$\int \frac{m{x}^{m+2n-1}-n{x}^{n-1}}{x^{2m+2n}+2x^{m+n}+1}dx$ is equal to 

 $$\begin{gathered} \int \frac{m{x}^{m+2n-1}-n{x}^{n-1}}{x^{2m+2n}+2x^{m+n}+1}dx \\ \mathrm{Devide} \mathrm{with} {\mathrm{x}}^{2\mathrm{n}} , \\ \mathrm{G}.\mathrm{I}=\int \frac{m{x}^{m-1}-n{x}^{-n-1}}{x^{2m}+2x^{m-n}+{\displaystyle x^{-2n}}}dx \\ =\int \frac{{\mathrm{mx}}^{\mathrm{m}-1}-n{x}^{-n-1}}{{\left({\mathrm{x}}^{\mathrm{m}}+x^{-n}\right)}^2}\mathrm{dx} \\ =\frac{-1}{\left({\mathrm{x}}^{\mathrm{m}}+{\mathrm{x}}^{-\mathrm{n}}\right)}+c=-\frac{x^n}{x^{m+n}+1}+C \end{gathered}$$