Both the numerator and denominator of this function strongly suggest that the way ahead is to make a trigonometric substitution. More than that, both would become considerably simpler if x = cosu ... the numerator, because that is the inverse of cosˉ¹x, and the denominator, because we would then be utilising the trigonometric identity sin²u + cos²u = 1.
Before making this substitution, however, I decided to use this video to discuss the importance of always noting the domain of a function (and the limits of a definite integral). I had thought to leave this as an exercise for you to do yourself but, upon reflection, this particular integral provided too good an opportunity to be missed! So, if you wish to bypass all my discussion concerning domains, you might jump to about 6:10 in the video (which is where I actually begin to evaluate this integral).
You will readily see that the original function only makes sense if x lies between -1 and 1 (believe it, or not, YouTube does not allow "less than" symbols in the description!). When we substitute x = cosu , or u = cosˉ¹x, we note that u must lie between 0 and π. This becomes very relevant when, in simplifying the integral after substitution, √sin²u = |sinu|. Normally, in dealing with an indefinite integral, we would need to take account of the domain of u that would cause sinu to be positive, and that which would cause it to be negative ... and present the two separate cases. In this integral, however, because we now know that u must lie between 0 and π, we observe that sinu must therefore always be positive! This means that we do not have to use the absolute value sign (nor take account of any negative value for sinu).
The rest of the integration proceeds very smoothly indeed. This is quite a lovely little integral, and I hope that my discussion of domains has inspired you to be very careful in noting them in your work. Sadly, in order to keep my videos shorter, I have opted NOT to discuss domains for most of the integrals in this series. I say this is sad because, if you are to integrate properly/thoroughly, then you should ALWAYS be watching out for this detail! This is certainly true at university level. In school courses, however, we are often more concerned that you simply learn the mechanics of HOW to integrate without also demanding that you check the CONDITIONS under which your integral is valid!