A good place to look is pages 405 and 413 of the Nachlass section of Gauss's Werke III, which can be found online through Google Books. On page 405, he gives the following formula for "$\text{arc sin lemn }x$":
$$\text{arc sin lemn }x= x+{1\over2}\cdot{1\over5}x^5 + {1\cdot3\over2\cdot4}{1\over9}x^9+{1\cdot3\cdot5\over2\cdot4\cdot6}{1\over13}x^{13}+{1\cdot3\cdot5\cdot7\over2\cdot4\cdot6\cdot8}{1\over17}x^{17}+\cdots$$
One page 413, he computes the value of $\int_0^1{dx\over\sqrt{1-x^4}}$, presumably using the expansion above, but explicitly citing the formula
$$\text{ arc sin lemn }{7\over23}+2\text{arc sin lemn }{1\over2}$$
obtaining
$$1.3110287771\quad460599052\quad320.7$$
He also notes there that Stirling had obtained the value $1.3110287771\ 4605987$. This is in reference to the calculations on pages 57-58 of Stirling's Methodus differentialis from 1730, which can also be found through Google Books. It might be worth noting that even Gauss was slightly off in the last couple of decimal places. A more accurate value, which I took from here, is
$$1.3110287771\quad460599052\quad324197949$$