  Hyperbolic beta integrals are analogues of Euler's beta integral in which the
role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma
function. They may be viewed as $(q,\widetilde{q})$-bibasic analogues of the
beta integral in which the two bases $q$ and $\widetilde{q}$ are interrelated
by modular inversion, and they entail $q$-analogues of the beta integral for
$|q|=1$. The integrals under consideration are the hyperbolic analogues of the
Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman
integral. We show that the hyperbolic Nassrallah-Rahman integral is a formal
limit case of Spiridonov's elliptic Nassrallah-Rahman integral.
