  The evaluation of multi-loop Feynman integrals is one of the main challenges
in the computation of precise theoretical predictions for the cross sections
measured at the LHC. In recent years, the method of differential equations has
proven to be a powerful tool for the computation of Feynman integrals. It has
been observed that the differential equation of Feynman integrals can in many
instances be transformed into a so-called canonical form, which significantly
simplifies its integration in terms of iterated integrals. The main result of
this thesis is an algorithm to compute rational transformations of differential
equations of Feynman integrals into a canonical form. Apart from requiring the
existence of such a rational transformation, the algorithm needs no further
assumptions about the differential equation. In particular, it is applicable to
problems depending on multiple kinematic variables and also allows for a
rational dependence on the dimensional regulator. First, the transformation law
is expanded in the dimensional regulator to derive differential equations for
the coefficients of the transformation. Using an ansatz in terms of rational
functions, these differential equations are then solved to determine the
transformation. This thesis also presents an implementation of the algorithm in
the Mathematica package CANONICA, which is the first publicly available program
to compute transformations to a canonical form for differential equations
depending on multiple variables. The main functionality and its usage are
illustrated with some simple examples. Furthermore, the package is applied to
state-of-the-art integral topologies appearing in recent multi-loop
calculations. These topologies depend on up to three variables and include
previously unknown topologies contributing to higher-order corrections to the
cross section of single top-quark production at the LHC.
