$$ \begin{array}{l} \int_ {0} ^ {1} \frac {1}{(x + 1) (x ^ {2} - 2 x + 2)} d x = \int_ {0} ^ {1} \left(\frac {A}{x + 1} + \frac {B x + C}{x ^ {2} - 2 x + 2}\right) d x \\ = \int_ {0} ^ {1} \left(\frac {\frac {1}{5}}{x + 1} + \frac {- \frac {1}{3} x + \frac {3}{5}}{x ^ {2} - 2 x + 2}\right) d x \\ = \frac {1}{5} \ln | 1 + x | \bigg | _ {0} ^ {1} - \frac {1}{1 0} \ln | x ^ {2} - 2 x + 2 | \bigg | _ {0} ^ {1} + \frac {2}{5} \arctan (x - 1) \bigg | _ {0} ^ {1} = \frac {3}{0} \ln 2 + \frac {1}{1 0} \pi . \\ \end{array} $$