Is there a CAS that can automatically apply the following simplification:
Given $I_k = \int_a^b f_k(x) dx$, rewrite
$$\sum_{k=1}^K I_k^m = \int_{[a,b]^m}\sum_{k=1}^K \prod_{i=1}^m f_k(x_i) dx_1\ldots dx_m$$
So e.g. $\Big(\int f(x)dx \Big)^2 + \Big(\int g(x)dx\Big)^2 =\iint [f(x)f(y)+g(x)g(y)] dxdy$