WolframAlpha can compute (simple) Laurent series:
https://www.wolframalpha.com/input/?i=series+sin(z%5E-1)
Series[Sin[z^(-1)], {z, 0, 5}]
1/z-1/(6 z^3)+1/(120 z^5)+O((1/z)^6) (Laurent series) (converges everywhere away from origin)
Unfortunately, more “complex” (pun intended) Laurent series are not possible for WolframAlpha.
Mathtuition88 
Wolfram aka. Alpha Mutt can do that simple example that you showed…the problem wasn’t Alpha Mutt…Try this…10xπ-√(987)..that shouldn’t be too difficult… Wolfram can handle more than you can imagine…it can also be beaten into submission.
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