If $\int e^{secx}\left(secx\tan xf\left(x\right)+\left(secx\tan x+se{c}^{2}x\right)\right)dx=e^{secx}f\left(x\right)+C,$ then a possible choice of $f\left(x\right)$ is:

$\int e^{\sec x}(\sec x\tan xf\left(x\right)+\left(\sec x\tan x+{\sec }^{2}x\right)dx=e^{\sec x}f\left(x\right)+C$

Differentiating both sides w.r.t $‘x’$ we get

$e^{secx}\left(secx\tan xf\left(x\right)+\left(secx\tan x+{\sec }^{2}x\right)\right)=\left(e^{\sec x}\cdot \sec x\tan x\right)f\left(x\right)+e^{\sec x}{f}^{\text{'}}\left(x\right)$

$\Rightarrow f^{\text{'}}\left(x\right)={\mathrm{s}\mathrm{e}\mathrm{c}}^{2}x+\mathrm{t}\mathrm{a}\mathrm{n}x\mathrm{s}\mathrm{e}\mathrm{c}x$

$\Rightarrow f\left(x\right)=\int \left({\sec }^{2}x+\tan x\sec x\right)dx$

$\Rightarrow f\left(x\right)=\mathrm{t}\mathrm{a}\mathrm{n}x+\mathrm{s}\mathrm{e}\mathrm{c}x+c, c\in R$
Hence, possible choice is $f\left(x\right)=\mathrm{s}\mathrm{e}\mathrm{c}x+\mathrm{t}\mathrm{a}\mathrm{n}x+\frac{1}{2}.$