Prove

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

Let $e^{x} x=t$

$\Rightarrow\left(e^{x} \cdot x+e^{x} \cdot 1\right) d x=d t$

$e^{x}(x+1) d x=d t$

$\therefore \int \frac{e^{x}(1+x)}{\sec ^{2}\left(x^{x}\right)} d x=\int \frac{d t}{\cos ^{2} t}$

$=\int \sec ^{2} t d t$

$=\tan t+\mathrm{C}$

$=\tan \left(e^{x} \cdot x\right)+\mathrm{C}$

Hence, the correct answer is B.