I am developing a monetary economics model with an exponential growth term embedded within an exponential function. When I solve the equation, I wind up either with everything cancelling out, or with $\int f(x) dx = \int \frac{e^{g(x)}}{g'(x)} dx$.
Life would be so much easier, if it were $\displaystyle\int e^{g(x)}g'(x)\, dx$, but, sadly, $g'(x)$ winds up in the denominator.
Any suggestions, even just hints, would be greatly appreciated!
If the starting problem is $$I=\int e^{-r\, e^{\,\rho \, t}} \,dt$$ let $$r\, e^{\,\rho \, t}=x \implies t=\frac{1}{\rho}{\log \left(\frac{x}{r}\right)}\implies dt=\frac{dx}{\rho x}$$ $$I=\frac{1}{\rho}\int \frac{e^{-x}}{ x}\,dx=\frac{1}{\rho}\text{Ei}(-x)$$ where appears the exponential integral function already mentioned by lulu. Back to $t$ $$I=\frac{1}{\rho}\text{Ei}\left(-r\, e^{\,\rho\, t}\right)$$
However, I do not see the connection with the problem in title.