$\text{ Evaluate }\int \frac{e^x(1+x)}{{\cos }^2\left(x{e}^x\right)}dx$

$\text{ Let }z=x{e}^x.\text{ Then }dz=\left(1e^x+x{e}^x\right)dx=e^x\left(1+x\right)dx\text{ . }$

$\begin{array}{l}Thus, \int \frac{e^x(1+x)}{{\cos }^2\left(x{e}^x\right)}dx=\int \frac{dz}{{\cos }^{2}z}\\ =\int {\sec }^{2}zdz\\ =\tan z+c=\tan \left(x{e}^x\right)+c\end{array}$