Is there a closed form for this definite integral? [duplicate]
Using the Reflection formula
$$\Gamma\left(1-z\right)\Gamma(z)=\frac{\pi}{\sin\pi z}$$ combined with what you found in that other answer: $$\int_{-\infty}^\infty \frac{1}{x^{2n} + 1} \, dx =\frac{\pi}{n\sin\frac \pi {2n}}$$