The Feynman propagator for the QM harmonic oscillator reads
$$\tag{A} \Delta_F(t)~=~ \frac{1}{2\omega} \sum_{\pm}\theta(\pm t)e^{\mp i\omega t}. $$
The time derivative is
$$\tag{B} \dot{\Delta}_F(t)~=~ \frac{1}{2i} \sum_{\pm}\pm\theta(\pm t)e^{\mp i\omega t}, $$
where the two contributions in eq. (B) proportional to the delta function $\delta(t)$ have cancelled out. Hence
$$\tag{C} \Delta^3_F(t)~=~ \frac{1}{8\omega^3} \sum_{\pm}\theta(\pm t)e^{\mp 3i\omega t}, $$
and
$$\tag{D} \dot{\Delta}^2_F(t)\Delta_F(t)~=~ \frac{-1}{8\omega} \sum_{\pm}\theta(\pm t)e^{\mp 3i\omega t}. $$
It is implicit understood that one should apply the Feynman $i\epsilon$ prescription. Here it means that we should substitute $\omega \to \omega-i\epsilon$. This implies that the $t=\pm \infty$ boundary terms in an $\int_{\mathbb{R}} \mathrm{d}t$ integration are exponentially suppressed, so that we arrive at the formulas (2.17) of Ref. 1 coming from the kinks at $t=0$:
$$\tag{2.17a} \int_{\mathbb{R}} \mathrm{d}t~\Delta^3_F(t)~=~ \frac{1}{12i\omega^4}, $$
and
$$\tag{2.17b} \int_{\mathbb{R}} \mathrm{d}t~\dot{\Delta}^2_F(t)\Delta_F(t)~=~ \frac{i}{12\omega^2}. $$
References:
- J.-L. Gervais and A. Jevicki, Point Canonical Transformations in Path Integral, Nucl. Phys. B110 (1976) 93. The pdf file is here.