SymPy

>>> from sympy import pprint, init_printing, Symbol, sin, cos, exp, sqrt, series, Integral, Function
>>>
>>> x = Symbol("x")
>>> y = Symbol("y")
>>> f = Function("f")
>>> # pprint will default to unicode if available
>>> pprint(x ** exp(x))
 ⎛ x⎞
 ⎝ℯ ⎠
x   
>>> # An output without unicode
>>> pprint(Integral(f(x), x), use_unicode=False)
  /       
 |        
 | f(x) dx
 |        
/        
>>> # Compare with same expression but this time unicode is enabled
>>> pprint(Integral(f(x), x), use_unicode=True)
⌠        
⎮ f(x) dx
⌡     
>>> # Alternatively, you can call init_printing() once and pretty-print without the pprint function.
>>> init_printing()
>>> sqrt(sqrt(exp(x)))
   ____
4 ╱  x 
╲╱  ℯ  
>>> (1/cos(x)).series(x, 0, 10)
     2      4       6        8         
    x    5⋅x    61⋅x    277⋅x     ⎛ 10⎞
1 + ── + ──── + ───── + ────── + O⎝x  ⎠
    2     24     720     8064

>>> from sympy import init_printing, Symbol, expand
>>> init_printing()
>>>
>>> a = Symbol("a")
>>> b = Symbol("b")
>>> e = (a + b) ** 3
>>> e
(a + b)³
>>> e.expand()
a³ + 3⋅a²⋅b + 3⋅a⋅b²  + b³

>>> from sympy import Rational, pprint
>>> e = 2**50 / Rational(10) ** 50
>>> pprint(e)
1/88817841970012523233890533447265625

>>> from sympy import init_printing, symbols, ln, diff
>>> init_printing()
>>> x, y = symbols("x y")
>>> f = x**2 / y + 2 * x - ln(y)
>>> diff(f, x)
 2⋅x    
 ─── + 2
  y 
>>> diff(f, y)
    2    
   x    1
 - ── - ─
    2   y
   y
>>> diff(diff(f, x), y)
 -2⋅x
 ────
   2 
  y

>>> from sympy import symbols, cos
>>> from sympy.plotting import plot3d
>>> x, y = symbols("x y")
>>> plot3d(cos(x * 3) * cos(y * 5) - y, (x, -1, 1), (y, -1, 1))
<sympy.plotting.plot.Plot object at 0x3b6d0d0>

>>> from sympy import init_printing, Symbol, limit, sqrt, oo
>>> init_printing()
>>> 
>>> x = Symbol("x")
>>> limit(sqrt(x**2 - 5 * x + 6) - x, x, oo)
-5/2
>>> limit(x * (sqrt(x**2 + 1) - x), x, oo)
1/2
>>> limit(1 / x**2, x, 0)
∞
>>> limit(((x - 1) / (x + 1)) ** x, x, oo)
 -2
ℯ

>>> from sympy import init_printing, Symbol, Function, Eq, dsolve, sin, diff
>>> init_printing()
>>>
>>> x = Symbol("x")
>>> f = Function("f")
>>>
>>> eq = Eq(f(x).diff(x), f(x))
>>> eq
d              
──(f(x)) = f(x)
dx         
>>>    
>>> dsolve(eq, f(x))
           x
f(x) = C₁⋅ℯ

>>>
>>> eq = Eq(x**2 * f(x).diff(x), -3 * x * f(x) + sin(x) / x)
>>> eq
 2 d                      sin(x)
x ⋅──(f(x)) = -3⋅x⋅f(x) + ──────
   dx                       x   
>>>
>>> dsolve(eq, f(x))
       C₁ - cos(x)
f(x) = ───────────   
            x³

>>> from sympy import init_printing, integrate, Symbol, exp, cos, erf
>>> init_printing()
>>> x = Symbol("x")
>>> # Polynomial Function
>>> f = x**2 + x + 1
>>> f
 2        
x  + x + 1
>>> integrate(f, x)
 3    2    
x    x     
── + ── + x
3    2     
>>> # Rational Function
>>> f = x / (x**2 + 2 * x + 1)
>>> f
     x      
────────────
 2          
x  + 2⋅x + 1

>>> integrate(f, x)
               1  
log(x + 1) + ─────
             x + 1
>>> # Exponential-polynomial functions
>>> f = x**2 * exp(x) * cos(x)
>>> f
 2  x       
x ⋅ℯ ⋅cos(x)
>>> integrate(f, x)
 2  x           2  x                         x           x       
x ⋅ℯ ⋅sin(x)   x ⋅ℯ ⋅cos(x)      x          ℯ ⋅sin(x)   ℯ ⋅cos(x)
──────────── + ──────────── - x⋅ℯ ⋅sin(x) + ───────── - ─────────
     2              2                           2           2    
>>> # A non-elementary integral
>>> f = exp(-(x**2)) * erf(x)
>>> f
   2       
 -x        
ℯ   ⋅erf(x)
>>> integrate(f, x)

  ___    2   
╲╱ π ⋅erf (x)
─────────────
      4

>>> from sympy import Symbol, cos, sin, pprint
>>> x = Symbol("x")
>>> e = 1 / cos(x)
>>> pprint(e)
  1   
──────
cos(x)
>>> pprint(e.series(x, 0, 10))
     2      4       6        8         
    x    5⋅x    61⋅x    277⋅x     ⎛ 10⎞
1 + ── + ──── + ───── + ────── + O⎝x  ⎠
    2     24     720     8064          
>>> e = 1/sin(x)
>>> pprint(e)
  1   
──────
sin(x)
>>> pprint(e.series(x, 0, 4))
           3        
1   x   7⋅x     ⎛ 4⎞
─ + ─ + ──── + O⎝x ⎠
x   6   360

>>> from sympy import *
>>> x = Symbol("x")
>>> y = Symbol("y")
>>> facts = Q.positive(x), Q.positive(y)
>>> with assuming(*facts):
...     print(ask(Q.positive(2 * x + y)))
True

>>> from sympy import *
>>> x = Symbol("x")
>>> # Assumption about x
>>> fact = [Q.prime(x)]
>>> with assuming(*fact):
...     print(ask(Q.rational(1 / x)))
True