When I type in Wolfram Alpha, say,
or (OK this next one can easily be obtained in Mathematica, still, I have run into Bessel function expressions that cannot be simplified by Mathematica, but get a decent "alternative expression" through W|A)
D[A^mu BesselK[mu, A r], A]
I get several alternative forms of these, but I cannot seem to obtain them simply from pure Mathematica (i.e. no W|A integration stuff). Am I missing something or is Wolfram Alpha just smarter in these things? The alternative expressions have helped me quite a lot, so it would be nice to have them in my toolbox without having to access Alpha.