thank you for the answer !
its so weird, sometimes it works sometimes doesnt.
now i upgraded to sage 9.4, and the last example that i showed perfectly worked:
sage: f = -(x - 1)*((x - 1)^2/((x - 1)^2 + 1)^2 + 1/((x - 1)^2 + 1)^2)/sqrt((x - 1)^2 + 1)
sage: integrate(f, x)
1/sqrt(x^2 - 2*x + 2)
thank you for telling about numerical computation of integral, i didnt knew such thing existed and i will definitely use it!
but im also interesting into knowing the integral itself, although the last example is solved, but now i have a new very familiar problem:
sage: a,b = var('a,b')
sage: integrate((((a - b)^2/(abs(-a + b)^2 + 1) - 1)^2 + (a - b)^2/(abs(-a + b)^2 + 1)^2)/(abs(-a + b)^2 + 1)^(3/2), b)
integrate((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 - 2*a^2*abs(a - b)^2 + 4*a*b*abs(a - b)^2 - 2*b^2*abs(a - b)^2 + abs(a - b)^4 - a^2 + 2*a*b - b^2 + 2*abs(a - b)^2 + 1)/(abs(a - b)^2 + 1)^(7/2), b)
sage: integrate((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 - 2*a^2*abs(a - b)^2 + 4*a*b*abs(a - b)^2 - 2*b^2*abs(a - b)^2 + abs(a - b)^4 - a^2 + 2*a*b - b^2 + 2*abs(a - b)^2 + 1)/(abs(a - b)^2 + 1)^(7/2), b)
(outputs some good long answer that i can actually use)
i dont understand why when i run this the first time it cant find antiderivative, but when i run its output it does solves the integral.
its exactly the same expression just in a different form, then why it cant work on the first time ? (maybe its a bug ¿)