I have an integral $$F(S,a,k)=\int_0^S\mathrm{d}z_1\int_0^S\mathrm{d}z_2\,\mathrm{e}^{-k a(z_1+z_2)}\mathrm{e}^{k^2f(R, z_1, z_2)}\left(a^2+g(z_1-z_2)-k\left(\frac{\partial}{\partial z_1}+\frac{\partial}{\partial z_2}\right)f(z_1, z_2)\right)~,$$ where $f(R, z_1, z_2)$ and $g(R, z_1-z_2)$ are real valued function such that $$f(R, z_1, z_2)=\frac{1}{2}\int_0^{z_1}\mathrm{d}z'\int_0^{z_1}\mathrm{d}z''\,g(0, z'-z'')+\frac{1}{2}\int_0^{z_2}\mathrm{d}z'\int_0^{z_2}\mathrm{d}z''\,g(0, z'-z'')+\int_0^{z_1}\mathrm{d}z'\int_0^{z_2}\mathrm{d}z''\,g(R, z'-z'')~.$$
Is there any way to reduce $F(S,a,k)$ to something simpler,either by integration by parts or any other useful technique?