Does $\int{\frac{x}{\cos(x)}dx}$ have an elementary solution? I need to solve the following integral: $\displaystyle\int\frac{x}{\cos(x)}\,dx$. My procedure is the following:
\begin{align*}\int\frac{x}{\cos(x)}\,dx &= \int x\sec(x)\,dx\\
&=x\ln(\tan(x)+\sec(x))-\int\ln(\tan(x)+\sec(x))\,dx.
\end{align*}
But, I'm stuck at this step, after using integration by parts I have $\ln(\tan(x)+\sec(x))$ inside the new integral and then I do not know how to solve it, I was trying using by parts again but it gets more complicated.
Any advice on how to continue? I looked for related questions to this problem here in math.stackexchange but did not find anything useful.

Perhaps the expanding of $e^x$ be useful, well
\begin{align}
\int\dfrac{x}{\cos x}dx
&= \int\dfrac{2xe^{-ix}}{1+e^{-2ix}}dx \\
&= \int 2xe^{-ix}\sum_{n\geq0}e^{-2inx}dx \\
&= \sum_{n\geq0}\int 2xe^{-i(2n+1)x}dx \\
&= \sum_{n\geq0}2e^{-i(2n+1)x}\left(\dfrac{ix}{2n+1}+\dfrac{1}{(2n+1)^2}\right) \\
\end{align}
