A friend of mine, obtained a lower bound for the trace norm of matrices described in [this question](https://mathoverflow.net/questions/302424/is-this-lower-bound-for-a-norm-of-some-complex-matrices-true) (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where
$$
f(n) := \int_0^\infty \log\left( \frac{(1+t)^n +(1-t)^n}{2} +n(n-1) t(1+t)^{n-2}\right)t^{- 3/2} \ \mathrm{d}t
$$
Numerical computaions suggest that 
$$ f(n) = 4 \pi n + o(n) $$
How to justify it? Moreover, is it possible to obtain a good rate of convergence?