To my best knowledge, elliptic integrals can not be solved without methods from complex analysis.
Also integrals involving the residue theorem to solve them seem to be hard to solve with other methods but some of them can also be solved without using the residue theorem.
But your questions isn't well-posed: Just because we don't know yet how to solve an integral without complex analysis doesn't mean that there cannot be any way to do so.
You should therefore probably ask: Are there integrals for which the only known ways to solve it are methods from complex analysis?
Also note that if there is a nice way to solve an integral using complex analysis, people will most likely not search for a way to solve these integrals by real (and probably very complicated) methods.