There is no known algorithm for finding what you defined as "computationally efficient form" [1, Ch.2], except of course simply computing and comparing all possible factorisations. This method overall of course is not very "computationally efficient".
The best one can do is to use a clever heuristic to find "good" factorisations. I implemented a greedy heuristic similar to the one described in  in the Python package multivar_horner.
In this context one always has to balance the "investment" of computing a good factorisation of a multivariate polynomial with the benefits of having such a factorisation (cf. the speed tests of my package on Github).
Remark: Depending on the Hardware and Language in use it is actually not always best to just look for a factorisation with the minimal amount of multiplications. Exponentiation for example often is much more expensive than a multiplication. In the case of Python this does not seem to be the case (cf. how-is-exponentiation-implemented-in-python). These considerations affect the choice of "clever factorisation heuristic".
Here a pointer to papers on that subject (with different proposed heuristics):
 Greedy Algorithms for Optimizing Multivariate Horner Schemes, 2004
 On the multivariate Horner scheme,2000