Integral of $$$\frac{x^{n - 1}}{x^{2 n} + 1}$$$ with respect to $$$x$$$

Find $$$\int \frac{x^{n - 1}}{x^{2 n} + 1}\, dx$$$.

This integral does not have a closed form:

$${\color{red}{\int{\frac{x^{n - 1}}{x^{2 n} + 1} d x}}} = {\color{red}{\frac{x^{n} {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, 1 \\ \frac{3}{2} \end{matrix}\middle| {- x^{2 n}} \right)}}{n}}}$$

Therefore,

$$\int{\frac{x^{n - 1}}{x^{2 n} + 1} d x} = \frac{x^{n} {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, 1 \\ \frac{3}{2} \end{matrix}\middle| {- x^{2 n}} \right)}}{n}$$

Simplify:

$$\int{\frac{x^{n - 1}}{x^{2 n} + 1} d x} = \frac{\operatorname{atan}{\left(x^{n} \right)}}{n}$$

Add the constant of integration:

$$\int{\frac{x^{n - 1}}{x^{2 n} + 1} d x} = \frac{\operatorname{atan}{\left(x^{n} \right)}}{n}+C$$

$$$\int \frac{x^{n - 1}}{x^{2 n} + 1}\, dx = \frac{\operatorname{atan}{\left(x^{n} \right)}}{n} + C$$$A