Are there any ellipses with known exact circumferences and major and minor axes $a$ and $b$ and $a \neq b$?

Question

I'm new to the world of ellipses and have come to understand that calculating the exact circumference of an ellipse is not a trivial matter that involves elliptic integrals and elliptic functions.

However, does elliptic integration produce the exact circumference, are they workable? Are there any known and proven exact ellipse measurements out there?