I wonder how to obtain the second equality as follows in Eq. (44) of http://www.tandfonline.com/doi/abs/10.1080/00018732.2015.1055918?journalCode=tadp20

\begin{eqnarray} K_{eff}^{(1)}[t_0](t)&=&\frac{1}{\hbar}\int_{t_0}^t dt'(H(t')-H_F^{(1)}[t_0]) \nonumber \\ &=& -\frac{1}{2\hbar}[\int_{t}^{t+T} dt'H(t')(1+2\frac{t-t'}{T})-\int_{t_0}^{t_0+T} dt'H(t')(1+2\frac{t_0-t'}{T})] \end{eqnarray}

I wonder how to obtain the second equality as follows in Eq. (44) of

Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. M Bukov, L D'Alessio & A Polkovnikov. Adv. Phys. 64, 139 (2015), arXiv:1407.4803.

which reads

\begin{eqnarray} K_\mathrm{eff}^{(1)}[t_0](t)&=&\frac{1}{\hbar}\int_{t_0}^t dt'(H(t')-H_F^{(1)}[t_0]) \nonumber \\ &=& -\frac{1}{2\hbar} \left[\int_{t}^{t+T} dt'H(t') \left(1+2\frac{t-t'}{T} \right)-\int_{t_0}^{t_0+T} dt'H(t') \left(1+2\frac{t_0-t'}{T} \right) \right] . \end{eqnarray}