  In this paper, we connect rectangular free probability theory and spherical
integrals. In this way, we prove the analogue, for rectangular or square
non-Hermitian matrices, of a result that Guionnet and Maida proved for
Hermitian matrices in 2005. More specifically, we study the limit, as $n,m$
tend to infinity, of the logarithm (divided by $n$) of the expectation of
$\exp[\sqrt{nm}\theta X_n]$, where $X_n$ is the real part of an entry of $U_n
M_n V_m$, $\theta$ is a real number, $M_n$ is a certain $n\times m$
deterministic matrix and $U_n, V_m$ are independent Haar-distributed orthogonal
or unitary matrices with respective sizes $n\times n$, $m\times m$. We prove
that when the singular law of $M_n$ converges to a probability measure $\mu$,
for $\theta$ small enough, this limit actually exists and can be expressed with
the rectangular R-transform of $\mu$. This gives an interpretation of this
transform, which linearizes the rectangular free convolution, as the limit of a
sequence of log-Laplace transforms.
