I am reading a paper Albrecher, Constantinescu and Loisel "2011Explicit ruin formulas for models with dependence among risks" and getting stuck at one integral (Example 2.4): $$\int _{\frac{\lambda }{c}}^{\infty }\frac{\lambda }{\theta c}e^{-u \theta } e^{\frac{\lambda c}{ u}} \frac{\alpha }{2\sqrt{\pi \theta ^3}}e^{-\frac{\alpha ^2}{4\theta }}d\theta.$$ The $\theta^{-\frac{5}{2}}$ in the integrand kills my attempts. Three authors claim that it should be: \begin{equation} \frac{\lambda }{\alpha ^2 c}e^{-\frac{\alpha ^2c}{4\lambda }}\left\{-\frac{2 \alpha }{\sqrt{\pi \frac{\lambda }{c}}}+e^{\frac{\left(\alpha c-2 \lambda \sqrt{u}\right)^2}{4c \lambda }}\left(1+\alpha \sqrt{u}\right)\text{erfc}\left[\sqrt{\frac{u \lambda }{c}}-\frac{\alpha }{2\sqrt{\frac{\lambda }{c}}}\right]+e^{\frac{\left(\alpha c+2 \lambda \sqrt{u}\right)^2}{4c \lambda }}\left(-1+\alpha \sqrt{u}\right)\text{erfc}\left[\sqrt{\frac{u \lambda }{c}}+\frac{\alpha }{2\sqrt{\frac{\lambda }{c}}}\right]\right\}, \end{equation} where $\operatorname{erfc}(t) = \frac{2}{\sqrt{\pi}} \int_t^\infty e^{-x^2} dx$.
I cannot see how to do this. I start from the above and end with the following: $$\frac{\lambda }{\alpha c}\left\{-\frac{2 }{\sqrt{\pi \frac{\lambda }{c}}}e^{-\frac{\alpha ^2c}{4\lambda }}+\frac{ e^{\frac{u \lambda }{ c }}}{\sqrt{\pi }}\int _{\frac{\lambda }{c}}^{\infty }e^{-u \theta -\frac{\alpha ^2}{4\theta }}\text{ }\left(\frac{1}{\theta \sqrt{\theta }}+\frac{2u}{\sqrt{\theta }}\right)d\theta \right\}.$$ I cannot see why this is the first equation too. Any helps? Thanks in advance.