#### arcsin integral (18.3.97)

This question refers to the integration of the square of arcsin(x). If you give Maple these
commands:

It will respond with the correct result, at least according to Schaum’s Mathematical Handbook of
Formulas and Tables, equation 14.471. If, instead you go with:

The unevalueted integral is returned. I found a way to reach the result (equation 14.476,
Schaum) trough intparts from the student package but I wonder, is there a special command
that I need to know? Have I reached the limit of the integration kernel with this special
case?

We both reached the same limit :o) A package written with the all the asumptions made for all
trigonometric functions would be awesome .

When I saw this posting, I thought I might try MuPAD 1.3 on the problem (I realise that this is rather
naughty, but not to worry). In the following, asin(x) is the same as arcsin(x):

which after only a short time ( 30 sec) produces:

2 2 1/2
-2x + x asin(x) + 2 asin(x) (-x + 1)

I have no idea of what assumptions MuPAD makes with this problem.

There is a simple way to tell Maple how to calculate

int(arcsin(x)^n,x).

The following code does the job:

and works for any positive integer power of arcsin(x).