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.

∫ (x^{2}+1) = ∫ x^{2} + ∫
1

(x^{4}+4) (x^{4}+4)
(x^{4}+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)

(x^{4}+4) = (x^{2} - 2x + 2)(x^{2} + 2x +2)

Integrate Term by Term

- for the 1st (x^{2}) term use partial fractions

x^{2} / (x^{2} - 2x + 2)(x^{2} + 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(x^{2}-2x+2) - ln(x^{2}+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.

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