Question: I am trying to solve the integration integrate( sqrt(1 + cos(2 * x)), (x, 0, pi) )  Clearly, through pen and paper this is not hard, and the result is:  But when doing this through Sympy, something does not seem correct. I tried the sympy codes as below. from sympy import * x = symbols("x", real=True) integrate(sqrt(1 + cos(2 * x)), (x, 0, pi)).doit()  It then gives me a ValueError saying something in the complex domain not defined. But I've already defined the symbol x as a variable in the real domain. Here is the full error: --------------------------------------------------------------------------- ValueError Traceback (most recent call last) Cell In[7], line 4 1 from sympy import * 3 x = symbols("x", real=True) ----> 4 integrate(sqrt(1 + cos(2 * x)), (x, 0, pi)).doit() File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\integrals\integrals.py:1567, in integrate(meijerg, conds, risch, heurisch, manual, *args, **kwargs) 1564 integral = Integral(*args, **kwargs) 1566 if isinstance(integral, Integral): -> 1567 return integral.doit(**doit_flags) 1568 else: 1569 new_args = [a.doit(**doit_flags) if isinstance(a, Integral) else a 1570 for a in integral.args] File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\integrals\integrals.py:499, in Integral.doit(self, **hints) 497 if reps: 498 undo = {v: k for k, v in reps.items()} --> 499 did = self.xreplace(reps).doit(**hints) 500 if isinstance(did, tuple): # when separate=True 501 did = tuple([i.xreplace(undo) for i in did]) File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\integrals\integrals.py:710, in Integral.doit(self, **hints) 707 uneval = Add(*[eval_factored(f, x, a, b) 708 for f in integrals]) 709 try: --> 710 evalued = Add(*others)._eval_interval(x, a, b) 711 evalued_pw = piecewise_fold(Add(*piecewises))._eval_interval(x, a, b) 712 function = uneval + evalued + evalued_pw File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\core\expr.py:956, in Expr._eval_interval(self, x, a, b) 953 domain = Interval(b, a) 954 # check the singularities of self within the interval 955 # if singularities is a ConditionSet (not iterable), catch the exception and pass --> 956 singularities = solveset(self.cancel().as_numer_denom()[1], x, 957 domain=domain) 958 for logterm in self.atoms(log): 959 singularities = singularities | solveset(logterm.args[0], x, 960 domain=domain) File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2252, in solveset(f, symbol, domain) 2250 if symbol not in _rc: 2251 x = _rc[0] if domain.is_subset(S.Reals) else _rc[1] -> 2252 rv = solveset(f.xreplace({symbol: x}), x, domain) 2253 # try to use the original symbol if possible 2254 try: File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2276, in solveset(f, symbol, domain) 2273 f = f.xreplace({d: e}) 2274 f = piecewise_fold(f) -> 2276 return _solveset(f, symbol, domain, _check=True) File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:1060, in _solveset(f, symbol, domain, _check) 1057 result = Union(*[solver(m, symbol) for m in f.args]) 1058 elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \ 1059 _is_function_class_equation(HyperbolicFunction, f, symbol): -> 1060 result = _solve_trig(f, symbol, domain) 1061 elif isinstance(f, arg): 1062 a = f.args[0] File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:612, in _solve_trig(f, symbol, domain) 610 sol = None 611 try: --> 612 sol = _solve_trig1(f, symbol, domain) 613 except _SolveTrig1Error: 614 try: File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:688, in _solve_trig1(f, symbol, domain) 685 if g.has(x) or h.has(x): 686 raise _SolveTrig1Error("change of variable not possible") --> 688 solns = solveset_complex(g, y) - solveset_complex(h, y) 689 if isinstance(solns, ConditionSet): 690 raise _SolveTrig1Error("polynomial has ConditionSet solution") File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2284, in solveset_complex(f, symbol) 2283 def solveset_complex(f, symbol): -> 2284 return solveset(f, symbol, S.Complexes) File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2252, in solveset(f, symbol, domain) 2250 if symbol not in _rc: 2251 x = _rc[0] if domain.is_subset(S.Reals) else _rc[1] -> 2252 rv = solveset(f.xreplace({symbol: x}), x, domain) 2253 # try to use the original symbol if possible 2254 try: File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:2276, in solveset(f, symbol, domain) 2273 f = f.xreplace({d: e}) 2274 f = piecewise_fold(f) -> 2276 return _solveset(f, symbol, domain, _check=True) File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:1110, in _solveset(f, symbol, domain, _check) 1106 result += _solve_radical(equation, u, 1107 symbol, 1108 solver) 1109 elif equation.has(Abs): -> 1110 result += _solve_abs(f, symbol, domain) 1111 else: 1112 result_rational = _solve_as_rational(equation, symbol, domain) File C:\Dev_Tools\Anaconda3\Lib\site-packages\sympy\solvers\solveset.py:918, in _solve_abs(f, symbol, domain) 916 """ Helper function to solve equation involving absolute value function """ 917 if not domain.is_subset(S.Reals): --> 918 raise ValueError(filldedent(''' 919 Absolute values cannot be inverted in the 920 complex domain.''')) 921 p, q, r = Wild('p'), Wild('q'), Wild('r') 922 pattern_match = f.match(p*Abs(q) + r) or {} ValueError: Absolute values cannot be inverted in the complex domain.  How do I properly integrate this using Sympy?

Answer: Adding a simplification in there will produce the correct result, but I'm not sure why it is having an issue in the first place. integrate(sqrt(1+cos(2*x)).simplify(), (x, 0, pi)) # 2*sqrt(2)