  In this paper, partly based on Zachos' PhD thesis, we show that the geodesic
X-ray transform is stably invertible near infinity on a class of asymptotically
conic manifolds which includes perturbations of Euclidean space. In particular
certain kinds of conjugate points are allowed. Further, under a global convex
foliation condition, the transform is globally invertible.
  The key analytic tool, beyond the approach introduced by Uhlmann and Vasy, is
the introduction of a new pseudodifferential operator algebra, which we name
the 1-cusp algebra, and its semiclassical version.
