Indefinite Integration

$\log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+\frac{x}{1+x \mathrm{e}^x}+c$,
(where c is a constant of integration)

$\quad \log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+\frac{\mathrm{e}^x}{1+x \mathrm{e}^x}+c$,
(where c is a constant of integration)

$\quad \log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|+\frac{1}{1+x \mathrm{e}^x}+c$,
(where c is a constant of integration)

$\log \left|\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right|-\frac{1}{1+x \mathrm{e}^x}+c$,
(where c is a constant of integration)