Khẳng định nào sau đây sai?

A. $\underset{}{\overset{}{\int }}{e}^{x}dx=\frac{e^{x+1}}{x+1}+C$

B. $\underset{}{\overset{}{\int }}\frac{1}{x}dx=\ln \left|x\right|+C$

C. $\underset{}{\overset{}{\int }}\frac{1}{{\cos }^{2}x}dx=\tan x+C$

D. $\underset{}{\overset{}{\int }}{x}^{e}dx=\frac{x^{e+1}}{e+1}+C\left(e\ne -1\right).$