FIREBurn[expr, {q1, q2, ...}, {p1, p2, ...}] reduces
loop integrals with loop momenta q1, q2, ... and external
momenta p1, p2, ... with integration-by-parts (IBP)
relations.
FIREBurn expects that the input does not contain any
loop integrals with linearly dependent propagators. Therefore, prior to
starting the reduction, use ApartFF.
The evaluation is done on a parallel kernel using A.V. Smirnov’s and V.A. Smirnov’s FIRE.
int = SFAD[{p, m^2, 2}, {{0, 2 p . k}, m^2, 3}]\frac{1}{(p^2-m^2+i \eta )^2.(2 (k\cdot p)-m^2+i \eta )^3}
FIREBurn[int, {p}, {k}, Timing -> False]-\frac{(D-5) (D-3) k^2 \left(D m^2-4 k^2-6 m^2\right)}{m^4 \left(m^2-4 k^2\right)^3 (2 (k\cdot p)-m^2+i \eta ).(p^2-m^2+i \eta )}-\frac{(D-2) \left(2 D^2 k^2-24 D k^2+66 k^2+m^2\right)}{2 m^4 \left(m^2-4 k^2\right)^3 (p^2-m^2+i \eta )}