I read on a French math forum that one has $ \int_{0}^{\infty}\dfrac{\log^{2}x}{1+x^2}dx=\dfrac{\pi^{3}}{8} $. As Cramer's conjecture in its strong form predicts that $ \lim\sup\dfrac{p_{n+1}-p_{n}}{\log^{2}p_{n}}\leq 1 $ , could a sufficiently accurate computation of $ I : =\int_{0}^{\infty}\dfrac{p_{\pi(x)+1}-p_{\pi(x)}}{1+x^{2}}dx $ shed a light on this conjecture ?
Edit : Assume $ p_{0}=1 $.