分子が対数で分母が多項式の定積分

\begin{align*} \int_{0}^{\infty}\frac{\log x}{x^{n}+1}dx & =\left[\frac{d}{dt}\int_{0}^{\infty}\frac{x^{t}}{x^{n}+1}dx\right]_{t=0}\\ & =\frac{1}{n}\left[\frac{d}{dt}\int_{0}^{\infty}\frac{y^{\frac{t}{n}}}{y+1}y^{\frac{1}{n}-1}dy\right]_{t=0}\\ & =\frac{1}{n}\left[\frac{d}{dt}\int_{0}^{\infty}\frac{y^{\frac{1+t}{n}-1}}{y+1}dy\right]_{t=0}\\ & =\frac{1}{n}\left[\frac{d}{dt}B\left(\frac{1+t}{n}-1+1,1-\left(\frac{1+t}{n}-1\right)-1\right)\right]_{t=0}\\ & =\frac{1}{n}\left[\frac{d}{dt}B\left(\frac{1+t}{n},1-\frac{1+t}{n}\right)\right]_{t=0}\\ & =\frac{1}{n}\left[\frac{d}{dt}\frac{\Gamma\left(\frac{1+t}{n}\right)\Gamma\left(1-\frac{1+t}{n}\right)}{\Gamma\left(\frac{1+t}{n}+1-\frac{1+t}{n}\right)}\right]_{t=0}\\ & =\frac{1}{n}\left[\frac{d}{dt}\Gamma\left(\frac{1+t}{n}\right)\Gamma\left(1-\frac{1+t}{n}\right)\right]_{t=0}\\ & =\frac{\pi}{n}\left[\frac{d}{dt}\sin^{-1}\left(\frac{1+t}{n}\pi\right)\right]_{t=0}\\ & =\frac{\pi}{n}\left[-\frac{\pi}{n}\sin^{-2}\left(\frac{1+t}{n}\pi\right)\cos\left(\frac{1+t}{n}\pi\right)\right]_{t=0}\\ & =-\frac{\pi^{2}}{n^{2}}\sin^{-2}\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right) \end{align*}