D\psi \exp\left\{ - \frac 1 2 \psi \hat B \psi \right\} \exp \left\{ - \frac{1}{2} \mathrm{Tr} \log \left[ \hat A + \psi \hat C \psi \right]\right\} $$ but I don't know how to treat the Trace in this case. Would this procedure be acceptable, and how should I deal with the trace operator?