Integration by popular opinion

How do first year maths students evaluate the integral

\( I = \displaystyle\int \frac{x}{\sqrt{x^2 - 1}} \,dx \)?

A good idea is to use a substitution. But which substitution? Some options are $x = sec(u)$, $x = cosh(u)$, $u^2 = x^2 - 1$ or $u = x^2 - 1$. Students in Higher Mathematics 1B were asked to try each of the substitution, and vote for their preferred substitution. They were given two additional options of "all these substitutions" or "I prefer algebra". Here is what they said:

Staff opinions were divided on the best approach to use. Norman Wildberger prefers the substitution $u = x^2 - 1$ because it's rational (and it has to be rational for Norman), Daniel Mansfield prefers the substitution \(u^2 = x^2 - 1\) because the integration is trivial \(I = \int \, du = u + C\); but Wolfgang Schief says this approach is cheating because the substitution presumes knowledge of the answer. Chris Tisdell backed the hyperbolic underdog \(x = cosh(u)\) because the derivative is nice to compute, and Catherine Greenhill confirmed that she does prefer algebra (but would choose the same substitution as the majority of the MATH1241 students).

Most staff avoided the circle trig substitution \(x = sec(u)\) because, in this case, it's more trouble than it's worth.