$\begin{aligned} & I=\int_{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x \\ & =x \log _e\left(x+\sqrt{x^2+1}\right)-\int_{-1}^2\left(\frac{1+\frac{x}{\sqrt{x^2+1}}}{x+\sqrt{x^2+1}}\right) d x \\ & =x \log _e\left(x+\sqrt{x^2+1}\right)-\int_{-1}^2 \frac{x}{\sqrt{x^2+1}} d x\end{aligned}$
$\begin{aligned} & =x \log _e\left(x+\sqrt{x^2+1}\right)-\left.\sqrt{x^2+1}\right|_{-1} ^2 \\ & =\left(2 \log _e(2+\sqrt{5})-\sqrt{5}\right) \\ & -\left(-\log _e(-1+\sqrt{2})-\sqrt{2}\right) \\ & =\log _e(2+\sqrt{5})^2-\sqrt{5}+\log _e(\sqrt{2}-1)+\sqrt{2} \\ & =\log _e(2+\sqrt{5})^2-\sqrt{5}+\log _e(\sqrt{2}-1)+\sqrt{2}\end{aligned}$
$\begin{aligned} & =\sqrt{2}-\sqrt{5}+\log _e\left(\frac{(2+\sqrt{5})^2}{\sqrt{2+1}}\right) \\ & =\sqrt{2}-\sqrt{5}+\log _e\left(\frac{9+4 \sqrt{5}}{\sqrt{2+1}}\right)\end{aligned}$