Date: 08/13/15-09:32:18 PM Z

For the benefit of the others let me answer this 5-years old question:

By default and if possible, TID attempts to perform full reduction of
1-loop tensor integrals into the basis of the PaVe scalar function A0,
B0, C0, D0.

However, for rank 4 4-point functions with fully general kinematics as
provided by Johan, such a reduction results into a very large number of
terms with huge kinematic prefactors. Also it takes TID a lot of time to
do it*

The situation, of course, improves, when the loop momenta are all
contracted and the kinematics of the external momenta is fixed. Still,
to avoid such huge expressions, people usually prefer to do the
reduction in terms of the PaVe tensor functions. This gives you
relatively compact expressions suitable for numerical evaluation.
This mode can be activated via the option UsePaVeBasis.

Then one can quickly obtain the needed results:

Amp = FVD[q, \[Mu]] FVD[q, \[Nu]] FVD[q, \[Rho]] FVD[
q, \[Sigma]] FAD[{q, m}, {q + Subscript[k, 4],
m}, {q - Subscript[k, 1] - Subscript[k, 2],
m}, {q - Subscript[k, 2], m}]

Amp2 = FVD[q, mu] FVD[q, nu] FVD[q, al] FVD[q,
be] FAD[{q, mD}, {q, Lambda}, {q, Lambda}, {p3 + q, m1}, {q - p4,
m2}]

This takes only couple of second on my 4-years old laptop:

r1 = TID[Amp, q, UsePaVeBasis -> True]
r2 = TID[Amp2, q, UsePaVeBasis -> True]

* In principle one could speed it up by using a look-up table like TIDL
for tensor decompositions. However, I'm not sure if this is really
needed for any practical purposes...

Cheers,