Question: does anybody know any good examples of integrals that 1. at a glance look simple but turn out to have unexpectedly complicated solutions, and 2. don't require higher math (or weird special functions) to solve?
Ups, there is sec, but even so is easier by variable change. You start with the variable change, do this easy job, and return to the initial variable and done.
@Chris'ssis well, if you write it like that, instead of $$\log\left(\sin\left(\frac{x}{2}\right)+\cos\left(\frac{x}{2}\right)\right)- \log \left(\cos\left(\frac{x}{2}\right)-\sin\left(\frac{x}{2}\right)\right)$$
Let's say I'm a beginner and want to evaluate that one. Then I might proceed like that $$\int \frac{1}{\cos(x)} \ dx=\int \frac{\cos(x)}{\cos^2(x)} \ dx=\int \frac{\cos(x)}{1-\sin^2(x)} \ dx$$
The rest is a piece of cake.
We can do the same for $\displaystyle \int \frac{1}{\sin(x)} \ dx$.
@Chris'ssis as someone who has repeatedly seen $$\require{cancel}\frac{\sin \cancel{x}}{\cos \cancel{x}}=\frac{\sin}{\cos}$$ i would imagine what i wrote is not far from the truth