Archimax copula models can account for any type of asymptotic dependence between extremes and at the same time capture joint risks at medium levels. An Archimax copula C ψ, is characterized by two functional parameters, the stable tail dependence function , and the Archimedean generator ψ which acts as a distortion of the extreme-value dependence model. This article develops semiparametric inference for Archimax copulas: a nonparametric estimator of and a moment-based estimator of ψ assuming the latter belongs to a parametric family. Conditions under which ψ and are identifiable are derived. The asymptotic behavior of the estimators is then established under broad regularity conditions; performance in small samples is assessed through a comprehensive simulation study. The Archimax copula model with the Clayton generator is then used to analyze monthly rainfall maxima at three stations in French Brittany. The model is seen to fit the data very well, both in the lower and in the upper tail. The nonparametric estimator of reveals asymmetric extremal dependence between the stations, which reflects heavy precipitation patterns in the area. Technical proofs, simulation results and R code are provided in the Appendix.