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When I searched for "pane docked" using the search box, I got 3 answers. When I repeated the search in Google using "site:mathematica.stackexchange.com pane docked", I got 3 pages.

What would it take to have an option in the search box to use Google engine?

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    "What would it take to have an option in the search box to use Google engine?" — the chances of that happening are little to none. There are alternatives (such as directly searching in Google) and plugins such as this one (for Chrome) that lets you search the current site via Google (essentially, saves you the trouble of typing site:foo.com each time). – rm -rf Sep 26 '13 at 23:35
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I wouldn't recommend trying to replace the site search with Google, even if you can get a userscript that does this. There are a number of search flags that will work with the integrated search that will not work with Google. You can search for specific tags with [tagname], restrict the search to questions (is:question) or answers (is:answer), search within code fields (code:), search only accepted answers (isaccepted:1) etc. See How do I search? for more.

Instead I suggest you make searching using Google more convenient. rm -rf referenced a Chrome plugin for site searches. You can also (in Firefox at least, but I believe similarly elsewhere) use "keywords" such that typing mma mysearch in the location bar will search the site for mysearch. In Firefox this is done by creating a bookmark with the keyword mma and the URL:

http://www.google.com/search?&q=site:mathematica.stackexchange.com%20%s

Note the %s: this will be filled by e.g. mysearch as described above. Multi-word searches are possible.

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