I realize that general results on the hitting times of a curve are practically nonexistant, but I am hoping that someone can string together a sequence of tricks to tell me what
$$ \Pr\left( \inf\left\{ t \ge 0: B_t + \mu t \le \log( A + Ce^{bt} ) \right\} \le x \right) $$
where $B_s$ is a standard Brownian motion, $A>0, C>0$ is. I would like $\mu \lessgtr 0$ and $b \lessgtr 0$ if possible, but would be interested in any result in this direction. Alternately, if someone can simply offer the expectation of
$$ \int_0^\tau e^{rt} d t $$
where $\tau$ is the hitting time discussed above, this is how I will use the distribution. This problem is derived from the first time that a geometric Brownian motion with drift hits the curve $A + C^{bs}$. If the more general case of the first time that $Ae^{aB_s} + Ce^{cB_s} \ge \Xi$ is available, I would also be very interested.
Many thanks.