# What do I need to know to integrate any rational function?

My analysis book makes the following statement:

Every rational function with real coefficients can be integrated in terms of

• rational functions,
• logarithm functions,
• arctangent functions.

Does it mean that after polynomial division, partial fraction expansion, completing the square, substituting where needed, and knowing that

\begin{align} \\ \int \frac{dx}{(x-a)^n} &= -\frac{1}{n-1} \frac{1}{(x-a)^{n-1}} \ \ \ (n \neq 1) \\ \int \frac{dx}{x-a} &= \log|x-a| \\ \int \frac{dx}{x^2+1} &= \arctan x \end{align}

one can integrate any rational function?

• In principle. In reality, you will often not be able to split the polynomials into irreducible factors. – Daniel Fischer Jun 28 '13 at 18:22
• You also need to be able to compute $$\int \frac{dx}{(x^2+1)^n}$$ for $n > 1$. – mrf Jun 28 '13 at 18:55
• @mrf: I saw this one in many materials, yet I didn't quite understand why should we pay special attention to the integral if it can be quickly reduced to the 1st case after substitution and partial integration? – Leo Jun 28 '13 at 19:07
• @Leo If you know complex numbers you can do it even without arctangent... You can just reduce it to the first and second formula (always). – Kartik Aug 11 '16 at 14:56