Consider the equation
\begin{equation}
-\hbar^2\frac{d^2 f(t)}{dt^2} + \omega(t)f(t) = 0\,.
\end{equation}
I added an $\hbar$ as a power counting parameter (just set $\hbar = 1$ at the end of the calculations to recover your equation).
The WKB ansatz is of the form
\begin{equation}
f(t) = \exp\left(\frac{i}{\hbar}\int^t \mathcal{P}(t^\prime) dt^\prime\right),.
\end{equation}
Here $\mathcal{P}(t)$ is sometimes called the quantum momentum in the context of quantum mechanics.
Substituting the ansatz in the ODE you get the Riccati equation
\begin{equation}\label{eq:exact wkb riccati}
\mathcal{P}^2(t) - i \hbar \frac{d\mathcal{P}(t)}{dt} = -\omega(t)\,.
\end{equation}
Now you expand $\mathcal{P}(t)$ in a formal series in $\hbar$, this series is in general not convergent, but asymptotic
\begin{equation}
\mathcal{P}(t) = \sum_{n=0}^\infty p_n(t) \hbar^n\,.
\end{equation}
Then you put this into the Riccati equation and equate each order of the resulting series to 0. You get the initial condition
\begin{equation}
p_0(t) = \sqrt{-\omega(t)}\,,
\end{equation}
notice the two branches of the square root correspond to two different independent solutions of the original second order equation. Also you obtain the recursion relation
\begin{equation}
p_{n}(t) = \frac{1}{2 p_0(t)}\left(i\frac{d p_{n-1}(t)}{dt} - \sum_{m=1}^{n-1} p_{m}(t)p_{n-m}(t)\right)\,.
\end{equation}
In particular one can divide the formal series in the even and odd powers, call $W(t)$ the even series i.e. $W(t) = \sum_n p_{2n}(t)\hbar^{2n}$ and $W_{\text{odd}}(t)$ the odd series. It turns out that
\begin{equation}
W_{\text{odd}}(t) = \frac{i\hbar}{2}\frac{d}{dt}\log W(t)\,.
\end{equation}
Thus we can express $f$ in terms only of the even powers of $\hbar$
\begin{equation}
f(t) = \frac{1}{\sqrt{W(t)}}\exp\left(\frac{i}{\hbar}\int^t W(t^\prime) dt^\prime\right)\,.
\end{equation}
I do not see the problem in the recursive formula, you can use it in principle to get any to any finite order, it is pretty pointless to be lazy, the formula is pretty easy and the first few orders can be easily done by hand. Also even if you found it on a resource double-checking papers result is basically a fundamental step in reading most papers since you may find typos often (and I invite you to double check my formulas too, the ideas are right, the calculations hopefully are too).
There are many references on what I explained here, a good one which goes way deeper than what I wrote is https://arxiv.org/abs/1811.04812 and the references therein. Here also the points I made about the series being asymptotic and how to make sense of it via Borel resummations is explained as well as the framework of resugent quantum mechanics. The parts on the Thermodynamic Bethe Ansatz are interesting but most likely not too relevant
