If  $f\left(x\right)={\displaystyle \int \frac{x^{2}dx}{(1+x^2)\left(1+\sqrt{1+x^2}\right)}}$   and $f\left(0\right)=0$ then $f\left(1\right)=$

           $x=\tan \theta ,dx={\sec }^2\theta d\theta$ 

$f\left(x\right)={\displaystyle \int \frac{{\tan }^2\theta {\sec }^2\theta }{{\sec }^2\theta \left(1+\sec \theta \right)}}d\theta$

           $={\displaystyle \int \left(\sec \theta -1\right)}d\theta$

           $=\log \left|\sec \theta +\tan \theta \right|-\theta +c$

$f\left(0\right)=\log \left(1\right)-0+c\Rightarrow c=0$

$f(x=1)=f(\theta =\frac{\pi }{4})$

                  $=\log \left|\sqrt{2}+1\right|-\frac{\pi }{4}+0$