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in this integral $a=\psi_{1}(\mu), \ b=\psi_{2} (\mu)$. I expanded the function in Taylor series (3 terms) around ($\gamma= \frac{b}{2}$), numerically (for varioud values of $\mu$, and other constants I didn't include here for better readability/simplicity) it gave a fairly good approximation (although a bit worse than Simpson/Riemann quadrature).

When I solve this integral symbolically though, both Taylor series expansion and its integration are just gigantic ($ \approx 20$ terms).

If anyone could help me out with simplification of this expression, I'd massively appreciate it. This is an expectation of the first hitting time.

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