My curiousity is to get more integrals about the constant $\pi$ using special functions. I have used some special functions as shown below in the integral:
$$5\int_{0}^{\infty}\exp\left(-x^2 \text{erf}(x)\right)x^{\sin x+\frac12}dx$$
Wolfram alpha shows the value of that integral is a rational number; however all used function are far to contribute rational values. Now my question: Does $5\int_{0}^{\infty}\exp(-x^2 \text{erf}(x))x^{\sin x+\frac12}dx$ really result in a rational number? And does it close to $\pi$ if it is?
Note: It is very hard to compute that integral in closed form, since it has used some complicated special functions.
There's no indication that this integral evaluates to a rational number. There seems to be a bug in the mobile version of Wolfram|Alpha that you link to, in that it omits the ellipsis at the end of the approximation upon timing out. If you enter the integral in the desktop version of the site, it times out with an ellipsis at the end of the approximation. It also has a “More digits” button that the mobile version lacks.