  We present a direct, geometric derivation of the generalized Smarr formula
for the stationary axially symmetric black holes with nonlinear electromagnetic
fields. The additional term is proven to be proportional to the integral of the
trace of the electromagnetic energy-momentum tensor and can be written as a
product of two conjugate variables. From the novel relation we can deduce all
previously proposed forms of the generalized Smarr formula, which were derived
only for the spherically symmetric black holes, and provide the lowest order
quantum correction to the classical relation from the Euler-Heisenberg
Lagrangian.
