This is something I've considered before as well; in fact, I wonder why we aren't forced to learn it in school, considering we spend years in school, going to lectures and taking down notes of people talking as quickly as we can write.
In terms of computing, though, I think it would suffer many of the problems of handwriting recognition - wobbly lines, different sizes of people's writings, pens lifting off the page at different times - and some more of it's own.
For instance, Pitman shorthand comes in different versions - every few years they change the standard to try to make it more concise. Many of the shapes are in relative positions so the same thing above/on/below the line have different meanings, and the extra informational signals such as dots and dashes can be left off by more advanced people without changing the meaning.
Worse than that is that it's short because it's phonetic - so not only does the computer trying to recognise it have to do everything it does for handwriting recognition, but it also has another layer to do first - what sounds are you using and what words are you trying to write from them?
Instead of (pen strokes -> words), you get (pen strokes -> phonetic sounds -> words). Also, Pitman shorthand is pressure sensitive - light strokes for unvoiced sounds and heavy strokes for voiced sounds, IIRC.
Worse even than that, it's short because it allows you to leave information out where you know what it will be and to develop your own shortcuts where useful to you. You might abbreviate your name to "Sh" with {name indicating symbol}, or use "em eff ing" for manufacturing, or have a symbol for "Our newest product" or write "would like know b prices" for "we would like to know your best prices".
So now you have pen strokes -> sounds -> filling in missing sounds -> words -> filling in missing words -> text.
Seems useful for taking notes in a hurry and then writing them up in full yourself the same day or week, less useful for writing for other people or future reference, and even less useful for computer recognition.