is the natural father of the Magnus expansion method [@Magnus]. We eventually derive the Magnus series with the help of Rota–Baxter theory. For the purpose, the Riemann integral is treated in this paper as a particular Rota–Baxter operator of weight zero.

We first show in Section 7 that skewderivations and Rota–Baxter operators (of the same weight) are natural inverses. The ordinary Spitzer formula is revisited in Section 8, together with a nonlinear CBHD recursion due to one of us. The latter is instrumental in obtaining the *noncommutative Spitzer formula*. Also, inspired by the work of Lam, we give a noncommutative generalization of the Bohnenblust–Spitzer formula. In the next two sections, the Magnus expansion is arrived at as a limiting case of that formula. All along, we try to distinguish carefully which statements are valid for general Rota–Baxter operators, which for Rota–Baxter operators of vanishing weight, and which just for the Riemann integral. The main alternative integration method, the Dyson–Chen ‘expansional’ [@Dyson; @Chen; @Araki], flows from the Magnus series, and vice versa, by our Hopf algebraic means in Section 11. In turn, it reveals itself useful to understand the quirks of the Magnus expansion, and to solve the weight-zero CBHD recursion. 