Integration in question could not be resolved. 
I do not know how to solve this integration

$e^x$ and $\log x$ are inverse functions, in the sense that $e^{\log x} = \log(e^x) = x$. And so you really just have $\displaystyle \int \sec^2 x \mathrm d x$.
As an aside, since you know the derivative of $\tan x$, and antiderivatives are unique up to constants, you know that the integrand must be $\sec^2 x$ up to a constant. 
