I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:

$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$

where $a > 0$ and $d \geq 3$. My guess for the mean:

$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d} )$

The reference I checked (Ledoux & Talagrand, 1991) helpfully told me that this calculation is "standard". The argument apparently depends on integration by parts, but I can't figure out the trick.