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?
On
sympy cannot compute this integral.
You're looking at a manually created indefinite integral of:
which is now complex.
Or a maxima created integal of:
Either way, all algorithms and options have been exhausted, and it failed on every try. Some integrals are just too difficult for sympy to solve.
On
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.
To simplifying integral you can use:
To solve integral you can use: