Example 1. Find $\int \frac{1}{x(1+x)\left(1+x+k^{2}\right)} d x$
\begin{aligned} \text { Original expression } & =\int-\frac{\left(x^{2}+x+1\right)-x(1+x)}{x(1+x)} d x \\ & =\int\left[\frac{1}{x(1+x)}-\frac{1}{1+x+x^{2}}\right] d x \\ & =\ln \left\lvert\, \frac{x}{x+1}-\frac{2}{\sqrt{3}} \operatorname{arctg} \frac{2 x+1}{\sqrt{3}}+C\right.\end{aligned}