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For instance

$$\int \dfrac{1}{x^2 - a^2}dx$$

The easiest method is fractions, but suppose not. Most of the time we use $x=a\tan(\theta)$, but we could also use $x = a\cosh(t)$

Most of the time the latter substitution is often omitted in many calculus classes. Why?

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I don't have any definitive evidence of this, but I'd say it likely stems in part from the familiarity most students have with the regular trigonometric functions and the relative unfamiliarity that they have with the hyperbolic trigonometric functions. Trig is typically a course that students will take before they take a calculus sequence. It's not uncommon for professors to omit discussions of hyperbolic trig functions in introductory calc to save on time (in my experience -- ymmv) –  anonymous Jan 6 '13 at 23:10
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But then most people graduate without learning them, in fact I attest most people in my institutions graduate without even know what a hyperbolic function is! –  sidht Jan 6 '13 at 23:11
    
I'd guess that most professors take the a stance like "those who need them will come to them at some point. Those who don't will appreciate that I skipped them. I have X amount of material to get through, and there's never enough time in a semester. I need to cut something out. Hyperbolic functions aren't necessary for going on in calculus." There are a lot of topics that could be covered in intro to calc. There's just not enough time to do everything. Presumably, someone who understands calc well would have little trouble picking up hyperbolic trig functions later as needed. –  anonymous Jan 6 '13 at 23:14
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In your example, probably $a\sec\theta$ is intended. The main advantage of trig functions is that we are familiar with double angle identities, and much less familiar with their analogues. –  André Nicolas Jan 6 '13 at 23:17
    
I probably picked a bad example, but anyone have a good example where a hyperbolic sub is much better than a trig sub? –  sidht Jan 6 '13 at 23:28
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