Semiflexible polymers comprised of active Brownian particles (ABPOs) exhibit intriguing activity-driven conformational and dynamical features. Analytically, the generic properties of ABPOs can be obtained in a mean-field description applying the Gaussian semiflexible polymer model. In this article, we derive a path integral representation of the stationary-state distribution function of such ABPOs, based on the stationary-state distribution function of the normal mode amplitudes following from the Langevin equation of motion. The path integral includes characteristic semiflexible polymer contributions from entropy and bending energy, with activity dependent coefficients, and, in addition, activity-induced torsional and higher order correlations along the polymer contour. Focusing on a semiflexible polymer approximation, we determine various properties such as the tangent-vector correlation function, effective persistence length, and the mean-square end-to-end distance. The latter reflects the characteristic features of ABPOs, and good quantitative agreement is obtained with the full solution for larger activities, specifically for flexible polymers. Moreover, the approximation indicates the relevance of torsional and higher order contour correlations for the ABPO conformations. In general, the ABPO path integral illustrates how colored noise (active fluctuations) affects semiflexible polymer conformations in comparison to white noise thermal fluctuations.