If your fraction stack is growing out of control, you can just get rid of the fraction entirely - write
\frac{\frac{1}{x}+e^x}{1+\frac{1}{\sin^{-1} x}}
as
\left( \frac{1}{x}+e^x \right) \left( 1+\frac{1}{\sin^{-1} x} \right) ^{-1}.
Another useful trick is
e^{\frac{1}{x} + \sin x} \to...