In [34]: timeit clear_cache();mellin_transform(besselj(a,x)*besselk(a,x), x, s) 1 loops, best of 3: 1.65 s per loop In [35]: timeit clear_cache();mellin_transform(besselj(a,x)*besselj(b,x), x, s) 1 loops, best of 3: 1.98 s per loop In [36]: mellin_transform(besselj(a,x)*besselk(a,x), x, s) Out[36]: ⎛ s ⎛s⎞ ⎛a s⎞ ⎞ ⎜2 ⋅Γ⎜─⎟⋅Γ⎜─ + ─⎟ ⎟ ⎜ ⎝2⎠ ⎝2 4⎠ ⎟ ⎜────────────────, (Max(-2*re(a), 0), ∞), True⎟ ⎜ ⎛a s ⎞ ⎟ ⎜ 8⋅Γ⎜─ - ─ + 1⎟ ⎟ ⎝ ⎝2 4 ⎠ ⎠ In [37]: mellin_transform(besselj(a,x)*besselj(b,x), x, s) Out[37]: ⎛ s - 1 ⎛a b s⎞ ⎞ ⎜ 2 ⋅Γ(-s + 1)⋅Γ⎜─ + ─ + ─⎟ ⎟ ⎜ ⎝2 2 2⎠ ⎟ ⎜────────────────────────────────────────────────────, (-re(a) - re(b), 1), True⎟ ⎜ ⎛ a b s ⎞ ⎛a b s ⎞ ⎛a b s ⎞ ⎟ ⎜Γ⎜- ─ + ─ - ─ + 1⎟⋅Γ⎜─ - ─ - ─ + 1⎟⋅Γ⎜─ + ─ - ─ + 1⎟ ⎟ ⎝ ⎝ 2 2 2 ⎠ ⎝2 2 2 ⎠ ⎝2 2 2 ⎠ ⎠ In [61]: simplify(integrate(besselj(a,x)*besselj(b,x)/x, (x,0,oo), conds='none')) Out[61]: ⎛π⋅a π⋅b⎞ 2⋅sin⎜─── - ───⎟ ⎝ 2 2 ⎠ ───────────────── π⋅(a - b)⋅(a + b) In [62]: timeit clear_cache(); integrate(besselj(a,x)*besselj(b,x)/x, (x,0,oo), conds='none') 1 loops, best of 3: 2.08 s per loop