Anti differentiation and the calculation of Feynman amplitudes This volume comprises review papers presented at the Conference on Antidifferentiation and the Calculation of Feynman Amplitudes held in Zeuthen Germany in October 2020 and a few additional invited reviews. The book aims at comprehensive surveys and new innovative results of the analytic integration methods of Feynman integrals in quantum field theory. These methods are closely related to the field of special functions and their function spaces the theory of differential equations and summation theory. Almost all of these algorithms have a strong basis in computer algebra. The solution of the corresponding problems are connected to the analytic management of large data in the range of Giga to Terabytes. The methods are widely applicable to quite a series of other branches of mathematics and theoretical physics. Graph complexes and Cutkosky rules Differential equations and dispersion relations for Feynman amplitudes with elliptic functions Elliptic integrals and the two loop ttbar production in QCD Solutions of 2nd and 3rd order differential equations with more singularities Analytic continuation of Feynman diagrams with elliptic solutions Twisted elliptic multiple zeta values and non planar one loop open string amplitudes Genus one superstring amplitudes and modular forms Difference field methods in Feynman diagram calculations Feynman integrals and iterated integrals of modular forms Iterated elliptic and hypergeometric integrals for Feynman diagrams. Feynman integrals L series and Kloosterman moments.