$\int\limits_{1}^{2}\frac{1-x^2}{x+x^3}dx=\int\limits_{1}^{2}\frac{1+x^2-2x^2}{x(1+x^2)}dx$$=\int\limits_{1}^{2}(\frac{1}{x}-\frac{2x}{1+x^2})dx$
$=\int\limits_{1}^{2}\frac{1}{x}dx-\int\limits_{1}^{2}\frac{2xdx}{1+x^2}$
$=\int\limits_{1}^{2}\frac{1}{x}dx-\int\limits_{1}^{2}\frac{d(1+x^2)}{1+x^2}$
$=lnx|_1^2-ln(1+x^2)|_1^2$
$=ln2-ln1-(ln5-ln2)$
$=2ln2-ln5=ln4-ln5=ln\frac{4}{5}$