Partial Fraction Decomposition Problem 6, repeated irreducible quadratic factors (Calculus 2)

This one is brutal: a partial fraction decomposition with a repeated irreducible quadratic factor. This leads to a system of five equations with five unknowns! We'll cover ALL of the steps in detail without rushing. This is about as hard as it gets for a partial fraction decomposition question in calculus 2. More problems are linked below.

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► Artist Attribution
Music By: "After The Fall"
Track Name: "Pieces"
Music Published by: Chill Out Records LLC
License: Creative Commons Attribution 4.0 International (CC BY - 4.0)

[shadowking]

this was awesome and very helpful thanks

[cejkisondrej]

How would you integrate it when E wasnt zero? There could be sth like integrate(1/(x^2+1)^2) and u-substitution couldnt be used here...