Holographic free energy and integrated correlators of $\mathcal{N}=1^{\ast }$ theory

We consider the 2nd integrated correlators of $\mathcal{N}=4$, $D=4$ super Yang-Mills theory, especially those which can be associated with $\mathcal{N}=1^{\ast }$ mass-deformed theories. We provide an analytic derivation of the integrals at supergravity tree level, which has not been available so far. Our result agrees with the previous results from the study of BPS solutions in the dual supergravity models.