I have an integral of the form
$$ I=\int\_0^{\infty}{dx}\ln \bigg(1+\exp(-\frac{f(x)}{a})\bigg) $$ where $a$ is a positive constant and $f(x)$ is a regular and positive function such that $I$ is finite (for example: $f(x)=x$); moreover $f(x)=x+O(x^3)$ for $x \rightarrow 0$. I have to expand this integral $I=c\_0+c\_1a+..+c\_na^n+O(a^{n+1})$ for $a \rightarrow 0^+$, with the value of the integral beeing dominated by the behaviour of $f(x)$ near $0$.
Can anyone give me some hints or references to compute the first coefficients of this expansion?



Expansion of an integral
Let $f(x)$ have the expansion $f(x)=x+c\_3 x^3+c\_4 x^4 +\cdots$, then define $y=x/a$ and you have
$$I=\int\_0^{\infty}{dx}\ln \bigg(1+\exp\left(-\frac{f(x)}{a}\right)\bigg)$$
$$=\int\_0^\infty \left[a\ln \left(1+e^{-y}\right)-\frac{a^3 c\_3 y^3}{e^y+1}-\frac{a^4 c\_4 y^4}{e^y+1}+{\cal O}(a^5)\right]\,dy$$
$$=\frac{1}{12} \pi ^2 a-\frac{7}{120} \pi ^4 c\_3 a^3-\frac{45}{2} \zeta (5)c\_4 a^4+{\cal O}(a^5).$$
The term of order $a^p$ has coefficient $-c\_p\left(1-2^{-p}\right)p! \zeta (p+1)$, when $c\_p$ is the coefficient of order $x^p$ in the expansion of $f(x)$.


