Lot's of great discussion in the other answers, and the comments by Dave Renfro were on point as usual. But, I'll add my two cents here:
I think the reason that hyperbolic functions are largely ignored is multifaceted and likely to worsen as more and more educators buy into tempting arguments to downplay topics like trigonometric substitution and partial fractions because those calculations are not interesting and can be easily done with technology. Also, most calculators don't have hyperbolic trig functions, so why would you use them as an example when using the calculator is baked into the teaching of many.
My own experience with calculus was that hyperbolic functions were not emphasized and perhaps not even mentioned. Now, probably if I went back and looked I could find hyperbolic sine and cosine somewhere in the textbooks I used. But, it didn't really get my notice until in the midst of my course on Special Relativity. The professor used cosh and sinh and tanh as follows:
$$ \tanh \phi = \beta \ \ \ \ \& \ \ \ \ \cosh \phi = \gamma \ \ \ \ \& \ \ \ \ \sinh \phi = \gamma \beta $$
where $\beta = v/c$ and $c$ is the speed of light whereas $v$ is the speed of the inertial frame. Here $\gamma = \frac{1}{\sqrt{1-\beta^2}}$ so the assignment of rapidity is reasonable as
$$\cosh^2 \phi - \sinh^2 \phi = \gamma^2-\gamma^2\beta^2 = \gamma^2(1-\beta^2)=1. $$ The quantity $\phi$ is called the rapidity. As a student, the rapidity was fascinating, new, and followed the simple rule rapidities add. In contrast, velocities do not add in Special Relativity. Much to the horror of our professor, we explained didn't know what "cosh" and "sinh" were. That was my introduction. Later, as I taught Calculus and Differential Equations, and Complex Analysis I learned that not only were cosh and sinh essential topics, they are necessary to take a complete view of the exponential function.
Let me give an example from the problem of inverse Laplace transforms.
$$ \mathcal{L}^{-1} \left( \frac{s}{s^2+4s+13}\right) = \mathcal{L}^{-1} \left( \frac{s+2-2}{(s+2)^2+9}\right) = e^{-2t}\cos (3t) -\frac{2}{3}e^{-2t}\sin (3t) $$
Typically, when the denominator factors over $\mathbb{R}$ then partial fractions is used in the standard course. But, there is no need for a problem which parallels the one above. In fact,
$$ \mathcal{L}^{-1} \left( \frac{s}{s^2+4s-5}\right) = \mathcal{L}^{-1} \left( \frac{s+2-2}{(s+2)^2-9}\right) = e^{-2t}\cosh (3t) -\frac{2}{3}e^{-2t}\sinh (3t) $$
So, with both cos and cosh as well as sin and sinh, the technique of completing the square leads to a completely symmetric treatment for the non-repeated root case.
It turns out there are generalizations of both trigonometric and hyperbolic trigonometric functions which appear in connection with the exponential over an algebra which is characteristic of a given $n$-th order ODE. The component functions of this exponential form a fundamental solution set. Cosine and sine and cosh and sinh are just the lowest order examples of this pattern.
There is much that can be said about Calculus II as well, we see trigonometric substitution used as a method to eliminate radicals. Hyperbolic functions can be leveraged to make the same reduction. There are problems which are simple with the hyperbolic technique which are quite complicated in the trig. technique. Likewise, there are other problems which are easy in trig. substitution which are dastardly in the hyperbolic approach. Both methods should be used as the reinforce one another.
There is a whole theory of hyperbolic trigonometry which mirrors the commonly known theory of circular trigonometry. I think Saul Stahl's text A Gateway to Modern Geometry is a good place to look if you want to learn more. I suspect working through hyperbolic trigonometry would give a deeper understanding of the usual circle-based trig.
