This is very much what's in the Carr-Lee paper "Robust Replication of Volatility Derivatives", but it wasn't so easy to obtain in practice:
The formulas as written in the paper are not usable as is: they can be simplified (not too difficult, but intimidating at first)
The numerical integration is not trivial: a simple Gauss-Laguerre is not precise enough (maybe if I had an implementation with more points), a Gauss-Kronrod is not either (maybe if we split it in different regions). Funnily a simple adaptive Simpson works ok (but my boundaries are very basic: 1e-5 to 1e5).
