To solve the expression 1/(X+Y) + 2X/(X^2+Y^2) + 4X^3Y^4/(Y^8-X^8), we can simplify it step by step.

1. Rewrite the expression:

The given expression is:

1/(X+Y) + 2X/(X^2+Y^2) + 4X^3Y^4/(Y^8-X^8)

2. Factor the denominator of the third term:

Notice that (Y^8 - X^8) can be factored as a difference of squares:

Y^8 - X^8 = (Y^4 + X^4)(Y^4 - X^4) = (Y^4 + X^4)(Y^2 + X^2)(Y^2 - X^2) = (Y^4 + X^4)(Y^2 + X^2)(Y + X)(Y - X)

3. Rewrite the expression using the factored form:

1/(X+Y) + 2X/(X^2+Y^2) + 4X^3Y^4/((Y^4+X^4)(Y^2+X^2)(Y+X)(Y-X))

4. Combine the first two terms:

Find a common denominator for the first two terms, which is (X+Y)(X^2+Y^2):

[1/(X+Y)] + [2X/(X^2+Y^2)] = [(X^2+Y^2) + 2X(X+Y)] / [(X+Y)(X^2+Y^2)]

= (X^2 + Y^2 + 2X^2 + 2XY) / ((X+Y)(X^2+Y^2))

= (3X^2 + 2XY + Y^2) / ((X+Y)(X^2+Y^2))

5. Look for a pattern or further simplification:

Now we have:

(3X^2 + 2XY + Y^2) / ((X+Y)(X^2+Y^2)) + 4X^3Y^4/((Y^4+X^4)(Y^2+X^2)(Y+X)(Y-X))

This expression is quite complex and does not appear to simplify easily into a compact form.

6. Alternative approach

Consider combining the first and second terms like so:

1/(X+Y) + 2X/(X^2+Y^2) = (X^2+Y^2+2X(X+Y))/((X+Y)(X^2+Y^2))

= (X^2+Y^2+2X^2+2XY)/((X+Y)(X^2+Y^2))

= (3X^2+2XY+Y^2)/((X+Y)(X^2+Y^2))

Which doesn't lead to a clear simplification when combined with the third term.

Given the complexity, it's likely that the problem is designed to be simplified using a specific substitution or context that isn't provided. If there's no further context, the expression remains as is or might require numerical methods for specific X and Y values.