I need to evaluate some (one-variable) integrals that neither SAGE nor Mathematica can do symbolically. As far as I can tell, I have two options:


(a) Use GSL (via SAGE), Maxima or Mathematica to do numerical integration. This is really a non-option, since, if I understand correctly, the "error bound" they give is not really a guarantee.


(b) Cobble together my own programs using the trapezoidal rule, Simpson's rule, etc., and get rigorous error bounds using bounds I have for the second (or fourth, or what have you) derivative of the function I am integrating. This is what I have been doing.


Is there a third option? Is there standard software that does (b) for me?



Rigorous numerical integration
[Interval arithmetic](http://en.wikipedia.org/wiki/Interval_arithmetic) methods will permit rigorous bounds. You might try [INTLAB](http://www.ti3.tu-harburg.de/rump/intlab/). There are various books on rigorous numerics, e.g., [Warwick Tucker's *Validated Numerics*](http://books.google.com/books/about/Validated_Numerics.html?id=IEN56sqHtR8C), and [the journal *Reliable Computing*](http://interval.louisiana.edu/reliable-computing-journal/) is dedicated to such things.


