Symbolic Integration of Hyperexponential 1-Forms

Let H be a hyperexponential function in n variables x=(x₁,…,x_n) with coefficients in a field ℚ, [ℚ :ℚ ] < ∞, and ω a rational differential 1-form. Assume that Hømega is closed and H transcendental. We prove using Schanuel conjecture that there exist a univariate function f and multivariate rational functions F,R such that ∫ Hω= f(F(x))+H(x)R(x). We present an algorithm to compute this decomposition. This allows us to present an algorithm to construct a basis of the cohomology of differential 1-forms with coefficients in Hℚ [x,1/(SD)] for a given H, D being the denominator of dH/H and S ∈ ℚ [x] a square free polynomial. As an application, we generalize a result of Singer on differential equations on the plane: whenever it admits a Liouvillian first integral I but no Darbouxian first integral, our algorithm gives a rational variable change linearising the system.