$$ \log \left|c x e^x\left(1+x e^x\right)\right| $$

$$ \log \left|c x e^x\left(1+x e^x\right)\right| $$

$$ \log \left|\frac{c\left(1+x e^x\right)}{x e^x}\right| $$

$$ \log \left|\frac{c x e^x}{1+x e^x}\right| $$

$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2+1}{x \sqrt{2}}\right)+C $$

$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2-1}{\sqrt{2} x}\right)+C $$

$$ -\frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C $$

$$ \frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C $$