#### 6.53 arcsin integral (18.3.97)

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

>assume(x>0,x<1):
>int(arcsin(x),x);

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:

>assume(x>0,x<1):
>int((arcsin(x))^2,x);

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):

>> int(asin(x)^2,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:

> restart:with(student): # Restart and load student package
> assume(cos(u)>0): # used later on
> Int(arcsin(x)^2,x): # arcsin integral (inert form)
> changevar(arcsin(x)=u,%,u): # change variables x->u
> value(simplify(%)): # calculate integral
> subs(u=arcsin(x),%): # change u back to x
> simplify(%); # final simplification

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