In [1]: from sympy.core.cache import clear_cache

In [2]: t = Symbol('t', positive=True)

In [3]: integrate(sin(x)/x, (x, 0, t), meijerg=True)
Out[3]: Si(t)

In [4]: timeit clear_cache();integrate(sin(x)/x, (x, 0, t), meijerg=True)
10 loops, best of 3: 79.6 ms per loop

In [5]: integrate(-cos(x)/x, (x, t, oo), meijerg=True)
Out[5]: 
       ⎛   ⎛    ⎛     ⎛   ⅈ⋅π⎞              ⎞               ⎞                                    ⎞
       ⎜   ⎜    ⎜     ⎜   ───⎟              ⎟               ⎟                                    ⎟
       ⎜   ⎜    ⎜     ⎜    2 ⎟           ⅈ⋅π⎟               ⎟                                    ⎟
       ⎜   ⎜  4⋅⎜- log⎝t⋅ℯ   ⎠ + Ci(t) + ───⎟               ⎟                                    ⎟
       ⎜ 2 ⎜    ⎝                         2 ⎠   4⋅EulerGamma⎟      ⎛ 2⎞                          ⎟
       ⎜t ⋅⎜- ─────────────────────────────── + ────────────⎟      ⎜t ⎟                          ⎟
       ⎜   ⎜                  2                       2     ⎟   log⎜──⎟                          ⎟
   ⎽⎽⎽ ⎜   ⎝                 t                       t      ⎠      ⎝4 ⎠   2⋅log(2)   2⋅EulerGamma⎟
-╲╱ π ⋅⎜───────────────────────────────────────────────────── - ─────── - ──────── - ────────────⎟
       ⎜                           ⎽⎽⎽                             ⎽⎽⎽       ⎽⎽⎽          ⎽⎽⎽    ⎟
       ⎝                       2⋅╲╱ π                            ╲╱ π      ╲╱ π         ╲╱ π     ⎠
──────────────────────────────────────────────────────────────────────────────────────────────────
                                                2                                                 

In [6]: expand(_)
Out[6]: Ci(t)

In [7]: timeit clear_cache();integrate(-cos(x)/x, (x, t, oo), meijerg=True)
1 loops, best of 3: 380 ms per loop

In [8]: integrate(exp(-x)/x, (x, t, oo))
Out[8]: 
  ⎛       ⎛   ⅈ⋅π⎞     ⎛   ⅈ⋅π⎞             ⎞                      
  ⎜  - log⎝t⋅ℯ   ⎠ + Ei⎝t⋅ℯ   ⎠   EulerGamma⎟                      
t⋅⎜- ────────────────────────── + ──────────⎟ - log(t) - EulerGamma
  ⎝              t                    t     ⎠                      

In [9]: expand(_)
Out[9]: 
    ⎛   ⅈ⋅π⎞      
- Ei⎝t⋅ℯ   ⎠ + ⅈ⋅π

In [10]: _.rewrite(expint)
Out[10]: E₁(t)

In [11]: timeit clear_cache();integrate(exp(-x)/x, (x, t, oo))
1 loops, best of 3: 233 ms per loop

In [12]: integrate(exp(-z*x)/x**y, (x, 1, oo), meijerg=True, conds='none')
Out[12]: 
 y - 1           2⋅ⅈ⋅π⋅(-y + 1)  2⋅ⅈ⋅π⋅(y - 1)                                                                    
z     ⋅(-y + 1)⋅ℯ              ⋅ℯ             ⋅Γ(y - 1)⋅γ(-y + 1, z)    y - 1  ⅈ⋅π⋅(y - 1)  -ⅈ⋅π⋅y + ⅈ⋅π          
──────────────────────────────────────────────────────────────────── + z     ⋅ℯ           ⋅ℯ            ⋅Γ(-y + 1)
                                Γ(y)                                                                              

In [16]: simplify(unpolarify(_12.rewrite(expint).expand(func=True)))
Out[16]: expint(y, z)

In [17]: timeit clear_cache();integrate(exp(-z*x)/x**y, (x, 1, oo), meijerg=True, conds='none')
1 loops, best of 3: 323 ms per loop

In [18]: integrate(sinh(x)/x, x, meijerg=True)
Out[18]: Shi(x)

In [19]: u = Symbol('u', polar=True)

In [20]: integrate(cosh(x)/x, meijerg=True)
Out[20]: 
             ⎛ 2⎞         
             ⎜x ⎟         
          log⎜──⎟         
             ⎝4 ⎠         
-log(x) + ─────── + Chi(x)
             2            

In [21]: expand(_)
Out[21]: 
             ⎛ 2⎞                  
          log⎝x ⎠                  
-log(x) + ─────── + Chi(x) - log(2)
             2                     

In [22]: integrate(cosh(u)/u, meijerg=True)
Out[22]: 
             ⎛ 2⎞         
             ⎜u ⎟         
          log⎜──⎟         
             ⎝4 ⎠         
-log(u) + ─────── + Chi(u)
             2            

In [23]: expand(_)
Out[23]: Chi(u) - log(2)

In [24]: integrate(si(x), x, meijerg=True)
Out[24]: x⋅Si(x) + cos(x)

In [25]: _.diff(x)
Out[25]: Si(x)

In [26]: integrate(_24, x, meijerg=True)
Out[26]: 
 2                          
x ⋅Si(x)   x⋅cos(x)   sin(x)
──────── + ──────── + ──────
   2          2         2   

In [27]: _.diff(x, 2)
Out[27]: Si(x)

In [31]: laplace_transform(ci(x), x, s)
Out[31]: 
⎛    ⎛ 2    ⎞         ⎞
⎜-log⎝s  + 1⎠         ⎟
⎜────────────, 0, True⎟
⎝    2⋅s              ⎠

In [32]: laplace_transform(si(x), x, s)
Out[32]: 
⎛     ⎛      ⅈ⋅π                  ⎞             ⎞
⎜     ⎜      ───                  ⎟             ⎟
⎜     ⎜       2                   ⎟             ⎟
⎜     ⎜   s⋅ℯ              1      ⎟             ⎟
⎜ⅈ⋅log⎜──────────── + ────────────⎟             ⎟
⎜     ⎜     ⅈ⋅π            ⅈ⋅π    ⎟             ⎟
⎜     ⎜     ───            ───    ⎟             ⎟
⎜     ⎜      2              2     ⎟             ⎟
⎜     ⎝- s⋅ℯ    + 1   - s⋅ℯ    + 1⎠   π         ⎟
⎜────────────────────────────────── + ─         ⎟
⎜                2                    2         ⎟
⎜──────────────────────────────────────, 0, True⎟
⎝                  s                            ⎠


In [33]: inverse_laplace_transform(_31[0], s, u)
Out[33]: Heaviside(u)⋅Ci(u)

In [35]: timeit clear_cache(); laplace_transform(ci(x), x, s)
1 loops, best of 3: 413 ms per loop

In [36]: timeit clear_cache(); inverse_laplace_transform(_31[0], s, u)
1 loops, best of 3: 437 ms per loop

In [38]: integrate(e1(x)*besselj(0, x), (x, 0, oo), meijerg=True)
Out[38]: 
       ⎛ ⅈ⋅π⎞
       ⎜ ───⎟
       ⎜  2 ⎟
-ⅈ⋅asin⎝ℯ   ⎠

In [39]: integrate(e1(x)*besselj(1, x), (x, 0, oo), meijerg=True)
Out[39]: 
   ⎛       ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎞
   ⎜      ╱      ⅈ⋅π ⎟
   ⎜1   ╲╱  1 - ℯ    ⎟
log⎜─ + ─────────────⎟
   ⎝2         2      ⎠

In [40]: timeit clear_cache(); integrate(e1(x)*besselj(1, x), (x, 0, oo), meijerg=True)
1 loops, best of 3: 394 ms per loop

In [78]: integrate(e1(x)**2, (x, 0, oo), meijerg=True)
Out[78]: 2⋅log(2)

In [79]: timeit clear_cache();integrate(e1(x)**2, (x, 0, oo), meijerg=True)
1 loops, best of 3: 383 ms per loop

In [8]: integrate(si(x)/x**3, (x, 1, oo), meijerg=True)
Out[8]: 
      ⎛            ⎛cos(1)   sin(1)   3⋅Si(1)⎞⎞
      ⎜    ⎽⎽⎽   8⋅⎜────── + ────── + ───────⎟⎟
  ⎽⎽⎽ ⎜  ╲╱ π      ⎝  8        8         8   ⎠⎟
╲╱ π ⋅⎜- ───── + ─────────────────────────────⎟
      ⎜    2                   ⎽⎽⎽            ⎟
      ⎝                      ╲╱ π             ⎠
───────────────────────────────────────────────
                       4                       

In [9]: expand(_)
Out[9]: 
  π   cos(1)   sin(1)   3⋅Si(1)
- ─ + ────── + ────── + ───────
  8     4        4         4