Unfortunately, there is no closed-form solution for maximizing the log-likelihood (or minimizing the inverse, the logistic cost function); at least it has not been found, yet.

There’s the exception where you only have 2 obervations, and there is this paper

Lipovetsky, Stan. “Analytical closed-form solution for binary logit regression by categorical predictors.” Journal of Applied Statistics 42.1 (2015): 37-49. (Analytical closed-form solution for binary logit regression by categorical predictors)

which “shows that for categorical explanatory variables, it is possible to present the solution in the analytical closed-form formulae.” The problem is that the logistic sigmoid function is non-linear – in case of linear regression, you are assuming independent Gaussian noise.



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