Let
$$
f(n) := \int_0^\infty \log\left( \frac{(1+t)^n +(1-t)^n}{2} +n(n-1) t(1+t)^{n-2}\right)t^{- 3/2} \ \mathrm{d}t
$$
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?