Difficulty to compute an integral Have somebody ideas to evaluate the following integral ? $$J_n=\int_{-\infty}^{+\infty} \left(\frac{\pi^2}{4}-\arctan(x)^2\right)^n\,dx$$ I'm trying this because I have shown that the empiric median $\widehat\theta_n$ in the Cauchy distribution located in $\theta$ verifies : $$\mathbb E_\theta(\widehat\theta_n)=(2n+1)\frac{\binom{2n}{n}}{\pi^{2n}}J_n\theta$$ So if the factor could be one, the empiric median will be a good estimator of $\theta$...

For sure, this does **not** provide an answer to the question; so forgive me if I am off topic.

Assuming $n$ to be a positive integer, a CAS found some nice expressions such as $$J_2=6 \pi \zeta (3)$$ $$J_3=45 \pi \zeta (5)-3 \pi ^3 \zeta (3)$$ $$J_4=630 \pi \zeta (7)-60 \pi ^3 \zeta (5)$$ $$J_5=15 \left(\pi ^5 \zeta (5)-105 \pi ^3 \zeta (7)+945 \pi \zeta (9)\right)$$

Looking at the numerical values for $2 \leq n \leq 20$, it seems that $$J_n \approx e^{0.874368 n+0.146319}$$ is a quite good approximation.

I hope this could be helping you somehow.