  The hook length formula for $d$-complete posets states that the $P$-partition
generating function for them is given by a product in terms of hook lengths. We
give a new proof of the hook length formula using $q$-integrals. The proof is
done by a case-by-case analysis consisting of two steps. First, we express the
$P$-partition generating function for each case as a $q$-integral and then we
evaluate the $q$-integrals. Several $q$-integrals are evaluated using partial
fraction expansion identities and others are verified by computer.
