Problem with symbolic math integration of besselk times exponent

According to Mathematica and the G&R book, the integral

where K0 is the modified bessel function of the second kind, has a closed form solution that can be expressed using the Meijer G function or the hypergeometric function. However, Matlab's symbolic integrations outputs,

syms x
>> I = int(besselk(0,x/a) * exp(-x/a),x)
 
I =
 
x*exp(-x/a)*(besselk(0, x/a) - besselk(1, x/a))

which is obviously the wrong answer. Matlab itself confirms that, by the way, since,

diff(I,x)
 
ans =
 
exp(-x/a)*(besselk(0, x/a) - besselk(1, x/a)) + x*exp(-x/a)*(besselk(0, x/a)/a - besselk(1, x/a)/a + besselk(1, x/a)/x) - (x*exp(-x/a)*(besselk(0, x/a) - besselk(1, x/a)))/a

What am I doing wrong?

Thanks!

Answer 1

[Walter Roberson]

MATLAB is correct, that is one of the ways of expressing the integral.

convert(BesselK(v, z), MeijerG);
         1        /          [[1      1  ]    ]  1  2\
         - MeijerG|[[], []], [[- v, - - v], []], - z |
         2        \          [[2      2  ]    ]  4   /

What am I doing wrong?

You failed to simplify() the differentiation.