An integral for the Prime

[Alberto Candel]

The following integral is from the 2005 Putnam competition:
$$\int_0^1 \dfrac{\ln(1+x)}{1+x^2} dx =\dfrac{\pi}{8} \ln 2.$$
In CAS, the input integrate(ln(1+x)/(1+x^2),x,0,1)} returns itself, while approx(integrate(ln(1+x)/(1+x^2),x,0,1)) returns 0.272198261288.

In WolframAlpha, the input (log is ln) integrate(log(1+x)/(1+x^2),x,0,1)} returns 0.272198261288 first and quickly changes to the exact value \((\pi/8)\log 2.\) Mathematica returns the exact value directly.

It would be nice if the Prime would return exact values for these logarithmic integrals. Maybe a "Dictionary or Real Numbers" (like the book of J and P Borwein) could be included in it (as part of the units) and used as a look-up table for finding the exact value.