TITLE: What are some difficult integrals done by substitution and elementary functions?
QUESTION [11 upvotes]: 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.

REPLY [2 votes]: Some More Questions $$\displaystyle \int \sqrt{\tan x}dx\;,\int\sqrt{\cot x}dx\;,\int \left(\sqrt{\tan x}+\sqrt{\cot x}\right)dx\;,\int \left(\sqrt{\tan x}-\sqrt{\cot x}\right)dx$$
$$\displaystyle \int\frac{1+x^4}{1+x^6}\;\;, \int\frac{1}{1+x^6}dx$$