= Late-time scalar gradient vertex integral
{title2=$\operatorname{Im}I=c_s[-K+s_2/K+s_3/K^2]$}
For $K=k_1+k_2+k_3$, $s_2=k_1k_2+k_1k_3+k_2k_3$ and $s_3=k_1k_2k_3$, the convergent <in-in formalism> integral is $I=\int_{-\infty(1-i0)}^{0^-}\tau^{-2}\prod_i(1-ic_sk_i\tau)e^{ic_sK\tau}d\tau$. Its real endpoint divergence drops out of the expectation value, but the finite imaginary endpoint term $-c_sK$ must be kept. The remaining terms contribute $c_ss_2/K+c_ss_3/K^2$. Reversing the exponential requires conjugating both the contour and the outside prefactor.
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