Elliptic integrals

[AlexFekken]

In the first pair of calculations you are integrating over a singularity at the end of the integration internval when you use the non-trig defiinition (i.e. at x=t=1). The trig version does not have that singularity. So I suspect that the integration algorithm gives you a less accurate result because of that singularity, i.e. it is not so good at guessing the limit of the improper Riemann integral.

This seems to be comfirmed by the second pair of calculations where you are staying away from the singularity and the results are essentially the same.