Evaluating :$\int \frac{1}{x^{10} + x}dx$

Not really different, but even simpler: $$\begin{align} \int\frac{1}{x^{10}+x} dx=&\int\frac{x^{-10}}{1+x^{-9}} dx =-\frac 1 9 \log |1+x^{-9}| + C \end{align}$$

Let we generalise the problem with a slightly different way. Consider $$\int\frac{\mathrm dx}{x^n+x}$$ Making substitution $x=\frac{1}{y}$ and $z=y^{n-1}$ then reverse it back we get \begin{align} \int\frac{\mathrm dx}{x^n+x}&=-\int\frac{y^{n-2}}{1+y^{n-1}}\mathrm dy\\[9pt] &=-\frac{1}{n-1}\int\frac{\mathrm dz}{1+z}\\[9pt] &=-\frac{\ln |1+z|}{n-1}+C\\[9pt] &=\bbox[8pt,border:3px #FF69B4 solid]{\color{red}{\large\ln |x|-\frac{\ln \left|1+x^{n-1}\right|}{n-1}+C}} \end{align} In your case $$\int\frac{\mathrm dx}{x^{10}+x}=\ln |x|-\frac{\ln \left|1+x^{9}\right|}{9}+C$$