Accurate real-time rendering of specular surfaces is a challenging task when considering area light source illumination. The difficulty resides in the evaluation of a surface integral for which no practical solution exists except using expensive Monte Carlo methods. Recent techniques like the most representative point approach [Drobot2014] alleviate this problem but make some accuracy trade-off to achieve real-time performance. We introduce analytic approximations for accurate real-time rendering of specular surfaces illuminated by polygonal light sources. Our solution is based on a reformulation of the contour integral [Arvo 1995] we approximate analytically with simple peak functions. In addition, using simple geometric operations, we extend the solution to handle more physically plausible BRDFs. Our solution works without any assumption on light source shape nor surface roughness, bringing real-time performances with a quality close to the ground truth.