Python: Integrating Function Using Sympy

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I have a function that I want to integrate symbolically using the Sympy library. I followed the steps outlined in the website: https://docs.sympy.org/latest/modules/integrals/integrals.html

However, when I try to integrate, I just get back the integral in symbolic form: the Sympy integrate function doesn’t seem to have integrated at all.

I am not sure what is wrong with my code:

s = Symbol('s')
s0 = Symbol('s0')
k = Symbol('k')
u = Symbol('u')
E = Symbol('E')
init_printing(use_unicode=False, wrap_line=False)

func = (np.pi - 2)*s/s0*(1/((1/s0**2)*(1 - k*u/E - s**2*u**2))**0.5)

integrate(func,(u,0,1/a))

I believe that I followed everything correctly, but I’m not getting the output that should show up like in the website when integrating simple functions. Was wondering if someone can help me out with this issue.

  • s0 completely cancels out of that. 1/((1/s0**2)**0.5) is s0.

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  • Works for me: I set a = Symbol('a', real=True) and integrate(func,(u,0,1/a)) returned some huge expression involving a bunch of logarithms and no integrals.

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  • 1

    Why np.pi? I would use pi (that is sympy.pi). Anyway, I also get a result after I declared a = Symbol('a'): with sympy 1.12 on my local machine I get a piecewise because of an extreme particular case, while with live.sympy.org that uses sympy 1.11.1 I get the same main formula, without the “paranoid” special case.

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When in SymPy-land, use SymPy. Like @kikon said, no need to use a pi other than the one that SymPy provides. Since you are dividing by s0 and a it would be nice to tell SymPy that you know those are not zero. The following works for me:

from sympy import symbols, integrate
s, k, u, E = Symbol('s k u E')
a,s0 = symbols('a s0', nonzero=True)
func = (pi - 2)*s/s0*(1/((1/s0**2)*(1 - k*u/E - s**2*u**2))**0.5)
integrate(func,(u,0,1/a))

Piecewise((-s*(-2 + pi)*log(2*sqrt(-s**2) - k/E)/sqrt(-s**2) + s*(-2 + pi)*log(2*sqrt(-s**2)*sqrt(1 - s**2/a**2 - k/(E*a)) - 2*s**2/a - k/E)/sqrt(-s**2), s0 >= 0), (-s*(-s0)**1.0*(-2 + pi)*log(2*sqrt(-s**2) - k/E)/(s0*sqrt(-s**2)) + s*(-s0)**1.0*(-2 + pi)*log(2*sqrt(-s**2)*sqrt(1 - s**2/a**2 - k/(E*a)) - 2*s**2/a - k/E)/(s0*sqrt(-s**2)), True))

If you use sqrt instead of **0.5 you will get a different form:

>>> from sympy import nsimplify
>>> integrate(nsimplify(func), (u,0,1/a))
Piecewise((-s*(-2 + pi)*log(2*sqrt(-s**2) - k/E)/sqrt(-s**2) + s*(-2 + pi)*log(2*sqrt(-s**2)*sqrt(1 - s**2/a**2 - k/(E*a)) - 2*s**2/a - k/E)/sqrt(-s**2), ((s0 >= 0) & Ne(s**2, 0)) | ((s0 >= 0) & Ne(s**2, 0) & Ne(k/E, 0))), (-2*E*s*(-2 + pi)*sqrt(1 - k/(E*a))/k + 2*E*s*(-2 + pi)/k, (s0 >= 0) & Ne(k/E, 0)), (s*(-2 + pi)/a, s0 >= 0), (s*(-2 + pi)*log(2*sqrt(-s**2) - k/E)/sqrt(-s**2) - s*(-2 + pi)*log(2*sqrt(-s**2)*sqrt(1 - s**2/a**2 - k/(E*a)) - 2*s**2/a - k/E)/sqrt(-s**2), ((s > -oo) & (s < oo) & Ne(s, 0)) | (Ne(s**2, 0) & Ne(k/E, 0))), (2*E*s*(-2 + pi)*sqrt(1 - k/(E*a))/k - 2*E*s*(-2 + pi)/k, Ne(k/E, 0)), (-s*(-2 + pi)/a, True))

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