Prove that if $f(x) = lnx$ ,
$\int_{1}^{e^2}\frac{f(xe^{x+1})dx}{(x+1)^2\ +\ f^2(x^x)} > \frac{\pi}{4}$
My attempt : I simplified the integrand and got the below integral.
$\int_{1}^{e^2}\frac{(lnx\ +\ x\ +\ 1)dx}{(x+1)^2\ +\ x^2(lnx)^2}$
After this, I thought of applying inequalities to compare this integral with the integral $\int_{1}^{e^2}\frac{2}{x((lnx)^2+4)}dx$ which is equal to $\frac{\pi}{4}$. But I do not know how to compare these integrals.
Also, please let me know if there are any other methods of proving this inequality.
(Please provide a solution which does not require the use of a graphing calculator / website)