This antiderivative has arguable the highest ratio of “really hard to compute” to “really easy to write”:

To compute this integral, I will use the technique of partial fractions. This requires factoring the denominator over the real numbers, which can be accomplished by finding the roots of the denominator. In other words, I need to solve

,

or

.

I switched to the letter since the roots will be complex. The four roots of this quartic equation can be found with De Moivre’s Theorem by writing

,

where is a real number, and

By De Moivre’s Theorem, I obtain

.

Matching terms, I obtain the two equations

and

or

and

or

and .

This yields the four solutions

Therefore, the denominator can be written as the following product of linear factors over the complex plane:

or

or

or

or

.

We have thus factored the denominator over the real numbers:

,

and the technique of partial fractions can be applied.

I’ll continue the calculation of this integral with tomorrow’s post.

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