Show that

$\frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)}$

Let $e^{x}=t \Rightarrow e^{x} d x=d t$

$\Rightarrow \int \frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)} d x=\int \frac{d t}{(t+1)(t+2)}$

$=\int\left[\frac{1}{(t+1)}-\frac{1}{(t+2)}\right] d t$

$=\log |t+1|-\log |t+2|+\mathrm{C}$

$=\log \left|\frac{t+1}{t+2}\right|+\mathrm{C}$

$=\log \left|\frac{1+e^{x}}{2+e^{x}}\right|+\mathrm{C}$