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.

Substitute $1-x^2e^{2\sin x} = u$
This gives $$-[2xe^{2\sin x} + 2x^2 \cos x e^{2\sin x}] dx = du \\
-2xe^{2 \sin x} (1+x\cos x) dx = du$$
Putting the value of $dx$ in your integral, we get 
$$\int \frac{du}{-2x^2e^{2\sin x}(u)}$$
We know that $$\begin{align} 1-x^2 e^{2\sin x} &= u \\
x^2 e^{2\sin x} &= 1-u \\
-2x^2 e^{2\sin x} &= 2(u-1)\end{align}$$
This turns the integral to 
$$\int \frac{du}{2u(u+1)}$$
Use partial fractions to go further.
