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There is a file gauss_legendre.pyx in src/sage/numerical/ which curiously does not result in a gauss_legendre entry in the reference manual under sage/numerical.

There Edited to take into account suggestions and observations by @FrédéricC and @nbruin.

Sage's numerical_integral function uses a heuristic to guess an error bound, provide no certified error bound.

The good way to go is a to use Arb, which is easy from Sage.

Example (from http://fredrikj.net/math/scan2018.pdf):

sage: C = ComplexBallField(100)
sage:  C.integral(lambda x, _: cos(x) * sin(x), 0, 1)
[0.35403670913678559674939205737 +/- 8.68e-30]

[Side observation: the file [gauss_legendre.pyx ](https://github.com/sagemath/sage/blob/master/src/sage/numerical/gauss_legendre.pyx) in [src/sage/numerical/ which curiously ](https://github.com/sagemath/sage/blob/master/src/sage/numerical/) strangely does not result in a gauss_legendre entry in the reference manual under [sage/numerical.](http://doc.sagemath.org/html/en/reference/numerical/sage/numerical/).]

Edited to take into account suggestions and observations by @FrédéricC and @nbruin.

Sage's numerical_integral function uses a heuristic to guess an error bound, provide no certified error bound.

The good way to go is to use Arb, Arb, which is easy from Sage.

Example (from http://fredrikj.net/math/scan2018.pdf):

sage: C = ComplexBallField(100)
sage:  C.integral(lambda x, _: cos(x) * sin(x), 0, 1)
[0.35403670913678559674939205737 +/- 8.68e-30]

[Side observation: the file [gauss_legendre.pyx](https://github.com/sagemath/sage/blob/master/src/sage/numerical/gauss_legendre.pyx) in [src/sage/numerical/](https://github.com/sagemath/sage/blob/master/src/sage/numerical/) strangely does not result in a gauss_legendre entry in the reference manual under [sage/numerical](http://doc.sagemath.org/html/en/reference/numerical/sage/numerical/).]

Edited to take into account suggestions and observations by @FrédéricC and @nbruin.

Sage's numerical_integral function uses a heuristic to guess an error bound, provide no certified error bound.

The good way to go is to use Arb, which is easy from Sage.

Example (from http://fredrikj.net/math/scan2018.pdf):

sage: C = ComplexBallField(100)
sage:  C.integral(lambda x, _: cos(x) * sin(x), 0, 1)
[0.35403670913678559674939205737 +/- 8.68e-30]

[Side Side observation: the file [gauss_legendre.pyx](https://github.com/sagemath/sage/blob/master/src/sage/numerical/gauss_legendre.pyx) in [src/sage/numerical/](https://github.com/sagemath/sage/blob/master/src/sage/numerical/) strangely does not result in a gauss_legendre entry in the reference manual under [sage/numerical](http://doc.sagemath.org/html/en/reference/numerical/sage/numerical/).].