You may take the help of these formula to solve above problem

Solution:

Integral [x^4(1-x)^4]/(1+x^2),x,0,1 = integral [-4/(x^2+1) + x^6 -4x^5 + 5x^4 - 4x^2 + 4],x,0,1

= -4arctan(x) +x^7/7 - 2x^6/3 + x^5 -4x^3/3 + 4x],0,1 = [22-7π]/7

Note, since the binomial is only of degree 4, might as well expand and then collect terms.