<s>[INST] Problem on indefinite integration $$ f(x) = \int\frac{x^2}{(1+x^2)(1 + \sqrt{1+x^2})}dx$$
where $f(0)=0$, find $f(1)$.
Despite many tries by taking $$\sqrt{1+x^2}=t $$  or 
$$ 1+\sqrt{1+x^2}$$ as $t$, I could not get it solved.
 [/INST] Hint:
$$I = \int \frac{x^2}{(1+x^2)(1+\sqrt{1+x^2})} \mathrm{d}x = \frac{x^2(\sqrt{1+x^2}-1)}{(1+x^2)(1+\sqrt{1+x^2})(\sqrt{1+x^2}-1)} \mathrm{d}x =\int \frac{\sqrt{x^2+1}-1}{1+x^2}\mathrm{d}x = \int \frac{1}{\sqrt{1+x^2}}-\frac{1}{1+x^2}\mathrm{d}x$$
Hope you can take it from here.
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