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We derive analytic results for the symbol of certain two-loop Feynmanintegrals relevant for seven- and eight-point two-loop scattering amplitudes inplanar $\mathcal{N}=4$ super-Yang--Mills theory. We use a bootstrap inspiredstrategy, combined with a set of second-order partial differential equationsthat provide powerful constraints on the symbol ansatz. When the completesymbol alphabet is not available, we adopt a hybrid approach. Instead of thefull function, we bootstrap a certain discontinuity for which the alphabet isknown. Then we write a one-fold dispersion integral to recover the completeresult. At six and seven points, we find that the individual Feynman integralslive in the same space of functions as the amplitude, which is described by the9- and 42-letter cluster alphabets respectively. Starting at eight pointshowever, the symbol alphabet of the MHV amplitude is insufficient forindividual integrals. In particular, some of the integrals require algebraicletters involving four-mass box square-root singularities. We point out thatthese algebraic letters are relevant at the amplitude level directly startingwith N$^2$MHV amplitudes even at one loop.

Bootstrapping two-loop Feynman integrals for planar N=4 sYM