Integrals of rational functions
These are examples of rational functions integrated by Symtegration. Here, rational functions mean the ratio of two polyomials, not functions of rational numbers.
\(\int \frac{x}{1 + x} \, dx = -\log \left(1 + x\right) + x\)
\(\int \frac{x^{2}}{1 + x^{2}} \, dx = \tan^{-1} \left(-x\right) + x\)
\(\int \frac{x^{2} + x}{x - 1} \, dx = 2 \log \left(-1 + x\right) + 2 x + \frac{1}{2} x^{2}\)
\(\int \frac{1}{x^{3} - x^{5}} \, dx = -\frac{1}{2} \log \left(\frac{3}{2} - \frac{3}{2} x^{2}\right) - \frac{1}{2 x^{2}} + \log \left(9 x\right)\)
\(\int \frac{x^{6} + x + 1}{x^{2} - 1} \, dx = -\frac{1}{2} \log \left(2 + 2 x\right) + x + \frac{1}{3} x^{3} + \frac{1}{5} x^{5} + \frac{3}{2} \log \left(2 - 2 x\right)\)
\(\int \frac{x^{4} - 3 x^{2} + 6}{x^{6} - 5 x^{4} + 5 x^{2} + 4} \, dx = \tan^{-1} x + \tan^{-1} x^{3} + \tan^{-1} \frac{x - 3 x^{3} + x^{5}}{2}\)
Unsupported integrals
Some rational functions cannot be symbolically integrated yet. Some may not be feasible to derive if they require solutions to polynomials of degree 5 or more. Others would be feasible, but would require support for deriving real roots of simultaneous polynomials beyond those that are rational numbers, or support for solving general quartic polynomials.
\(\int \frac{1}{1 + x^{4}} \, dx = \bot\)
\(\int \frac{x}{1 + x^{10}} \, dx = \bot\)
\(\int \frac{x^{5} + 4 x^{2}}{x^{3} + 2 x^{2} + 3} \, dx = \bot\)