  Bertolini-Darmon and Mok proved a formula of the second derivative of the
two-variable $p$-adic $L$-function of a modular elliptic curve over a totally
real field along the Hida family in terms of the image of a global point by
some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and
$p$-adic multiplicative integrals on $p$-adic upper half planes plays an
important role in their work. In this paper, we generalize these integrals for
$p$-adic measures which are not necessarily $\mathbf{Z}$-valued, and prove a
formula of the second derivative of the two-variable $p$-adic $L$-function of
an abelian variety of ${\rm GL}(2)$-type associated to a Hilbert modular form
of weight 2.
