When integrating rational functions, a polynomial divided by a polynomial, there are many things to consider that will aid you in deciding the technique of integration to use:
1) Is the denominator a monomial (one term)? If so, write each term in the numerator over the common denominator, simplify each fraction, then integrate term by term.
2) Is the degree of the numerator the same or higher than the degree of the denominator? If so, use long division to write the rational function in
quotient + reminder/divisor form. Then integrate the quotient term by term and the reminder/divisor portion using these steps.
3) If the degree of the numerator is less than the degree of the denominator
a) check to see if choosing the entire denominator as u gives the correct form for the du replacement.
If not,
b) check to see if the denominator is factorable to carry out the method of partial fractions and integrate term by term
If not,
c) check to see if the denominator can be written as a constant plus the square of a variable expression to consider integrating to inverse tangent. This may require completing the square and would need the correct form for the du replacement.