late 14c., from Old English (Anglian) hlæhhan, earlier hliehhan, hlihhan "to laugh, laugh at; rejoice; deride," from Proto-Germanic *klakhjan (source also of Old Norse hlæja, Danish le, Old Frisian hlakkia, Old Saxon hlahhian, Middle Dutch and Dutch lachen, Old High German hlahhan, German lachen, Gothic hlahjan), from PIE *kleg-, of imitative origin (compare Latin cachinnare "to laugh aloud," Sanskrit kakhati "laughs," Old Church Slavonic chochotati "laugh," Lithuanian klagėti "to cackle," Greek kakhazein).
Originally with a "hard" -gh- sound, as in Scottish loch; the spelling remained after the pronunciation shifted to "-f."
If laugh were written as it is pronounced, laaff, there would be nothing in the word itself to put us in mind of the thing signified. The imitation begins to be felt in the guttural ach of G. lachen, and is clearly indicated in the reduplicate form of the Du. lachachen, to hawhaw or laugh loud, preserved by Kilian. [Hensleigh Wedgwood, introduction to "A Dictionary of English Etymology," 1878]
To laugh in one's sleeve is to laugh inwardly so as not to be observed:
If I coveted nowe to avenge the injuries that you have done me, I myght laughe in my slyve. [John Daus, "Sleidanes Commentaries," 1560]
"The phrase generally implies some degree of contempt, and is used rather of a state of feeling than of actual laughter" [Century Dictionary]. Related: Laughed; laugher; laughing.