It looks like it can do it if you complete the square manually. So we just
need to have better support for doing this in the algorithm(s).
In [7]: integrate(sqrt(((x + S(1)/2)**2 + S(3)/4)), x)
Out[7]:
⎛2⋅√3⋅(x + 1/2)⎞
3 3⋅asinh⎜──────────────⎟
(x + 1/2) 3⋅(x + 1/2) ⎝ 3 ⎠
───────────────────── + ─────────────────────── + ───────────────────────
__________________ __________________ 8
╱ 2 ╱ 2
╲╱ 4⋅(x + 1/2) + 3 4⋅╲╱ 4⋅(x + 1/2) + 3
Aaron Meurer
On Sun, Feb 14, 2016 at 7:46 PM, Oscar Benjamin <[email protected]>
wrote:
> On 14 February 2016 at 15:17, Andrew Corrigan <[email protected]>
> wrote:
> > I'm having trouble computing a definite integral involving the sqrt of a
> > non-negative expression, as implemented below (computing the length of a
> > quadratic line in 2D). It seems to fail. I've generally had success
> using
> > sympy for integration, except for when a sqrt is present.
> >
> > If anyone has any advice on how to make this work, I would appreciate it
> > tremendously. Thanks in advance!
> >
> > from sympy import *
> > x0,x1,x2,y0,y1,y2,xi = symbols('x0 x1 x2 y0 y1 y2 xi', real=True)
> > f_squared = (-4*x0*xi + 3*x0 - 4*x1*xi + x1 + 8*x2*xi - 4*x2)**2 +
> (4*xi*y0
> > + 4*xi*y1 - 8*xi*y2 - 3*y0 - y1 + 4*y2)**2
> > f = sqrt(f_squared)
> > integrate(f, (xi,0,1))
>
> Distilling this down you want to compute the integral of the square
> root of a quadratic. Sympy can do this in some simple cases:
>
> >>> sqrt(1 + x**2).integrate(x)
> ________
> ╱ 2
> x⋅╲╱ x + 1 asinh(x)
> ───────────── + ────────
> 2 2
>
> However it fails for slightly more complicated cases:
>
> >>> sqrt(1 + x + x**2).integrate(x)
> ⌠
> ⎮ ____________
> ⎮ ╱ 2
> ⎮ ╲╱ x + x + 1 dx
> ⌡
>
> This second form can always be reduced to the first by completing the
> square and changing variables. It seems that sympy is currently unable
> to do that though :(
>
> --
> Oscar
>
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