Comment #7 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 made some progress on this issue:
In [1]: from sympy.integrals.rationaltools import *
In [2]: integrate(1/(x**2+1), x)
Out[2]:
ⅈ⋅log(x + ⅈ) ⅈ⋅log(x - ⅈ)
──────────── - ────────────
2 2
In [3]: ratint(1/(x**2+1), x)
Out[3]: atan(x)
In [4]: A = x**4 - 3*x**2 + 6
In [5]: D = x**6 - 5*x**4 + 5*x**2 + 4
In [6]: time r = ratint(A/D, x)
CPU times: user 0.54 s, sys: 0.01 s, total: 0.55 s
Wall time: 0.57 s
In [8]: r
Out[8]:
⎛ 3 5⎞
⎛ 3⎞ ⎜x 3⋅x x ⎟
atan(x) + atan⎝x ⎠ + atan⎜─ - ──── + ──⎟
⎝2 2 2 ⎠
In [9]: time R = integrate(A/D, x)
CPU times: user 2.08 s, sys: 0.02 s, total: 2.09 s
Wall time: 2.16 s
In [11]: R
(here starts universe filling expression :)
In [12]: P = x**7-24*x**4-4*x**2+8*x-8
In [13]: Q = x**8+6*x**6+12*x**4+8*x**2
In [14]: integrate(P/Q, x)
Out[14]:
⎽⎽⎽ ⎽⎽⎽
3⋅ⅈ⋅╲╱ 2 3⋅ⅈ⋅╲╱ 2
1/2 - ───────── 1/2 + ─────────
⎽⎽⎽ ⎽⎽⎽
4 4 1 3⋅ⅈ⋅╲╱ 2
3⋅ⅈ⋅╲╱ 2
- ─────────────── - ─────────────── + log(x) + ─ - ──────────────── +
────────────────
⎽⎽⎽ ⎽⎽⎽ x
2 2
x + ⅈ⋅╲╱ 2 x - ⅈ⋅╲╱ 2 ⎛ ⎽⎽⎽⎞
⎛ ⎽⎽⎽⎞
4⋅⎝x - ⅈ⋅╲╱ 2 ⎠ 4⋅⎝x
+ ⅈ⋅╲╱ 2 ⎠
In [15]: ratint(P/Q, x)
Out[15]:
2 3
4 + 6⋅x + 8⋅x + 3⋅x
───────────────────── + log(x)
3 5
4⋅x + 4⋅x + x
However I still can't make 1/(x**5+1) to integrate (to something readable).
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