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I have spent some time analysing the pendulum problem, and hence the Jacobi elliptic functions recently, and have come across what seems to me to be a slight inconsitency.
I define my $mathrmam(t|k)$ as the inverse to the integral
$$int_0^t (1-k^2 sin^2x)^-1/2 , mathrmdx = mathrmam^-1(t|k) $$
And I'm interested in the special case of $k^2 = 2$

I understand the Jacobi elliptic functions to be defined as parametrising the ellipse
$$x^2 + fracy^2b^2 = 1$$
and $k^2 = 1-b^-2$. Then parametrise the ellipse by $x = mathrmcn(t|k)$, $y = b ,mathrmsn(t|k)$. In the special case of $k = 0$ it's simple to show that the ellipse becomes a circle, and these functions reduce to standard trig functions, and for $k = 1$ that the ellipse becomes two parallel lines, and that the elliptic functions reduce to $tanh$ and $mathrmsech$.

Now let $b = i beta$, and the ellipse becomes a hyperbola. $k^2 = 1 + beta^2$, and the special case of $beta = 1$ is the rectangular hyperbola
$$x^2 - y^2 = 1$$

This corresponds to $k^2 = 2$, so my claim is that it should be possible to solve the integrals for the elliptic functions in this case to give
$$ mathrmsn(t|sqrt2) = sinh(t)$$
$$ mathrmcn(t|sqrt2) = cosh(t)$$
Which would then agree with the parametrisation of the original ellipse equation, with $b = i$

However the defining integral reduces to
$$int_0^t sqrtsec(2x) , mathrmdx = mathrmam^-1(t|sqrt2) $$
Which I don't believe to be integrable in terms of non-special functions. (I have spent quite a long time trying different substitutions and trig rearrangements on this integral and not had much luck).

Does anyone have any thoughts on this?

Hmm, something's wrong here... The function $f(z)=operatornamesn(z,sqrt2)$ is definitely not equal to $sinh z$. There's a transformation formula which gives $f(z)=operatornamesn(zsqrt2,1/sqrt2)/sqrt2$, and this is a Jacobi sn function with $0<k<1$, which (as you probably know) implies that it is periodic on the real axis and also doubly periodic in the complex plane (unlike $sinh z$).

I haven't thought about what exactly goes wrong in your reasoning, but
I think that maybe you are putting too much emphasis on the formula $x^2 + fracy^2b^2=1$. That formula alone isn't enough to define the elliptic functions, since for example $(x,y)=(cos t,b sin t)$ is another perfectly good parametrization of the ellipse, with good old trig functions.