Integrate $ \int \frac{1+x\cos(x)}{x(1-x^2(e^{2\sin(x)}))}dx $ $$ \int \frac{1+x\cos x}{x(1-x^2e^{2\sin x})}dx  $$
Attempt:
I substituted $(1-xe^{2\sin(x)})$ by $u$ and tried from there by differentiating it. But I get stuck midway.

HINT:
Note that we can write
$$\begin{align}
\frac{1+x\cos(x)}{x(1-x^2e^{2\sin(x)})}&=(1+x\cos(x))\left(\frac{1}{x(1-x^2e^{2\sin(x)})}\right)\\\\
&=(1+x\cos(x))\left(\frac{1}{x}+\frac{xe^{2\sin(x)}}{1-x^2e^{2\sin(x)}}\right)\\\\
&=\frac1x+\cos(x)+\frac{(x+x^2\cos(x))e^{2\sin(x)}}{1-x^2e^{2\sin(x)}} \tag 1
\end{align}$$
Can you find a substitution for the third term in $(1)$ that facilitates quick integration?
