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I am trying to integrate various expressions with Bessel functions. In order to learn basics with Mathematica, I have decided to try a simple case to see what is going on.

I did :

Assuming[ Element[n, Integers] && n > 0 && c > 0 && Im[c] == 0, 
 Integrate[BesselJ[n, c r] r, {r, 0, 1}]] 

I was expecting to see an answer in terms of Bessel functions and their derivatives. Instead, I get an answer in terms of Hypergeometric functions. Is there any way to force Mathematica to express the answer in terms of Bessel functions?

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For general n, no, but if you choose a value for n and use FunctionExpand, you will get simpler answers.

$Assumptions = Element[n, Integers] && n > 0 && c > 0

int = Integrate[BesselJ[n, c r] r, {r, 0, 1}];

For example the first 3 values of n gives

Table[int, {n, 1, 3}] // FunctionExpand

(* {(Pi*StruveH[0, c]*BesselJ[1, c])/(2*c) - 
   (Pi*StruveH[1, c]*BesselJ[0, c])/(2*c), 
  2/c^2 - (2*BesselJ[0, c])/c^2 - BesselJ[1, c]/c, 
  ((3*Pi*c*StruveH[0, c] - 16)*BesselJ[1, c])/(2*c^2) + 
   ((8 - 3*Pi*StruveH[1, c])*BesselJ[0, c])/(2*c)} *)
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If you want a Maple 2018 answer:

Maple code:

sol := `assuming`([int(BesselJ(n, c*r)*r, r = 0 .. 1)], [n::posint, c > 0, Im(c) = 0]);

[-(1/2)*Pi*(BesselJ(0, c)*StruveH(1, c)-BesselJ(1, c)*StruveH(0, c))/c, - 
(c*BesselJ(1, c)+2*BesselJ(0, c)-2)/c^2, -(1/2)*(3*BesselJ(0, c)*StruveH(1, 
c)*Pi*c-3*StruveH(0, c)*BesselJ(1, c)*Pi*c-8*c*BesselJ(0, c)+16*BesselJ(1, 
c))/c^2]

enter image description here

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