В результате преобразования получилось
int (1 - x^2) dx/((1 + x^2) * (1 + 3 * x^2))
Знаю, что надо решать методом разложения на простейшие дроби, но как дальше???

int (1 - x^2) dx/((1 + x^2) * (1 + 3 * x^2)) =
= int (2 * (1 + x^2) - (1 + 3 * x^2)) dx/((1 + x^2) * (1 + 3 * x^2)) =
= 2 * int (1 + x^2) dx/((1 + x^2) * (1 + 3 * x^2)) -
- int (1 + 3 * x^2) dx/((1 + x^2) * (1 + 3 * x^2)) =
= 2 * int dx/(1 + 3 * x^2) - int dx/(1 + x^2) =
= 2 * 1/3 * int dx/(1/3 + x^2) - arctg x =
= | x = (1/3)^(1/2) * t; t = x/(1/3)^(1/2); dx = (1/3)^(1/2) dt | =
= 2/3 * int (1/3)^(1/2) dt/(1/3 + 1/3 * t^2) - arctg x =
= 2/3 * (1/3)^(1/2)/(1/3) * int dt/(1 + t^2) - arctg x =
= (2/3)/(1/3)^(1/2) * arctg t - arctg x + C = | t = x/(1/3)^(1/2) | =
= 2/3 * 3^(1/2) * arctg (3^(1/2) * x) - arctg x + C =
= 2/3^(1/2) * arctg (3^(1/2) * x) - arctg x + C