Well, mail was not invulnerable to bodkins. Perhaps plate (will this argument ever be agreed upon? ).
Quite a few of the archers which Muslim armies contained did use the composite bow, a weapon very capable of piercing mail. As with all infantry from the late 9th to late 13th centuries, archers were not utilized to their full capacity. In wet muddy conditions, the lightness of archers attire would enable them to "dance" around heavier opponents, they were good light infantry.