The shape shifting wrapper can encompass a number of different apparently magical wonders and a hospital full of scientists (and other worshippers of cause & effect) might have drastically different responses, depending on which wonders are immediately apparent.
If the patient transforms into a creature of drastically different mass and if the transformation does not make up for that mass by either absorbing or excreting appropriate quantities of matter; then a fundamental law of physics is being broken. You can expect the scientists to get a little emotional once you've violate conservation of mass and energy.
Similarly, if all of the patients clothing and possessions vanish while they are in animal form and reappear during their journey back to being bipedal; then a lesser known fundamental law joins CoM&E on the butcher block. Because that is the destruction and reconstruction of information and a slap in the face of entropy. Expect more pissed-off PHDs after that performance.
But if your shifter is well behaved and at least appears to be obeying the laws of nature, if he carefully disrobes before his bones and muscles begin changing into a comparably sized creature; then you can expect the surrounding professional care givers to show the patience, curiosity and kindness as they support, comfort and study this afflicted soul.
They will probably even give him a strong comforting sedative before the dissection begins.