We present an additive decomposition algorithm in a chain of exponential extensions. The algorithm decomposes such a chain as a direct sum of its integrable subspace and a $C$-linear subspace $V$, where $C$ is the constant subfield of the chain. Remainders are exactly elements of $V$. The minimality of remainders can be described by supports of a carefully-chosen $C$-basis for the chain. The algorithm is based on Hermite reduction for hyperexponential functions. It yields an alternative for computing elementary integrals over $F_n$. The alternative can compute the integrals that cannot be obtained by {\sc maple} nor {\sc mathematica}. This is joint work with Shaoshi Chen, Hao Du and Ziming Li