Reputation: 21

Integration using Sympy

I want to evaluate an integral of the form:

using python sympy, however, I am not able to do so. This is what I have implemented:

import sympy as sym
r,a,b = sym.symbols('r,a,b')
I = r/sym.sqrt((a-r)*(r-b))
I_1 = sym.integrate (I,r)
print (I_1)

The result that I am getting is: Integral(r/sqrt(-(-a + r)*(-b + r)), r)

Thus, it is not evaluating the integral, rather just simplifying the equation.

Can anyone help me with this?

Upvotes: 2

Answers (3)

Selimzhan Ozer

Reputation: 34

To simplifying integral you can use:

sympy.pprint(sympy.Integral(I, x))

To solve integral you can use:

sympy.pprint(integrate(I, x))

Upvotes: 0

Oscar Benjamin

Reputation: 14530

It is possible to compute an antiderivative with a substitution:

In [1]: import sympy as sym
   ...: r,a,b = sym.symbols('r,a,b')
   ...: I = r/sym.sqrt((a-r)*(r-b))
   ...: I_1 = sym.integrate (I,r)
   ...: print (I_1)
Integral(r/sqrt(-(-a + r)*(-b + r)), r)

In [2]: t = symbols('t')

In [3]: tsubs = r - (a+b)/2

In [4]: Integral(I, r)
Out[4]: 
⌠                        
⎮          r             
⎮ ──────────────────── dr
⎮   __________________   
⎮ ╲╱ (a - r)⋅(-b + r)    
⌡                        

In [5]: Integral(I, r).transform(tsubs, t)
Out[5]: 
⌠                                 
⎮           a   b                 
⎮           ─ + ─ + t             
⎮           2   2                 
⎮ ───────────────────────────── dt
⎮     _________________________   
⎮    ╱ ⎛a   b    ⎞ ⎛a   b    ⎞    
⎮   ╱  ⎜─ - ─ - t⎟⋅⎜─ - ─ + t⎟    
⎮ ╲╱   ⎝2   2    ⎠ ⎝2   2    ⎠    
⌡                                 

In [6]: Integral(I, r).transform(tsubs, t).doit()
Out[6]: 
  ⎛⎧    ⎛         _________________⎞                         ⎞     ⎛⎧    ⎛         _________________⎞                         ⎞                              
  ⎜⎪    ⎜        ╱        1        ⎟       2            2    ⎟     ⎜⎪    ⎜        ╱        1        ⎟       2            2    ⎟                              
a⋅⎜⎨asin⎜2⋅t⋅   ╱  ─────────────── ⎟  for a  - 2⋅a⋅b + b  > 0⎟   b⋅⎜⎨asin⎜2⋅t⋅   ╱  ─────────────── ⎟  for a  - 2⋅a⋅b + b  > 0⎟      ________________________
  ⎜⎪    ⎜      ╱    2            2 ⎟                         ⎟     ⎜⎪    ⎜      ╱    2            2 ⎟                         ⎟     ╱  2            2      2 
  ⎝⎩    ⎝    ╲╱    a  - 2⋅a⋅b + b  ⎠                         ⎠     ⎝⎩    ⎝    ╲╱    a  - 2⋅a⋅b + b  ⎠                         ⎠   ╲╱  a  - 2⋅a⋅b + b  - 4⋅t  
────────────────────────────────────────────────────────────── + ────────────────────────────────────────────────────────────── - ───────────────────────────
                              2                                                                2                                               2             

In [7]: Integral(I, r).transform(tsubs, t).doit().subs(t, tsubs)
Out[7]: 
  ⎛⎧    ⎛                     _________________⎞                         ⎞     ⎛⎧    ⎛                     _________________⎞                         ⎞        ____________________________________
  ⎜⎪    ⎜  ⎛  a   b    ⎞     ╱        1        ⎟       2            2    ⎟     ⎜⎪    ⎜  ⎛  a   b    ⎞     ╱        1        ⎟       2            2    ⎟       ╱                                  2 
a⋅⎜⎨asin⎜2⋅⎜- ─ - ─ + r⎟⋅   ╱  ─────────────── ⎟  for a  - 2⋅a⋅b + b  > 0⎟   b⋅⎜⎨asin⎜2⋅⎜- ─ - ─ + r⎟⋅   ╱  ─────────────── ⎟  for a  - 2⋅a⋅b + b  > 0⎟      ╱   2            2     ⎛  a   b    ⎞  
  ⎜⎪    ⎜  ⎝  2   2    ⎠   ╱    2            2 ⎟                         ⎟     ⎜⎪    ⎜  ⎝  2   2    ⎠   ╱    2            2 ⎟                         ⎟     ╱   a  - 2⋅a⋅b + b  - 4⋅⎜- ─ - ─ + r⎟  
  ⎝⎩    ⎝                ╲╱    a  - 2⋅a⋅b + b  ⎠                         ⎠     ⎝⎩    ⎝                ╲╱    a  - 2⋅a⋅b + b  ⎠                         ⎠   ╲╱                        ⎝  2   2    ⎠  
────────────────────────────────────────────────────────────────────────── + ────────────────────────────────────────────────────────────────────────── - ─────────────────────────────────────────
                                    2                                                                            2                                                            2                    

In [8]: factor(piecewise_fold(_), deep=True)
Out[8]: 
⎧      ⎛                    _________________⎞         ⎛                    _________________⎞                                           
⎪      ⎜                   ╱        1        ⎟         ⎜                   ╱        1        ⎟                                           
⎪a⋅asin⎜(-a - b + 2⋅r)⋅   ╱  ─────────────── ⎟   b⋅asin⎜(-a - b + 2⋅r)⋅   ╱  ─────────────── ⎟                                           
⎨      ⎜                 ╱    2            2 ⎟         ⎜                 ╱    2            2 ⎟                                           
⎪      ⎝               ╲╱    a  - 2⋅a⋅b + b  ⎠         ⎝               ╲╱    a  - 2⋅a⋅b + b  ⎠     ____________________             2    
⎪───────────────────────────────────────────── + ───────────────────────────────────────────── - ╲╱ -(-a + r)⋅(-b + r)   for (a - b)  > 0
⎩                      2                                               2

The result comes out with a condition that is basically equivalent to a != b (if a and b are assumed real). You can use .args[0][0] to extract just the expression part.

Upvotes: 3