Updates:
Owner: asmeurer
Labels: Integration
Comment #9 on issue 829 by asmeurer: Improve simplify() to handle result
from integrate(log(k**2-m**2), k)
http://code.google.com/p/sympy/issues/detail?id=829
This is what it is going to look like with the new Risch integrator:
In [17]: a = integrate_primitive(Poly(t, t), Poly(1, t), [Poly(1, k),
Poly(2*k/(k**2 - m**2), t)], [k, t], [lambda k: log(k**2 - m**2)])
In [18]: a
Out[18]:
⎛ ⌠ ⎞
⎜ ⎮ 2 ⎟
⎜ ⎛ 2 2⎞ ⎮ -2⋅k ⎟
⎜k⋅log⎝k - m ⎠ + ⎮ ─────── dk, True⎟
⎜ ⎮ 2 2 ⎟
⎜ ⎮ k - m ⎟
⎝ ⌡ ⎠
In [19]: a[0].doit()
Out[19]:
⎛ 2 2⎞
-2⋅k + k⋅log⎝k - m ⎠ + │m│⋅log(k + │m│) - │m│⋅log(k - │m│)
Note that this is with the default setting where m and k are integers. If
they are regular symbols, the result is a little more complicated for both
(though still simpler for the new algorithm):
In [20]: var('k m')
Out[20]: (k, m)
In [21]: a = integrate_primitive(Poly(t, t), Poly(1, t), [Poly(1, k),
Poly(2*k/(k**2 - m**2), t)], [k, t], [lambda k: log(k**2 - m**2)])
In [22]: a
Out[22]:
⎛ ⌠ ⎞
⎜ ⎮ 2 ⎟
⎜ ⎛ 2 2⎞ ⎮ -2⋅k ⎟
⎜k⋅log⎝k - m ⎠ + ⎮ ─────── dk, True⎟
⎜ ⎮ 2 2 ⎟
⎜ ⎮ k - m ⎟
⎝ ⌡ ⎠
In [23]: a[0].doit()
Out[23]:
⎽⎽⎽⎽ ⎛ ⎽⎽⎽⎽⎞ ⎽⎽⎽⎽ ⎛ ⎽⎽⎽⎽⎞
⎛ 2 2⎞ ╱ 2 ⎜ ╱ 2 ⎟ ╱ 2 ⎜ ╱ 2 ⎟
-2⋅k + k⋅log⎝k - m ⎠ + ╲╱ m ⋅log⎝k + ╲╱ m ⎠ - ╲╱ m ⋅log⎝k - ╲╱ m ⎠
In [24]: integrate(log(k**2-m**2), k)
Out[24]:
⎛ ⎽⎽⎽⎽⎞
2 ⎛ 2 2⎞ 2 ⎜ ╱ 2 ⎟
⎛ 2 2⎞ m ⋅log⎝k - m ⎠ 2⋅m ⋅log⎝k + ╲╱ m ⎠
-2⋅k + k⋅log⎝k - m ⎠ - ─────────────── + ─────────────────────
⎽⎽⎽⎽ ⎽⎽⎽⎽
╱ 2 ╱ 2
╲╱ m ╲╱ m
And for what it's worth, the new algorithm is superbly faster:
In [25]: %timeit integrate_primitive(Poly(t, t), Poly(1, t), [Poly(1, k),
Poly(2*k/(k**2 - m**2), t)], [k, t], [lambda k: log(k**2 - m**2)])[0].doit()
10 loops, best of 3: 150 ms per loop
In [26]: %timeit integrate(log(k**2-m**2), k)
1 loops, best of 3: 2.98 s per loop
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