  We present in detail the calculation of the O(alpha_s^3) inclusive total
cross section for the process pp -> t-tbar-h, in the Standard Model, at the
CERN Large Hadron Collider with center-of-mass energy sqrt(s_H)=14 TeV. The
calculation is based on the complete set of virtual and real O(alpha_s)
corrections to the parton level processes q-qbar -> t-tbar-h and gg ->
t-tbar-h, as well as the tree level processes (q,qbar)g -> t-tbar-h-(q,qbar).
The virtual corrections involve the computation of pentagon diagrams with
several internal and external massive particles, first encountered in this
process. The real corrections are computed using both the single and the two
cutoff phase space slicing method. The next-to-leading order QCD corrections
significantly reduce the renormalization and factorization scale dependence of
the Born cross section and moderately increase the Born cross section for
values of the renormalization and factorization scales above m_t.
