  We present a new method to solve the holomorphic anomaly equations governing
the free energies of type B topological strings. The method is based on direct
integration with respect to the non-holomorphic dependence of the amplitudes,
and relies on the interplay between non-holomorphicity and modularity
properties of the topological string amplitudes. We develop a formalism valid
for any Calabi-Yau manifold and we study in detail two examples, providing
closed expressions for the amplitudes at low genus, as well as a discussion of
the boundary conditions that fix the holomorphic ambiguity. The first example
is the non-compact Calabi-Yau underlying Seiberg-Witten theory and its
gravitational corrections. The second example is the Enriques Calabi-Yau, which
we solve in full generality up to genus six. We discuss various aspects of this
model: we obtain a new method to generate holomorphic automorphic forms on the
Enriques moduli space, we write down a new product formula for the fiber
amplitudes at all genus, and we analyze in detail the field theory limit. This
allows us to uncover the modularity properties of SU(2), N=2 super Yang-Mills
theory with four massless hypermultiplets.
