$\int {\mathrm{e}}^{\mathrm{x}}\left[\frac{1}{\sqrt{1+{\mathrm{x}}^2}}+\frac{1-2{\mathrm{x}}^2}{\sqrt{{\left(1+{\mathrm{x}}^2\right)}^5}}\right]\mathrm{dx}$

Solution:

$\begin{array}{l}I=\int e^x\left[\frac{1}{\sqrt{1+x^2}}-\frac{x}{\sqrt{{\left(1+x^2\right)}^3}}+\frac{x}{\sqrt{{\left(1+x^2\right)}^3}}+\frac{1-2x^2}{\sqrt{{\left(1+x^2\right)}^5}}\right]dx\\ I=e^x\frac{1}{\sqrt{1+x^2}}+e^x\frac{x}{\sqrt{{\left(1+x^2\right)}^3}}+C\\ I=e^x\left[\frac{1}{\sqrt{1+x^2}}+\frac{x}{\sqrt{{\left(1+x^2\right)}^3}}\right]+C\end{array}$

$\left[\because \text{ using }\int {\mathrm{e}}^{\mathrm{x}}\left[\mathrm{f}\left(\mathrm{x}\right)+{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\right]\mathrm{dx}={\mathrm{e}}^{\mathrm{x}}\mathrm{f}\left(\mathrm{x}\right)+\mathrm{C}\right]$