What are some difficult integrals done by substitution and elementary functions?

What are some examples of difficult integrals that are done using substitutions?

For example: $$\int{\frac{(1+x^{2})dx}{(1-x^{2})\sqrt{1+x^{4}}}}$$

Please no laplace and fourier transforms as I haven't studied those yet.

Here is a thread from AoPs that contains a good collection of indefinite integrals. You can find several non-trivial integrals done using substitution.

Indefinite Integral Marathon

I can't tell if it is at the accepted answer's list, but $$\int\sqrt{\tan{x}}\,dx$$ is a good one. It's pretty concise, and perhaps at first it feels like either it is going to be very easy or not doable with elementary functions. It is doable with elementary functions and techniques, but it takes a fair amount of effort.

Some More Questions $$\displaystyle \int \sqrt{\tan x}dx\;,\int\sqrt{\cot x}dx\;,\int \left(\sqrt{\tan x}+\sqrt{\cot x}\right)dx\;,\int \left(\sqrt{\tan x}-\sqrt{\cot x}\right)dx$$

$$\displaystyle \int\frac{1+x^4}{1+x^6}\;\;, \int\frac{1}{1+x^6}dx$$