Equivalent definition of locally integrable function

Question

If we know that $\int_{U} f\cdot \phi<\infty$ for all test functions $\phi\in C_c^\infty(U)$ where $U\subset \mathbb{R^n}$ is some open set, can we conclude that $\int_U |f\cdot \phi|<\infty$ for all test functions as well?