An integral for the Prime

[parisse]

I don't think it's useful if you get a series, I believe most of the time it's better to replace a series by an integral (and you could get infinite recursions in giac, since some series are solved by replacing them by integrals). Of course there are some exceptions where you could replace your integral by a series where the sum is known.
Here you could do that:

Code:


normal(subst(int(ln(x)/(1+x^2),x,1,inf),x=1/t)
assume(m>=0);
am:=int(x^(2*m)*ln(x),x,0,1);
S:=sum((-1)^m*am,m,0,inf);

Then giac does not find a closed formula for S, perhaps there is some trick to do that?