In [4]: laplace_transform(shi(x), x, s)
Out[4]: 
⎛                                  ⎛⎧                           2                                                                                                                                                                                      ⎞                            
⎜                                  ⎜⎪                            ⎛│                 ⎛          2      ⎞│⎞                                                                                        ⎛│                 ⎛          2      ⎞│⎞              ⎟                            
⎜                                  ⎜⎪                            ⎜│                 ⎜polar_lift (s)   ⎟│⎟                                                                                        ⎜│                 ⎜polar_lift (s)   ⎟│⎟              ⎟                            
⎜                                  ⎜⎪                            ⎜│periodic_argument⎜──────────────, ∞⎟│⎟                                                   ⎛          2      ⎞                  ⎜│periodic_argument⎜──────────────, ∞⎟│⎟    ⎽⎽⎽⎽⎽⎽    ⎟   ⎛│                 ⎛     
⎜⎧   acoth(s)             │ 2│     ⎜⎪                            ⎜│                 ⎝      4          ⎠│⎟ │ 2│                                              ⎜polar_lift (s)   ⎟                  ⎜│                 ⎝      4          ⎠│⎟   ╱ │ 2│     ⎟   ⎜│                 ⎜     
⎜⎪   ────────     for 1 < │s │, 0, ⎜⎪                       -sin ⎜──────────────────────────────────────⎟⋅│s │                         for periodic_argument⎜──────────────, ∞⎟ = 0 ≠ 0 ∨ 0 < cos⎜──────────────────────────────────────⎟⋅╲╱  │s │  - 1⎟ ∧ ⎜│periodic_argument⎜1 + ─
⎜⎪      s                          ⎜⎪                            ⎝                  2                   ⎠                                                   ⎝      4          ⎠                  ⎝                  2                   ⎠              ⎟   ⎜│                 ⎜     
⎜⎨                                 ⎜⎨                                                                                                                                                                                                                  ⎟   ⎝│                 ⎝    p
⎜⎪atanh(s)   ⅈ⋅π                   ⎜⎪   ⎛│                 ⎛          2      ⎞│⎞                                                                                                                                                                       ⎟                            
⎜⎪──────── - ───   otherwise       ⎜⎪   ⎜│                 ⎜polar_lift (s)   ⎟│⎟                                                                                                                                                                       ⎟                            
⎜⎩   s       2⋅s                   ⎜⎪   ⎜│periodic_argument⎜──────────────, ∞⎟│⎟    ⎽⎽⎽⎽⎽⎽     ⎛                 ⎛          2      ⎞⎞                                                                                                                  ⎟                            
⎜                                  ⎜⎪   ⎜│                 ⎝      4          ⎠│⎟   ╱ │ 2│      ⎜                 ⎜polar_lift (s)   ⎟⎟                                                                                                                  ⎟                            
⎜                                  ⎜⎪sin⎜──────────────────────────────────────⎟⋅╲╱  │s │ ⋅sign⎜periodic_argument⎜──────────────, ∞⎟⎟                   otherwise                                                                                      ⎟                            
⎜                                  ⎜⎪   ⎝                  2                   ⎠               ⎝                 ⎝      4          ⎠⎠                                                                                                                  ⎟                            
⎝                                  ⎝⎩                                                                                                                                                                                                                  ⎠                            

                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                  ⎛   ⎛│                 ⎛          2      ⎞│⎞                      
                                                                                                                                                                                                                  ⎜   ⎜│                 ⎜polar_lift (s)   ⎟│⎟                      
     ⅈ⋅π        ⎞│                                                           ⎞   ⎛      ⅈ⋅π                                     ⎞   ⎛                                                             ⅈ⋅π         ⎞   ⎜   ⎜│periodic_argument⎜──────────────, ∞⎟│⎟    ⎽⎽⎽⎽⎽⎽            
    ℯ           ⎟│       │                 ⎛     -ⅈ⋅π           2      ⎞│    ⎟   ⎜    -ℯ                 -ⅈ⋅π           2       ⎟   ⎜│                 ⎛     -ⅈ⋅π           2      ⎞│           -ℯ            ⎟   ⎜   ⎜│                 ⎝      4          ⎠│⎟   ╱ │ 2│             
─────────────, ∞⎟│ < π ∨ │periodic_argument⎝1 + ℯ    ⋅polar_lift (s), ∞⎠│ < π⎟ ∧ ⎜────────────── ≠ 1 ∨ -ℯ    ⋅polar_lift (s) ≠ 1⎟ ∧ ⎜│periodic_argument⎝1 + ℯ    ⋅polar_lift (s), ∞⎠│ < π ∨ ────────────── ≠ 1⎟ ∧ ⎜cos⎜──────────────────────────────────────⎟⋅╲╱  │s │  - 1 = 0 ∨ 0
         2      ⎟│                                                           ⎟   ⎜          2                                   ⎟   ⎜                                                                 2       ⎟   ⎝   ⎝                  2                   ⎠                      
olar_lift (s)   ⎠│                                                           ⎠   ⎝polar_lift (s)                                ⎠   ⎝                                                       polar_lift (s)    ⎠                                                                     
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    

                                                                                                                                            ⎞
      ⎛│                 ⎛          2      ⎞│⎞              ⎞                                                                               ⎟
      ⎜│                 ⎜polar_lift (s)   ⎟│⎟              ⎟                                                                               ⎟
      ⎜│periodic_argument⎜──────────────, ∞⎟│⎟    ⎽⎽⎽⎽⎽⎽    ⎟   ⎛│                 ⎛          ⅈ⋅π        ⎞│                                ⎞⎟
      ⎜│                 ⎝      4          ⎠│⎟   ╱ │ 2│     ⎟   ⎜│                 ⎜         ℯ           ⎟│         -ⅈ⋅π           2       ⎟⎟
 < cos⎜──────────────────────────────────────⎟⋅╲╱  │s │  - 1⎟ ∧ ⎜│periodic_argument⎜1 + ──────────────, ∞⎟│ < π ∨ -ℯ    ⋅polar_lift (s) ≠ 1⎟⎟
      ⎝                  2                   ⎠              ⎠   ⎜│                 ⎜              2      ⎟│                                ⎟⎟
                                                                ⎝│                 ⎝    polar_lift (s)   ⎠│                                ⎠⎟
                                                                                                                                            ⎟
                                                                                                                                            ⎟
                                                                                                                                            ⎟
                                                                                                                                            ⎟
                                                                                                                                            ⎟
                                                                                                                                            ⎟
                                                                                                                                            ⎠

In [5]: fourier_transform(sin(x)/x, x, s, noconds=False)
Out[5]: 
⎛                                                                             ⎛⎛                                                      ⎛⎛         ⅈ⋅π              │                 ⎛             ⅈ⋅π          ⎞│    ⎞                                                              
⎜⎧                                                                   │ 2  2│  ⎜⎜│                 ⎛ 2  -ⅈ⋅π           2      ⎞│       ⎜⎜       -ℯ                 │                 ⎜            ℯ             ⎟│    ⎟   ⎛    2  -ⅈ⋅π           2          │                 ⎛      
⎜⎪                           0                             for 1 < 4⋅│π ⋅s │, ⎜⎜│periodic_argument⎝π ⋅ℯ    ⋅polar_lift (s), ∞⎠│ = π ∧ ⎜⎜─────────────────── ≠ 1 ∧ │periodic_argument⎜1 + ───────────────────, ∞⎟│ < π⎟ ∨ ⎝-4⋅π ⋅ℯ    ⋅polar_lift (s) ≠ 1 ∧ │periodic_argument⎝1 + 4⋅
⎜⎪                                                                            ⎜⎜                                                      ⎜⎜   2           2          │                 ⎜       2           2      ⎟│    ⎟                                                              
⎜⎪⎧                                  π            │ 2  2│                     ⎝⎝                                                      ⎝⎝4⋅π ⋅polar_lift (s)       │                 ⎝    4⋅π ⋅polar_lift (s)   ⎠│    ⎠                                                              
⎜⎨⎪ⅈ⋅acoth(2⋅π⋅s) - ⅈ⋅atanh(2⋅π⋅s) + ─  for 1 < 4⋅│π ⋅s │      otherwise                                                                                                                                                                                                            
⎜⎪⎨                                  2                                                                                                                                                                                                                                              
⎜⎪⎪                                                                                                                                                                                                                                                                                 
⎜⎪⎩                 π                       otherwise                                                                                                                                                                                                                               
⎝⎩                                                                                                                                                                                                                                                                                  

                                 ⎞⎞                                                      ⎞   ⎛                                                     ⎛                                                     ⎛⎛         -ⅈ⋅π             │                 ⎛            -ⅈ⋅π          ⎞│
 2  -ⅈ⋅π           2      ⎞│    ⎞⎟⎟   │                 ⎛ 2  -ⅈ⋅π           2      ⎞│    ⎟   ⎜│                 ⎛ 2  ⅈ⋅π           2      ⎞│       ⎜│                 ⎛ 2  ⅈ⋅π           2      ⎞│       ⎜⎜       -ℯ                 │                 ⎜           ℯ              ⎟│
π ⋅ℯ    ⋅polar_lift (s), ∞⎠│ < π⎠⎟⎟ ∨ │periodic_argument⎝π ⋅ℯ    ⋅polar_lift (s), ∞⎠│ < π⎟ ∧ ⎜│periodic_argument⎝π ⋅ℯ   ⋅polar_lift (s), ∞⎠│ < π ∨ ⎜│periodic_argument⎝π ⋅ℯ   ⋅polar_lift (s), ∞⎠│ = π ∧ ⎜⎜─────────────────── ≠ 1 ∧ │periodic_argument⎜1 + ───────────────────, ∞⎟│
                                 ⎟⎟                                                      ⎟   ⎜                                                     ⎜                                                     ⎜⎜   2           2          │                 ⎜       2           2      ⎟│
                                 ⎠⎠                                                      ⎠   ⎝                                                     ⎝                                                     ⎝⎝4⋅π ⋅polar_lift (s)       │                 ⎝    4⋅π ⋅polar_lift (s)   ⎠│
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    

    ⎞                                                                                             ⎞⎞⎞⎞
    ⎟   ⎛│                 ⎛   2  ⅈ⋅π           2          ⎞│           2  ⅈ⋅π           2       ⎞⎟⎟⎟⎟
 < π⎟ ∨ ⎝│periodic_argument⎝4⋅π ⋅ℯ   ⋅polar_lift (s) + 1, ∞⎠│ < π ∧ -4⋅π ⋅ℯ   ⋅polar_lift (s) ≠ 1⎠⎟⎟⎟⎟
    ⎟                                                                                             ⎟⎟⎟⎟
    ⎠                                                                                             ⎠⎠⎠⎟
                                                                                                     ⎟
                                                                                                     ⎟
                                                                                                     ⎟
                                                                                                     ⎟
                                                                                                     ⎠

In [6]: laplace_transform(besselj(a, x), x, s)
Out[6]: 
⎛                           -a - 1                                                                                                                                                                                                                                                  
⎜        ⎛     ⎽⎽⎽⎽⎽⎽⎽⎽    ⎞       ⎛        ⎽⎽⎽⎽⎽⎽⎽⎽         ⎞                                                                                                                                                                                                                      
⎜ -a - 1 ⎜    ╱     1      ⎟       ⎜ 2     ╱     1      2    ⎟                                                                                                                                                                                                                      
⎜s      ⋅⎜   ╱  1 + ──  + 1⎟      ⋅⎜s ⋅   ╱  1 + ──  + s  + 1⎟                                                                                                                                                                                                                      
⎜        ⎜  ╱        2     ⎟       ⎜     ╱        2          ⎟     ⎛                │                 ⎛          2      ⎞│    ⎞                                                                                                                                                     
⎜        ⎝╲╱        s      ⎠       ⎝   ╲╱        s           ⎠     ⎜      ⎛a⎞       │                 ⎜polar_lift (s)   ⎟│    ⎟   ⎛      ⎛a⎞   1   │                 ⎛          2          ⎞│             -1          ⎞   ⎛│                 ⎛          2          ⎞│             -1
⎜─────────────────────────────────────────────────────────────, 0, ⎜0 < re⎜─⎟ + 1 ∨ │periodic_argument⎜──────────────, ∞⎟│ = π⎟ ∧ ⎜0 < re⎜─⎟ + ─ ∨ │periodic_argument⎝polar_lift (s) + 1, ∞⎠│ < π ∨ ────────────── ≠ 1⎟ ∧ ⎜│periodic_argument⎝polar_lift (s) + 1, ∞⎠│ < π ∨ ────────
⎜                             2                                    ⎝      ⎝2⎠       │                 ⎝      4          ⎠│    ⎠   ⎜      ⎝2⎠   2                                                              2       ⎟   ⎜                                                         
⎝                            s  + 1                                                                                               ⎝                                                                 polar_lift (s)    ⎠   ⎝                                                 polar_li

                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                       ⎛                │                 ⎛          2      ⎞│    ⎞                                                                                 
                   ⎛a⎞    ⎞   ⎛           2                ⎛a⎞       │                 ⎛          1          ⎞│    ⎞         ⎛a⎞   1   ⎜      ⎛a⎞   1   │                 ⎜polar_lift (s)   ⎟│    ⎟   ⎛      ⎛a⎞   1   │                 ⎛          2          ⎞│       │           
────── ≠ 1 ∨ 0 < re⎜─⎟ + 1⎟ ∧ ⎜-polar_lift (s) ≠ 1 ∨ 0 < re⎜─⎟ + 1 ∨ │periodic_argument⎜1 + ──────────────, ∞⎟│ < π⎟ ∧ 0 < re⎜─⎟ + ─ ∧ ⎜0 < re⎜─⎟ + ─ ∨ │periodic_argument⎜──────────────, ∞⎟│ = π⎟ ∧ ⎜0 < re⎜─⎟ + ─ ∨ │periodic_argument⎝polar_lift (s) + 1, ∞⎠│ < π ∨ │periodic_ar
  2                ⎝2⎠    ⎟   ⎜                            ⎝2⎠       │                 ⎜              2      ⎟│    ⎟         ⎝2⎠   2   ⎝      ⎝2⎠   2   │                 ⎝      4          ⎠│    ⎠   ⎜      ⎝2⎠   2                                                    │           
ft (s)                    ⎠   ⎝                                      │                 ⎝    polar_lift (s)   ⎠│    ⎠                                                                                  ⎝                                                                 │           

                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
                                                                                                                                                                                                                                                                                    
      ⎛          1          ⎞│    ⎞   ⎛           2                ⎛a⎞   1   │                 ⎛          1          ⎞│    ⎞   ⎛│                 ⎛          2          ⎞│             ⎛a⎞       │                 ⎛          1          ⎞│    ⎞   ⎛           2                -1  
gument⎜1 + ──────────────, ∞⎟│ < π⎟ ∧ ⎜-polar_lift (s) ≠ 1 ∨ 0 < re⎜─⎟ + ─ ∨ │periodic_argument⎜1 + ──────────────, ∞⎟│ < π⎟ ∧ ⎜│periodic_argument⎝polar_lift (s) + 1, ∞⎠│ < π ∨ 0 < re⎜─⎟ + 1 ∨ │periodic_argument⎜1 + ──────────────, ∞⎟│ < π⎟ ∧ ⎜-polar_lift (s) ≠ 1 ∨ ──────────
      ⎜              2      ⎟│    ⎟   ⎜                            ⎝2⎠   2   │                 ⎜              2      ⎟│    ⎟   ⎜                                                       ⎝2⎠       │                 ⎜              2      ⎟│    ⎟   ⎜                                
      ⎝    polar_lift (s)   ⎠│    ⎠   ⎝                                      │                 ⎝    polar_lift (s)   ⎠│    ⎠   ⎝                                                                 │                 ⎝    polar_lift (s)   ⎠│    ⎠   ⎝                      polar_lift

                                                                                                                                        ⎞
                                                                                                                                        ⎟
                                                                                                                                        ⎟
                                                                                                                                        ⎟
                                                                                                                                        ⎟
                 ⎛a⎞    ⎞   ⎛           2                ⎛a⎞   1         -1          ⎞         ⎛a⎞       ⎛      ⎛a⎞   1         ⎛a⎞    ⎞⎟
──── ≠ 1 ∨ 0 < re⎜─⎟ + 1⎟ ∧ ⎜-polar_lift (s) ≠ 1 ∨ 0 < re⎜─⎟ + ─ ∨ ────────────── ≠ 1⎟ ∧ 0 < re⎜─⎟ + 1 ∧ ⎜0 < re⎜─⎟ + ─ ∨ 0 < re⎜─⎟ + 1⎟⎟
2                ⎝2⎠    ⎟   ⎜                            ⎝2⎠   2             2       ⎟         ⎝2⎠       ⎝      ⎝2⎠   2         ⎝2⎠    ⎠⎟
 (s)                    ⎠   ⎝                                      polar_lift (s)    ⎠                                                  ⎠