Non-deterministic exception from integrals/poly

[oscarbenjamin]

The script below fails non-deterministically for me:

from sympy import *

t = Symbol('t')
f = t*exp(-3*t**2/2) + exp(t**2/2)
print(integrate(f, t))

Half the time I get

$ python z.py 
sqrt(2)*sqrt(pi)*erfi(sqrt(2)*t/2)/2 - exp(-3*t**2/2)/3

which I think is the right answer.

The other half of the time I get:

$ python z.py 
Traceback (most recent call last):
  File "z.py", line 5, in <module>
    print(integrate(f, t))
  File "/Users/enojb/current/sympy/sympy/sympy/integrals/integrals.py", line 1520, in integrate
    return integral.doit(**doit_flags)
  File "/Users/enojb/current/sympy/sympy/sympy/integrals/integrals.py", line 573, in doit
    antideriv = self._eval_integral(
  File "/Users/enojb/current/sympy/sympy/sympy/integrals/integrals.py", line 905, in _eval_integral
    result, i = risch_integrate(f, x, separate_integral=True,
  File "/Users/enojb/current/sympy/sympy/sympy/integrals/risch.py", line 1754, in risch_integrate
    ans, i, b = integrate_hyperexponential(fa, fd, DE, conds=conds)
  File "/Users/enojb/current/sympy/sympy/sympy/integrals/risch.py", line 1521, in integrate_hyperexponential
    i = p - (qd*derivation(qa, DE) - qa*derivation(qd, DE)).as_expr()/\
  File "/Users/enojb/current/sympy/sympy/sympy/integrals/risch.py", line 898, in derivation
    r += (d*pv.diff(v)).as_poly(t)
TypeError: unsupported operand type(s) for +=: 'Poly' and 'NoneType'

This happens with both Python 3.5 and 3.8 on OSX.

[jksuom]

There is a weakness in this construction where va*t1**i/vd is added to qa/qd

sympy/sympy/integrals/risch.py

Lines 1459 to 1460 in 1d3327b

	qa = qa*vd + va*Poly(t1**i)*qd
	qd *= vd

integrate_hyperexponential_polynomial actually has to deal with Laurent polynomials. Positive and negative and exponents are handled separately earlier in the loop

sympy/sympy/integrals/risch.py

Lines 1439 to 1447 in 1d3327b

	elif i < 0:
	# If you get AttributeError: 'NoneType' object has no attribute 'nth'
	# then this should really not have expand=False
	# But it shouldn't happen because p is already a Poly in t and z
	a = p.as_poly(z, expand=False).nth(-i)
	else:
	# If you get AttributeError: 'NoneType' object has no attribute 'nth'
	# then this should really not have expand=False
	a = p.as_poly(t1, expand=False).nth(i)

However, on line 1459 we now get polynomials with inverted generators for negative exponents, and those lead to problems later on:

In [6]: Poly(t**-3)
Out[6]: Poly((1/t)**3, 1/t, domain='ZZ')

It seems that we should put Poly(t1**-i) in the denominator qd in case i < 0.