Let $K(x)$ be the complete elliptic integral of the first kind (the argument is the parameter $m = k^2$).

Let $$ A = \int_0^1 \arcsin(K(x)) dx$$

With precision $1000$ decimal digits $\Re A = \frac{\pi}{2}$.

Is this true?

According to Wolfram Alpha for the indefinite integral there is no result in terms of standard mathematical functions.

$A=1.570796326794\ldots - 1.285983901951989\ldots i$.