For background: What’s the general application area of the code in question? Astronomy?
To my knowledge, complete elliptic integrals are not part of the CUDA math library, but I would suggest double checking the documentation. If you can’t find them in the docs, you might want to file an enhancement request with NVIDIA for adding them. An enhancement request will not guarantee that they will materialize any time soon, but without an enhancement request it is pretty much guaranteed they will not do so.
These functions were added in C++2017, and I am generally not sure where NVIDIA is with C++2017 support in CUDA. It is likely still a work in progress. Does the Boost library have GPU support? If so, that may be your your best bet for getting elliptic integrals to work in device code.
Looking at the Wikipedia description of these functions, it looks like creating accurate and robust implementations of these functions would take a considerable amount of work (man months).