In [63]: integrate(exp(-x**2)*besselj(0, x), (x,0,oo), meijerg=True) Out[63]: ⎽⎽⎽ -1/8 ╲╱ π ⋅ℯ ⋅besseli(0, -1/8) ──────────────────────────── 2 In [82]: timeit clear_cache();integrate(exp(-x**2)*besselj(0, x), (x,0,oo), meijerg=True) 1 loops, best of 3: 604 ms per loop In [9]: integrate(besselj(0,x)*besselj(1,x)*besselk(0,x), (x, 0, oo), meijerg=True) Out[9]: ⎛ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎞ ⎜ ╱ ⅈ⋅π ⎟ ⎜1 ╲╱ 1 - 4⋅ℯ ⎟ log⎜─ + ───────────────⎟ ⎝2 2 ⎠ ──────────────────────── 2 In [10]: _.n() Out[10]: 0.240605912529802 In [13]: timeit clear_cache();integrate(besselj(0,x)*besselj(1,x)*besselk(0,x), (x,0,oo), meijerg=True) 1 loops, best of 3: 1.03 s per loop In [1]: integrate(besselj(0,x)*besselj(1,x)*exp(-x**2), (x, 0, oo), meijerg=True) Out[1]: -1/2 ℯ ⋅besseli(0, -1/2) 1 - ────────────────────── + ─ 2 2 In [3]: timeit clear_cache();integrate(besselj(0,x)*besselj(1,x)*exp(-x**2), (x, 0, oo), meijerg=True) 1 loops, best of 3: 735 ms per loop In [4]: Integral(besselj(0,x)*besselj(1,x)*exp(-x**2), (x, 0, oo)).n() Out[4]: 0.177482364775425 In [5]: _1.n() Out[5]: 0.177482364775425 In [17]: integrate(erf(x)*besselj(0,x)**2/sqrt(x), (x,0,oo), meijerg=True) Out[17]: ╭─╮2, 3 ⎛1/4, 1/2, 0 3/4 │ ⎞ │╶┐ ⎜ │ 1⎟ ╰─╯4, 5 ⎝ -1/4, 0 0, 0, 0 │ ⎠ ────────────────────────────────── 2⋅π In [18]: hyperexpand(_, allow_hyper=True) Out[18]: ⎽⎽⎽ 2 ⎽⎽⎽ ┌─ ⎛1/4, 1/2, 3/4 │ ⎞ ╲╱ π ⋅Γ(1/4) ╲╱ π ⋅Γ(-1/4)⋅ ├─ ⎜ │ -1⎟ + ───────────── 3╵ 3 ⎝ 1, 1, 5/4 │ ⎠ 2 Γ(3/4) ─────────────────────────────────────────────────────── 2⋅π In [19]: _.n() Out[19]: 1.17432977134715 In [20]: timeit clear_cache();integrate(erf(x)*besselj(0,x)**2/sqrt(x), (x,0,oo), meijerg=True) 1 loops, best of 3: 917 ms per loop