It's perfectly standard in the context of Lebesgue intergration to refer to the "almost everywhere" antiderivative as the antiderivative. For an absolutely continuous integrand, the antiderivative is everywhere differentiable (just because an antiderivative is continuous does not mean it is everywhere differentiable), that derivative equals the integrand everywhere, and we reproduce the familiar results of Riemann integration. For less continuous functions, derivative of the antiderivative and the integrand will agree in regions where the integrand is absolutely continuous, and the antiderivative will not be differentiable elsewhere.

As for the substance of your complaint, I think it is a sad fact that all software packages have bugs. One way of validating the integrals is to use NIntegrate, substituting in random values for free parameters, and comparing against the numericized result for integrate. Hardly perfect, but catches a lot issues.