Comment #9 on issue 636 by mattpap: integrate(1/(x**2+1)) should return
arctan(x)
http://code.google.com/p/sympy/issues/detail?id=636
I've finally made ratint() integrate 1/(1+x^5):
In [1]: from sympy.integrals.rationaltools import *
In [2]: ratint(1/(1+x**1), x)
Out[2]: log(1 + x)
In [3]: simplify(diff(_, x))
Out[3]:
1
─────
1 + x
In [4]: ratint(1/(1+x**2), x)
Out[4]: atan(x)
In [5]: simplify(diff(_, x))
Out[5]:
1
──────
2
1 + x
In [6]: ratint(1/(1+x**3), x)
Out[6]:
⎛ ⎽⎽⎽ ⎽⎽⎽⎞
⎽⎽⎽ ⎜ ╲╱ 3 2⋅x⋅╲╱ 3 ⎟
⎛ 2⎞ ╲╱ 3 ⋅atan⎜- ───── + ─────────⎟
log⎝1 - x + x ⎠ log(1 + x) ⎝ 3 3 ⎠
- ─────────────── + ────────── + ───────────────────────────────
6 3 3
In [7]: simplify(diff(_, x))
Out[7]:
1
──────
3
1 + x
In [8]: ratint(1/(1+x**4), x)
Out[8]:
⎽⎽⎽ ⎛ ⎽⎽⎽⎞ ⎽⎽⎽ ⎛ ⎽⎽⎽ 2⎞ ⎽⎽⎽
⎛ ⎽⎽⎽⎞
⎽⎽⎽ ⎛ ⎽⎽⎽ 2⎞
╲╱ 2 ⋅atan⎝1 - x⋅╲╱ 2 ⎠ ╲╱ 2 ⋅log⎝1 - x⋅╲╱ 2 + x ⎠ ╲╱ 2 ⋅atan⎝1 +
x⋅╲╱ 2 ⎠
╲╱ 2 ⋅log⎝1 + x⋅╲╱ 2 + x ⎠
- ─────────────────────── - ─────────────────────────── +
─────────────────────── +
───────────────────────────
4 8 4
8
In [9]: simplify(diff(_, x))
Out[9]:
1
──────
4
1 + x
In [10]: ratint(1/(1+x**5), x)
Out[10]:
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
⎛ ⎽⎽⎽⎞ ⎛ ⎛ ⎽⎽⎽⎞ ⎞ ⎛
⎽⎽⎽⎞ ⎛
⎛ ⎽⎽⎽⎞ ⎞ ╱ ⎽
log(1 + x) ⎜ ╲╱ 5 ⎟ ⎜ ⎜ ╲╱ 5 ⎟ 2⎟ ⎜ ╲╱ 5
⎟ ⎜
⎜ ╲╱ 5 ⎟ 2⎟ ╱ ╲╱
────────── - ⎜1/20 - ─────⎟⋅log⎜1 - x⋅⎜1/2 - ─────⎟ + x ⎟ - ⎜1/20 +
─────⎟⋅log⎜1 -
x⋅⎜1/2 + ─────⎟ + x ⎟ + 2⋅ ╱ 1/40 - ───
5 ⎝ 20 ⎠ ⎝ ⎝ 2 ⎠ ⎠ ⎝ 20
⎠ ⎝
⎝ 2 ⎠ ⎠ ╲╱ 20
⎽⎽⎽
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
⎽⎽ ⎛ ⎽⎽⎽ ⎽⎽⎽ ⎞ ╱
⎽⎽⎽
⎛ ⎽⎽⎽ ⎽⎽⎽
5 ⎜ 800 + 160⋅╲╱ 5 8⋅x⋅╲╱ 5 ⎟ ╱
╲╱ 5
⎜ 800 - 160⋅╲╱ 5 8⋅x⋅╲╱ 5
── ⋅atan⎜- ────────────────────── + ───────────────────⎟ + 2⋅ ╱ 1/40 +
─────
⋅atan⎜────────────────────── + ──────────────
0 ⎜ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎟ ╲╱
200
⎜ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
⎜ ╱ ⎽⎽⎽ ╱ ⎽⎽⎽ ⎟
⎜ ╱ ⎽⎽⎽ ╱
⎝ 80⋅╲╱ 200 - 40⋅╲╱ 5 ╲╱ 200 - 40⋅╲╱ 5 ⎠
⎝80⋅╲╱ 200 + 40⋅╲╱ 5 ╲╱ 200 + 40⋅╲
⎞
⎟
─────⎟
⎽⎽⎽⎽⎽⎟
⎽⎽⎽ ⎟
╱ 5 ⎠
but here we can't simplify so ... hack, hack, hack:
In [11]: diff(_, x)
Out[11]:
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
⎛ ⎽⎽⎽⎞ ⎛ ⎽⎽⎽⎞ ⎛ ⎽⎽⎽⎞ ⎛ ⎽⎽⎽⎞
╱ ⎽⎽⎽
⎜ ╲╱ 5 ⎟ ⎜ ╲╱ 5 ⎟ ⎜ ╲╱ 5 ⎟ ⎜ ╲╱ 5 ⎟
⎽⎽⎽ ╱ ╲╱ 5
⎜1/20 + ─────⎟⋅⎜1/2 - 2⋅x + ─────⎟ ⎜1/20 - ─────⎟⋅⎜1/2 - 2⋅x - ─────⎟
16⋅╲╱ 5 ⋅ ╱ 1/40 - ─────
⎝ 20 ⎠ ⎝ 2 ⎠ ⎝ 20 ⎠ ⎝ 2 ⎠
╲╱ 200
────────────────────────────────── + ────────────────────────────────── +
───────────────────────────────────────────────────
⎛ ⎽⎽⎽⎞ ⎛ ⎽⎽⎽⎞ ⎛
⎜ ╲╱ 5 ⎟ 2 ⎜ ╲╱ 5 ⎟ 2
⎜ ⎛
⎽⎽⎽ ⎽⎽⎽
1 - x⋅⎜1/2 + ─────⎟ + x 1 - x⋅⎜1/2 - ─────⎟ + x
⎜ ⎜
800 + 160⋅╲╱ 5 8⋅x⋅╲╱ 5
⎝ 2 ⎠ ⎝ 2 ⎠
⎜1 + ⎜-
────────────────────── + ──────────────────
⎜ ⎜
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
⎜ ⎜
╱ ⎽⎽⎽ ╱ ⎽⎽⎽
⎝ ⎝
80⋅╲╱ 200 - 40⋅╲╱ 5 ╲╱ 200 - 40⋅╲╱ 5
⎽
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
╱
⎽⎽⎽
⎽⎽⎽ ╱ ╲╱ 5
16⋅╲╱ 5 ⋅ ╱ 1/40 +
─────
╲╱ 200
1
──────────────────────── +
───────────────────────────────────────────────────────────────────────── +
─────────
2⎞
⎛ 2⎞
5⋅(1 + x)
⎞ ⎟ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎜ ⎛ ⎽⎽⎽
⎽⎽⎽ ⎞ ⎟
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
⎟ ⎟ ╱ ⎽⎽⎽ ⎜ ⎜ 800 - 160⋅╲╱ 5 8⋅x⋅╲╱
5 ⎟ ⎟ ╱
⎽⎽⎽
─⎟ ⎟⋅╲╱ 200 - 40⋅╲╱ 5 ⎜1 + ⎜────────────────────── +
───────────────────⎟ ⎟⋅╲╱
200 + 40⋅╲╱ 5
⎽⎟ ⎟ ⎜ ⎜ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽
⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎟ ⎟
⎟ ⎟ ⎜ ⎜ ╱ ⎽⎽⎽ ╱
⎽⎽⎽ ⎟ ⎟
⎠ ⎠ ⎝ ⎝80⋅╲╱ 200 + 40⋅╲╱ 5 ╲╱ 200 + 40⋅╲╱
5 ⎠ ⎠
In [12]: n,d = _.as_numer_denom()
In [13]: n = n.expand()
In [14]: d = d.expand()
In [15]: N = n.subs(sqrt((200-40*sqrt(5))*(200+40*sqrt(5))), 80*sqrt(5))
In [16]: D = d.subs(sqrt((200-40*sqrt(5))*(200+40*sqrt(5))), 80*sqrt(5))
In [17]: Poly.cancel(N/D)
Out[17]:
⎽⎽⎽
17592186044416000000000000000000⋅╲╱ 5
──────────────────────────────────────────────────────────────────────────────────
⎽⎽⎽
⎽⎽⎽ 5
17592186044416000000000000000000⋅╲╱ 5 +
17592186044416000000000000000000⋅╲╱ 5 ⋅x
In [18]: trim(N/D)
Out[18]:
⎽⎽⎽
17592186044416000000000000000000⋅╲╱ 5
──────────────────────────────────────────────────────────────────────────────────
⎽⎽⎽
⎽⎽⎽ 5
17592186044416000000000000000000⋅╲╱ 5 +
17592186044416000000000000000000⋅╲╱ 5 ⋅x
In [19]: NN = N.subs(sqrt(5), 1)
In [20]: DD = D.subs(sqrt(5), 1)
In [21]: Poly.cancel(NN/DD)
Out[21]:
1
──────
5
1 + x
Implications of the example above are as follows:
1. Improve simplification of algebraic expressions (e.g. expand(power=True))
2. Implement efficient polynomials with coefficients in algebraic number
fields
I will send a series of patches soon, but first I need to polish them a bit.
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