1/(a+b)+2a/(a^2+b^)+4a^3/(a^4+b^)-8a^7/(a^8-b^8)
a^8-b^8=(a^4+b^4)(a^4-b^4)
=(a^4+b^4)(a^2+b^2)(a^2-b^2)
=(a^4+b^4)(a^2+b^2)(a-b)(a+b)
1/(a+b)+2a/(a^2+b^2)+4a^3/(a^4+b^4)-8a^7/(a^8-b^8)
=1/(a+b)+2a/(a^2+b^2)+(4a^3(a^4-b^4)/(a^8-b^8)-8a^7/(a^8-b^8)
=1/(a+b)+2a/(a^2+b^2)-(4a^3(a^4+b^4)/(a^8-b^8)
=1/(a+b)+2a/(a^2+b^2)-4a^3/(a^4-b^4)
=1/(a+b)+2a(a^2-b^2)/(a^4-b^4)-4a^3/(a^4-b^4)
=1/(a+b)-2a(a^2+b^2)/(a^4-b^4)
=1/(a+b)-2a/(a^2-b^2)
=(a-b)/(a^2-b^2)-2a/(a^2-b^2)
=(a-b-2a)/(a^2-b^2)
=-(a+b)/(a^2-b^2)
=b-a