In a time of 16 minutes and 28 seconds, I stopped the 15C and didn't get an answer as a result. Then I set the 15C to a fix of 4 and attempted to integrate the integral again.
In a time of 2 minutes and 42 seconds, I got the answer of 1.99. I set the HP-15C to a fix of 2 and integrated the integral from 0 to 1. Next I tested your integral on my Dad's (hardware 1987) HP-15C. I used the integral function that is on the key and I built the equation on the stack with RPN. I did not use the Equation Writer and I was in RPN mode. mode on the HP 50g with a fix of 4, I get 1.9999 in a matter of 3 to 4 minutes. In exact mode, I receive the answer of 2 immediately. I tested your integral on my (hardware) HP 50g.