I'm looking for a closed-form (or any numerically simpler) solution for the following integral:
$$ \int_{c_1}^{c_2}\phi(x,\mu_t,\sigma_t)\Phi(x,\mu_f,\sigma_f)^{(L-1)}dx $$
where
$\phi(x,\mu_t,\sigma_t)$ is the normal density function evaluated at $x$ with mean $\mu_t$ and standard deviation $\sigma_t$,
$\Phi(x,\mu_f,\sigma_f)$ is the cumulative normal evaluated at $x$ with mean $\mu_f$ and standard deviation $\sigma_f$,
and $L$ is a positive integer.
Note that the means and standard deviations of these two functions are different.
This formula is related to the probability that a sample drawn from the $N(\mu_t,\sigma_t)$ distribution will be higher than $L-1$ samples drawn from the $N(\mu_f,\sigma_f)$ distribution.
I'm also looking for a simplification of the conceptually-related integral:
$$ \int_{c_1}^{c_2}(L-1)\phi(x,\mu_f,\sigma_f)\Phi(x,\mu_f,\sigma_f)^{(L-2)}\Phi(x,\mu_t,\sigma_t)dx $$
It would also be useful to know if such a thing is not possible.
Thank you in advance.