Let  $f\left(x\right)=\frac{\log (1+x^2)}{(x^2-1)(x^2-25)}$  then

$f\left(x\right)=\frac{\log (1+x^2)}{(x^2-1)(x^2-25)}=\frac{\log (1+x^2)}{(x-1)(x+1)(x-5)(x+5)}$

since $[-1,\infty )$is continuous on $\log ( 1 + x 2 )$so f is continuous on $\Re -\{-5,-1,1,5\}$ which clearly contains the interval f is continuous on [6, 10]