Integrating out complex bosons

From here, I’m trying to integrate out the massive fields of the $D0$ brane action, expanded around a classical background. I have a ghost action with complex fields:

$$S_{GH} = i int dtau overline{c}_{1} (- partial_{tau}^{2} + r^{2}) c_{1} + overline{c}_{2} (- partial_{tau}^{2} + r^{2}) c_{2} – overline{c}_{3} partial_{tau}^{2} c_{3} + sqrt{g}epsilon^{abc}(partial_{tau} overline{c}^{a}) c^{b} A^{c} – sqrt{g}epsilon^{a3x} epsilon^{cbx} B^{i}_{3} overline{c}^{a} c^{b} Y^{i}_{c} $$

I know that there are two bosons with $m^{2} = r^{2}$ and one boson with $m^{2} = 0$. My question is: how do I get the determinant when the fields $c_{1}, c_{2}$ and their complex conjugates are integrated out, given that the terms are not in the simple form $phi (-partial_{tau}^{2} + m^{2}) phi$?