In standard calculus, many functions do not admit antiderivatives that can be written using elementary functions. Classical results in mathematics show that no algebraic manipulation can overcome this limitation. However, the absence of an elementary antiderivative does not mean that such integrals are without structure or representation. This work introduces a new symbolic approach for handling non-elementary integrals through the definition of a puncture operator. Rather than attempting to force an elementary closed form, the puncture operator compresses the infinite summation structure that naturally arises in these integrals into a single, well-defined symbolic object. This object fully encodes the antiderivative while avoiding the need to explicitly display long or impractical infinite series. The puncture operator is constructed explicitly and is shown to preserve convergence, remain invariant under partition refinement, and provide a canonical representation of series-based antiderivatives. The method is demonstrated in detail on the non-elementary integral ∫ �� ���� where the infinite expansion of the antiderivative is compressed into a compact symbolic form without loss of information. This framework does not contradict known impossibility results in symbolic integration. Instead, it offers a complementary perspective in which non-elementary antiderivatives are treated as structured symbolic objects rather than unsimplifiable expressions. The approach provides a new way to represent and manipulate integrals that lie beyond the reach of elementary calculus.