Elementary integration of trigonometric functions is wrong

[mleoni-pf]

I just stumbled upon this very surprising issue.

$ isympy
IPython console for SymPy 1.14.0 (Python 3.12.3-64-bit) (ground types: python)

These commands were executed:
>>> from sympy import *
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)
>>> init_printing()

Documentation can be found at https://docs.sympy.org/1.14.0/


In [1]: integrate(abs(sin(2*x)), (x, 0, 2*pi))
Out[1]: 2

while I am fairly convinced, by my calculations on paper and a quick check with WolframAlpha, that the result should be 4.

Edit: In fact, the result of integrate(abs(sin(k*x)), (x, 0, 2*pi)) is 4 for all k, but sympy is returning 4 / k instead.

[oscarbenjamin]

See also gh-27379 and gh-27396.

The antiderivative is computed correctly but has a Piecewise with the condition sin(2*x) > 0 in it. This is then handled with _solve_inequality that does not handle periodic solutions correctly:

In [4]: from sympy.solvers.inequalities import _solve_inequality

In [5]: _solve_inequality(sin(2*x) > 0, x)
Out[5]:
            π
0 < x ∧ x < ─
            2

That calls solve_univariate_inequality:

sympy/sympy/solvers/inequalities.py

Line 822 in c99a164

rv = solve_univariate_inequality(ie, s)

which gives the wrong result because is calls solvify which does not handle periodic equations properly:

sympy/sympy/solvers/inequalities.py

Line 520 in c99a164

solns = solvify(e, gen, domain)

In [5]: solvify(sin(2*x), x, Reals)
Out[5]:
⎡   π⎤
⎢0, ─⎥
⎣   2⎦

Really solve_univariate_inequality should not use solvify or on the other hand maybe _solve_inequality should not call solve_univariate_inequality.

[oscarbenjamin]

The simplest fix is changing _solve_inequality to not call solve_univariate_inequality:

diff --git a/sympy/solvers/inequalities.py b/sympy/solvers/inequalities.py
index f50f7a7572..a9a6b952ce 100644
--- a/sympy/solvers/inequalities.py
+++ b/sympy/solvers/inequalities.py
@@ -817,9 +817,9 @@ def classify(ie, s, i):
     except (PolynomialError, NotImplementedError):
         if not linear:
             try:
-                rv = reduce_rational_inequalities([[ie]], s)
+                rv = reduce_abs_inequality(ie.lhs - ie.rhs, ie.rel_op, s)
             except PolynomialError:
-                rv = solve_univariate_inequality(ie, s)
+                raise NotImplementedError
             # remove restrictions wrt +/-oo that may have been
             # applied when using sets to simplify the relationship
             okoo = classify(ie, s, oo)

Then the integral is not fully evaluated but is correct:

In [1]: integrate(abs(sin(2*x)), (x, 0, 2*pi))
Out[1]:
2⋅π
 ⌠
 ⎮  ⎧sin(2⋅x)   for sin(2⋅x) ≥ 0
 ⎮  ⎨                            dx
 ⎮  ⎩-sin(2⋅x)     otherwise
 ⌡
 0

That change causes some test failures because integrals that previously evaluate no longer evaluate. I think that the tests can just be changed though as they currently only succeed based on buggy code.