Determine this intergral by expanding it first
To expand you would expand it into a polynomial - if multiplication you would multiply by the distribution method (sometimes Wrongly called the foil method) - for division like here you would separate the terms of the numerator dividing each by a common denominator.
∫ (x2+1) = ∫ x2 + ∫ 1
(x4+4) (x4+4) (x4+4)
So now you can find the integral of each term
factor the denominators into linear irreducible quadratic terms
(note: you could do this 1st before separating terms but you said expand 1st)
(x4+4) = (x2 - 2x + 2)(x2 + 2x +2)
Integrate Term by Term
- for the 1st (x2) term use partial fractions
x2 / (x2 - 2x + 2)(x2 + 2x +2)
= x / 4(x2 - 2x + 2) - x / 4(x2 + 2x +2)
I did 20 more steps for each term plus some to put the terms together to get to
1/16 [ ln(x2-2x+2) - ln(x2+2x+20) - 6tan-1(1-x) + 6tan-1(x+1) ] + C
Sorry but it would take all day to type it into this foolish window that WyzAnt has with such poor character handling for math equations.