Step 1: Look at the integrand.
We want $\int e^{\tan x}\left(\sec^2 x+\sec^3 x\sin x\right)dx$. The exponent $\tan x$ is a strong hint to substitute.
Step 2: Simplify the second term.
Notice $\sec^3 x\sin x=\sec^2 x\cdot(\sec x\sin x)=\sec^2 x\cdot\tan x$, because $\sec x\sin x=\tan x$. So the bracket is $\sec^2 x(1+\tan x)$.
Step 3: Substitute.
Let $u=\tan x$, so $du=\sec^2 x\,dx$. The integral becomes $\int e^u(1+u)\,du$.
Step 4: Recognise a neat pattern.
Here $1+u=u+1$, and $\frac{d}{du}\left(u\,e^u\right)=e^u+u e^u=e^u(1+u)$. So the integrand is exactly the derivative of $u\,e^u$.
Step 5: Integrate.
Therefore $\int e^u(1+u)\,du=u\,e^u+c$.
Step 6: Substitute back.
Replacing $u=\tan x$ gives $\tan x\cdot e^{\tan x}+c$, which is option (A).
\[ \boxed{\,\tan x\cdot e^{\tan x}+c\,} \]