Q:

How would you integrate this?

If we had the following integral:
$$\int_{a}^{b} {\big(1+x^2 \big)^s} \space dx$$
Where $s$ is not given. Is there any general formula for this integration that works for all $s\in \mathbb{R}$?

A:

The substitution $x = \tan \theta$ leads to $\int_{\arctan a}^{\arctan b} \cos^{-2s-2} \theta \, d\theta$. According to Wolfram Alpha, this can be integrated using a hypergeometric function.
Namely, 
$$\int (1 + x^2)^s \, dx = x\,{}_2F_1(1/2, -s, 3/2, -x^2) + C,$$
where ${}_2F_1$ is the function described here.

