Isoparametric triangular finite elements can be formulated using "triangular coordinates" $(\zeta_1, \zeta_2, \zeta_3)$, also sometimes called area/areal/barycentric coords, or using "Cartesian coordinates" $(r,s)$ defined on a master element. These two systems can be related to each other without difficulty, and seem to be functionally equivalent.
It seems that several texts and journal papers use triangular coords $(\zeta_i)$ exclusively, particularly those that are more recent and/or mathematically oriented. On the other hand, I have older textbooks and use certain commercial FEA packages that consistently present element formulations with $(r,s)$ coordinates. I have a feeling that this might be due to historical reasons, but perhaps there is more to it.
Provided that the element geometry is restricted (straight sides, and any side nodes are uniformly spaced), I understand that triangular coords may be useful because they lend themselves to analytical integration, and thus avoid potentially costly numerical integration procedures. However, when these conditions are not met, numerical integration must be used. In that case, is there any benefit from using triangular coordinates, conceptual or computational?