# Error during integration: not of type (UNSIGNED-BYTE 15)

Hi everyone, I am new to Sage and coding, and I am trying to solve a tricky integral from equation 13 in this physics paper: http://www.nature.com/articles/srep11876 (www.nature.com/articles/srep11876)

Here is the integrand in code form (I would upload a screenshot of the full equation, but I do not have enough karma points):

```
sage: var ('alpha beta E_o a d_o z_o n u')
(alpha, beta, E_o, a, d_o, z_o, n, u)
sage: j = CC.0
sage: integrand = e**(-j * n * u) / (((a + d_o + 0.5 * z_o * (((16 * pi) / (alpha * beta)) ** (-3))) + 0.5 * z_o * (((16 *
....: pi) / (alpha * beta)) ** (-3)) * sin(u)) ** 3) - 1
```

The integral is part of a Fourier coefficient, so it is a definite integral with limits -pi and pi. I want to integrate with respect to u and then solve the full equation for beta. All of the other variables are known.

However, when I try to integrate, I get an error:

```
sage: integrate(integrand, u, -pi, pi)
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call last)
<ipython-input-32-84f9192f9549> in <module>()
----> 1 integrate(integrand, u, -pi, pi)
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/misc/functional.py in integral(x, *args, **kwds)
751 """
752 if hasattr(x, 'integral'):
--> 753 return x.integral(*args, **kwds)
754 else:
755 from sage.symbolic.ring import SR
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.integral (build/cythonized/sage/symbolic/expression.cpp:64474)()
12434 R = ring.SR
12435 return R(integral(f, v, a, b, **kwds))
> 12436 return integral(self, *args, **kwds)
12437
12438 integrate = integral
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/integration/integral.py in integrate(expression, v, a, b, algorithm, hold)
929 return indefinite_integral(expression, v, hold=hold)
930 else:
--> 931 return definite_integral(expression, v, a, b, hold=hold)
932
933
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/function.pyx in sage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:12262)()
1136 res = self._evalf_try_(*args)
1137 if res is None:
-> 1138 res = super(BuiltinFunction, self).__call__(
1139 *args, coerce=coerce, hold=hold)
1140
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/function.pyx in sage.symbolic.function.Function.__call__ (build/cythonized/sage/symbolic/function.cpp:6938)()
595 for i from 0 <= i < len(args):
596 vec.push_back((<Expression>args[i])._gobj)
--> 597 res = g_function_evalv(self._serial, vec, hold)
598 elif self._nargs == 1:
599 res = g_function_eval1(self._serial,
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/function.pyx in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ (build/cythonized/sage/symbolic/function.cpp:13412)()
1224 res = self._evalf_try_(*args)
1225 if res is None:
-> 1226 return self._eval0_(*args)
1227 else:
1228 return res
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/integration/integral.py in _eval_(self, f, x, a, b)
203 for integrator in self.integrators:
204 try:
--> 205 A = integrator(*args)
206 except (NotImplementedError, TypeError, AttributeError):
207 pass
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/symbolic/integration/external.py in maxima_integrator(expression, v, a, b)
44 result = maxima.sr_integral(expression,v)
45 else:
---> 46 result = maxima.sr_integral(expression, v, a, b)
47 return result._sage_()
48
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/interfaces/maxima_lib.py in sr_integral(self, *args)
788 """
789 try:
--> 790 return max_to_sr(maxima_eval(([max_integrate],[sr_to_max(SR(a)) for a in args])))
791 except RuntimeError as error:
792 s = str(error)
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/libs/ecl.pyx in sage.libs.ecl.EclObject.__call__ (build/cythonized/sage/libs/ecl.c:7794)()
803 """
804 lispargs = EclObject(list(args))
--> 805 return ecl_wrap(ecl_safe_apply(self.obj,(<EclObject>lispargs).obj))
806
807 def __richcmp__(left, right, int op):
/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/libs/ecl.pyx in sage.libs.ecl.ecl_safe_apply (build/cythonized/sage/libs/ecl.c:5456)()
375 if ecl_nvalues > 1:
376 s = si_coerce_to_base_string(ecl_values(1))
--> 377 raise RuntimeError("ECL says: {}".format(
378 char_to_str(ecl_base_string_pointer_safe(s))))
379 else:
RuntimeError: ECL says: 1.8189894035458565e-12 is not of type (UNSIGNED-BYTE 15).
```

I have no idea where 1.8189894035458565e-12 came from, and I get the same error after assigning values to all of the variables except beta and u.

Has anyone encountered this error before? Is there a better way I should evaluate this?

A few remarks:

$\LaTeX$ing (i. e.

`view`

ing) your code show that your integrand has not the right form.Do

notreplace exact constants (1/2,i) by floating-point approximations (0.5,`CC.0`

). This may bias some tests (e. nullity tests) and mislead the algorithms.To avoid drowning in details, simplify your expression by replacing expression not involving the integration variable by constants.

Explicitely add assumptions : this often allows the integration algorithms to progress...

HTH,