<s>[INST] Is it possible to solve this integral? I can't manage to solve this integral. Does it have an analytical solution?
$$\int\left(\frac{e^{x}(a-1)-1+\frac{1}{a}}{e^{x}(1-b)+1-\frac{1}{a}}\right)e^{-\frac{(x-(\mu-\frac{\sigma^{2}}{2})t)^{2}}{2t\sigma^{2}}}dx$$
 [/INST] $$\int\left(\frac{e^{x}(a-1)-1+\frac{1}{a}}{e^{x}(1-b)+1-\frac{1}{a}}\right)e^{-\frac{(x-c)^2}{d^2}}\,dx$$
If $b=1$, the integral is
$$-\frac{ \sqrt{\pi }}{2} \, d \left(a e^{c+\frac{d^2}{4}} \text{erf}\left(\frac{2
   c+d^2-2 x}{2 d}\right)+\text{erf}\left(\frac{x-c}{d}\right)\right)$$
If $b$ is "close" to $1$
$$\frac{e^{x}(a-1)-1+\frac{1}{a}}{e^{x}(1-b)+1-\frac{1}{a}}=\sum_{n=0}^\infty \left(\frac{a}{a-1}\right)^n\, e^{nx}\,(a e^x-1)\,(b-1)^n$$
$$I_n=\int e^{nx}\,(a e^x-1)\,e^{-\frac{(x-c)^2}{d^2}}\,dx$$
$$I_n=\frac{\sqrt{\pi }}{2} \, d \,e^{c n+\frac{d^2 n^2}{4}}\left(\text{erf}\left(\frac{2 c+d^2 n-2 x}{2 d}\right)-a e^{c+\frac{1}{4} d^2
   (2 n+1)} \text{erf}\left(\frac{2 c+d^2 (n+1)-2 x}{2 d}\right)\right)$$
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