What is the value of this integral whose integrand is a mixture of gamma and gaussian functions?

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I am wondering the value of the following integral $$\int_0^{\infty} \frac{1}{\sqrt{t(t+1)}} \exp{ \left\lbrace -\left[t+ \left(\frac{1}{\sqrt{t+1}} - \frac{1}{2} \right)^2 \right] \right\rbrace } dt.$$ The exponential part is hard to deal with.