And here is the latest and greatest
In[4]:= $Version
Out[4]= "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"
In[5]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, 1/2, 3/2}]
Out[5]= 2/9 (8 + \[Pi]^(
3/2) (9 MeijerG[{{1/2, 1}, {3/4, 5/4, 2}}, {{1, 1, 1}, {0, 0, 3/4,
5/4}}, 1/2, 1/2] -
MeijerG[{{1/2, 1}, {3/4, 5/4, 2}}, {{1, 1, 1}, {0, 0, 3/4, 5/
4}}, 3/2, 1/2]))
In[6]:= % // N
Out[6]= 1.45921
In[7]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, .5, 1.5}]
Out[7]= 1.45921
Can you explain this?
It's a neverending story to explain software (implementations). Nevertheless Mathematica has an overwhelming strong relation to science and there it's quite usual to switch to numerical results (in contrast to symbolic ones) if the user chooses to give numerical input. Compare
In[8]:= 1/2
Out[8]= 1/2
In[9]:= 1/2.
Out[9]= 0.5
nothing obscure in that.