f(x):=1/(x*sqrt(1+x^2));F(x):=integrate(f(x),x)$ F(x);

Asinh(x):=log(x+sqrt(x^2+1));is(equal(Asinh(x),asinh(x)));

F(x):=-Asinh(1/abs(x))$ F(x);

full:logcontract(F(3) - F(1));

inside:%e^full;

n:num(inside);
d:denom(inside);
complement:(sqrt(10)-1);

N:expand(n*complement);D:expand(d*complement);

result:log(ratsimp(N/D));

is((0 <= result) and (result<=sqrt(2)));

float(result);