TypeError: 'NoneType' object is not subscriptable from call to integrate

[nasser1]

Copied from

https://stackoverflow.com/questions/45173987/typeerror-nonetype-object-is-not-subscriptable-from-call-to-integrate

Error using integrate. Here is the MWE

python
Python 3.6.1 |Anaconda custom (64-bit)| (default, May 11 2017, 13:09:58)
[GCC 4.4.7 20120313 (Red Hat 4.4.7-1)] on linux
Type "help", "copyright", "credits" or "license" for more information.

from sympy import *
A,B,y=symbols('A B y')
integrate(-(A2+B2*(-y2+1))(1/2)/(-y**2+1),y)

Traceback (most recent call last):
File "", line 1, in
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/integrals.py", line 1295, in integrate
risch=risch, manual=manual)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/integrals.py", line 486, in doit
conds=conds)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/integrals.py", line 919, in _eval_integral
result = manualintegrate(g, x)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 1223, in manualintegrate
return _manualintegrate(integral_steps(f, var))
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 1013, in integral_steps
fallback_rule)(integral)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 85, in do_one_rl
result = rl(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 85, in do_one_rl
result = rl(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 65, in null_safe_rl
result = rule(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 212, in _alternatives
result = rule(integral)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 33, in conditioned_rl
return rule(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 176, in _rewriter
substep = integral_steps(rewritten, symbol)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 1013, in integral_steps
fallback_rule)(integral)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 85, in do_one_rl
result = rl(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 65, in null_safe_rl
result = rule(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 95, in switch_rl
return rl(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 85, in do_one_rl
result = rl(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 65, in null_safe_rl
result = rule(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 335, in mul_rule
next_step = integral_steps(f, symbol)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 1013, in integral_steps
fallback_rule)(integral)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 85, in do_one_rl
result = rl(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 85, in do_one_rl
result = rl(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/strategies/core.py", line 65, in null_safe_rl
result = rule(expr)
File "/home/me/anaconda3/lib/python3.6/site-packages/sympy/integrals/manualintegrate.py", line 743, in trig_substitution_rule
a = match[a]
TypeError: 'NoneType' object is not subscriptable

Interesting thing is that, if I issue the same command right again, the error do not show up

integrate(-(A2+B2*(-y2+1))(1/2)/(-y2+1),y)
Integral((A2 - B2*y2 + B**2)**0.5/((y - 1)*(y + 1)), y)

It only shows up first time it is used! It looks like using it first time loads something to memory and hence next time the error do not show up.

[jksuom]

There is a bug in manualintegrate.trig_substitution_rule:

def trig_substitution_rule(integral):
    integrand, symbol = integral
    a = sympy.Wild('a', exclude=[0, symbol])
    b = sympy.Wild('b', exclude=[0, symbol])
    theta = sympy.Dummy("theta")

    matches = integrand.find(a + b*symbol**2)
    if matches:
        for expr in matches:
            match = expr.match(a + b*symbol**2)
            a = match[a]
            b = match[b]

The wild symbols a and b are reassigned in the loop and, if the loop is repeated, expr.match will fail and return None.

[normalhuman]

The error no longer occurs (a and b have been changed to A and B in Wild matches), but the integral returns unevaluated:

>>> A, B, y = symbols('A B y', positive=True)
>>> integrate(-(A**2 + B**2*(-y**2 + 1))**(S(1)/2) / (-y**2 + 1), y)
Integral(sqrt(A**2 - B**2*y**2 + B**2)/((y - 1)*(y + 1)), y)

Wolfram Alpha can do this.