Simplifying the one loop five graviton amplitude in type IIB string theory

We consider the D^8 R^5 and D^{10} R^5 terms in the low momentum expansion of the five graviton amplitude in type IIB string theory at one loop. They involve integrals of various modular graph functions over the fundamental domain of SL(2,Z). Unlike the graphs which arise in the four graviton amplitude or at lower orders in the momentum expansion of the five graviton amplitude where the links are given by scalar Green functions, there are several graphs for the D^8 R^5 and D^{10} R^5 terms where two of the links are each given by a derivative of the Green function. Starting with appropriate auxiliary diagrams, we show that these graphs can be expressed in terms of those which do not involve any derivatives. This results in considerable simplification of the amplitude.