Delay introduced by parallel R/C components
Presumably your input signal level is in the 1V+ range peak to get 5V out, so the diffusion isolation junctions in the op-amp front end are conducting on negative half cycles (there's a nominal 400\$\Omega\$ resistor in series), which will shift the average input signal upwards (with a long time constant on decay due to R5*C1) and may cause some other weird behavior. 

You could try paralleling R5 with a Schottky diode, but I think a better solution is a rectifier circuit designed for proper single-supply operation, followed by a low-pass filter.