Integrals involving NORMALD_CDF - possible CAS bug?

[Albert Chan]

(02-23-2020 10:47 PM)Nigel (UK) Wrote:  I tried to evaluate the integral \[\int_{-10}^{10} \hbox{NORMALD}(X)*\hbox{NORMALD_CDF}(X)\,{\rm d}X \] in the Home screen, expecting to get an answer close to \(0.5\). Instead I got an error message - "Error: Undefined result". This was strange - the integrand is perfectly well-behaved and gives a smooth graph in the Function app.

XCas also unable to symbolically evaluate the integral.
But, numerical integration is easy, using either romberg, or gaussquad, both returning 0.5

BTW, this integral is trivial if you let c = normald_cdf(x) → dc = normald(x) dx

I(c) = ∫ c dc = c²/2

Let lower limit z = normald_cdf(-10), upper limit = normald_cdf(10) = 1-z

Integral = I(1-z) - I(z) = (1-z+z)(1-z-z)/2 = ½ - z ≈ ½