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Labels: ness987 NeedsReview
Comment #16 on issue 1304 by ness...@googlemail.com: Integrate sqrt(x**2 +
y**2) fails
http://code.google.com/p/sympy/issues/detail?id=1304
This works in gsoc-3 (https://github.com/sympy/sympy/pull/543):
In [18]: integrate(1/(x**2+y**2)**S('3/2'), x)
Out[18]:
1
─────────────────
⎽⎽⎽⎽⎽⎽⎽⎽
╱ 2
2 ╱ y
y ⋅ ╱ 1 + ──
╱ 2
╲╱ x
In [19]: _.diff(x)
Out[19]:
1
──────────────
3/2
⎛ 2⎞
3 ⎜ y ⎟
x ⋅⎜1 + ──⎟
⎜ 2⎟
⎝ x ⎠
The other integrals mentioned above can also be done, but are messier:
In [25]: integrate((x**2+y**2)**S('1/2'), x)
Out[25]:
2 ⎛x⎞
3 y ⋅asinh⎜─⎟
x x⋅y ⎝y⎠
────────────────── + ──────────────── + ───────────
⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽ 2
╱ 2 ╱ 2
╱ x ╱ x
2⋅y⋅ ╱ ── + 1 2⋅ ╱ ── + 1
╱ 2 ╱ 2
╲╱ y ╲╱ y
In [26]: integrate((x**2-y**2)**S('1/2'), x)
Out[26]:
⎧ 2 ⎛x⎞
⎪ 3 y ⋅acosh⎜─⎟ │ 2│
⎪ x x⋅y ⎝y⎠ │x │
⎪ ────────────────── - ──────────────── - ─────────── for 1 < │──│
⎪ ⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽ 2 │ 2│
⎪ ╱ 2 ╱ 2 │y │
⎪ ╱ x ╱ x
⎪ 2⋅y⋅ ╱ ── - 1 2⋅ ╱ ── - 1
⎪ ╱ 2 ╱ 2
⎪ ╲╱ y ╲╱ y
⎨
⎪ 2 ⎛x⎞
⎪ 3 ⅈ⋅y ⋅asin⎜─⎟
⎪ ⅈ⋅x ⅈ⋅x⋅y ⎝y⎠
⎪- ──────────────────── + ────────────────── + ──────────── otherwise
⎪ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ 2
⎪ ╱ 2 ╱ 2
⎪ ╱ x ╱ x
⎪ 2⋅y⋅ ╱ - ── + 1 2⋅ ╱ - ── + 1
⎪ ╱ 2 ╱ 2
⎩ ╲╱ y ╲╱ y
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