I want to evaluate the following integral numerically in Mathematica, $$\int_0^{\infty}(\frac{1}{(1 - e^t)^{10}} - (\frac{1}{t^{10}} - \frac{5}{t^9} + \frac{145}{12 t^8} - \frac{75}{ 4 t^7} + \frac{3013}{144 t^6} - \frac{285}{16 t^5} + \frac{4523}{378 t^4} - \frac{6515}{ 1008 t^3} + \frac{7129}{2520 t^2} - \frac{1}{t}))dt$$ which is convergent since the part that is subtracted from $\frac{1}{(1-e^x)^{10}}$ is essentially the negative powers of the Laurent expansion of $\frac{1}{(1-e^x)^{10}}$ near 0, which is the divergent part.
However, if I directly use NIntegrate in Mathematica, I get a divergent result. This is not too surprising given the unrealistic plot of the integrand near zero:
So is there a good way to actually evaluate this numerical integral?