In [2]: laplace_transform(si(x), x, s)
Out[2]: 
⎛             π⋅s         ⎞
⎜-s⋅atan(s) + ───         ⎟
⎜              2          ⎟
⎜────────────────, 0, True⎟
⎜        2                ⎟
⎝       s                 ⎠

In [3]: integrate(e1(x)*besselj(0, x), (x, 0, oo), meijerg=True)
Out[3]: 
   ⎛      ⎽⎽⎽⎞
log⎝1 + ╲╱ 2 ⎠

In [4]: integrate(expint(2,x)*besselj(3,x), (x,0,oo), meijerg=True)
Out[4]: 
          ⎛      ⎽⎽⎽⎞       ⎽⎽⎽
-3 - 3⋅log⎝1 + ╲╱ 2 ⎠ + 4⋅╲╱ 2 

In [5]: laplace_transform(shi(x), x, s, noconds=True)
Out[5]: 
⎧   acoth(s)             │ 2│
⎪   ────────     for 1 < │s │
⎪      s                     
⎨                            
⎪atanh(s)   ⅈ⋅π              
⎪──────── - ───   otherwise  
⎩   s       2⋅s              

In [6]: integrate(1/x/sqrt(1-x**2), x)
Out[6]: 
⎧      ⎛1⎞          │1 │
⎪-acosh⎜─⎟  for 1 < │──│
⎪      ⎝x⎠          │ 2│
⎪                   │x │
⎨                       
⎪      ⎛1⎞              
⎪ⅈ⋅asin⎜─⎟   otherwise  
⎪      ⎝x⎠              
⎩                       



In [9]: integrate(x/sqrt(1-x**4), x, meijerg=True)
Out[9]: 
⎧        ⎛ 2⎞              
⎪-ⅈ⋅acosh⎝x ⎠          │ 4│
⎪────────────  for 1 < │x │
⎪     2                    
⎨                          
⎪      ⎛ 2⎞                
⎪  asin⎝x ⎠                
⎪  ────────     otherwise  
⎩     2