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We introduce and study the Wilson-loop ${\rm d}\log$ representation ofcertain Feynman integrals for scattering amplitudes in ${\cal N}=4$ SYM andbeyond, which makes their evaluation completely straightforward. Such arepresentation was motivated by the dual Wilson loop picture, and it can alsobe derived by partial Feynman parametrization of loop integrals. We firstintroduce it for the simplest one-loop examples, the chiral pentagon in fourdimensions and the three-mass-easy hexagon in six dimensions, which arerepresented by two- and three-fold ${\rm d}\log$ integrals that are nicelyrelated to each other. For multi-loop examples, we write the $L$-loopgeneralized penta-ladders as $2(L{-}1)$-fold ${\rm d}\log$ integrals of someone-loop integral, so that once the latter is known, the integration can beperformed in a systematic way. In particular, we write the eight-pointpenta-ladder as a $2L$-fold ${\rm d}\log$ integral whose symbol can be computedwithout performing the integration; we also obtain the last entries and thesymbol alphabet of these integrals. Similarly we compute and study the symbolof the seven-point double-penta-ladder, which is represented by a$2(L{-}1)$-fold integral of a hexagon; the latter can be written as a two-fold${\rm d}\log$ integral plus a boundary term. We comment on the relation of ourrepresentation to differential equations and resum the ladders by solvingcertain integral equations.

The Wilson-loop $d \log$ representation for Feynman integrals