Value of $\int _0^{+\infty }\:e^{-x²}\cdot \sqrt{\sinh^2\left(x\right)+1}dx$ Can somebody help me find the value of
$\int _0^{+\infty }\:e^{-x^2}\cdot \sqrt{\sinh^2\left(x\right)+1}dx$
without using any special integral (so no erf)
I keep getting stuck trying to solve the integral with the following integral
$\int _0^{+\infty }\:\frac{\sqrt{e^{-2x}+e^{2x}+2}}{2e^{x^2}}dx$
a different method gave me
$\int _0^{+\infty }\:\left(e^{-x^2+x}+e^{-x^2-x}\right)dx$

$\textbf{Hint}$: The function is even therefore
$$I = \frac{1}{2}\int_{-\infty}^\infty e^{-x^2}\cosh x\:dx = \frac{1}{2}\int_{-\infty}^\infty e^{-x^2+x}\:dx$$
Can you continue from here?
