  Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$
with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$,
and $\omega$ a rational differential $1$-form. Assume that $H\omega$ 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
$\int H\omega= f(F(x))+H(x)R(x)$. 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
$H\mathbb{K}[x,1/(SD)]$ for a given $H$, $D$ being the denominator of $dH/H$
and $S\in\mathbb{K}[x]$ 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.
