Question 1121154
https://www.symbolab.com/solver/step-by-step/%5Cint%5Cleft(%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%5E%7B2%7D%2B16%7D%7D%5Cright)dx
check this out. This may solve your problem. If you have any question about the proofs of the specific formulae, just ask the search engines-there are abundant of them awaiting.

Special mention: it seems that your question turns out to be arcsinh(x/4) on Methematica. But it seems that the result given by the two sites are equivalent. I derived both using both tools, and got the same result.