this one has a long answer:
$\displaystyle \int\sqrt{\tan{x}}dx$
What do i start with first?
need advices or hints
After the brute force applied on the problem, there's another way to solve this (elegant by the way).
Suppose it remains to compute
$\displaystyle \int\frac{x^2}{x^4+1}\,dx$
Let's use $\displaystyle (x^2+1)+(x^2-1)=2x^2$
Next, we split the integral in two pieces and in each one of them we divide by $\displaystyle x^2$
After some simple calculations we happily get that
$\displaystyle \int\frac{x^2}{x^4+1}\,dx=\frac12\left(\frac1{\sqr t2}\arctan\frac{x-x^{-1}}{\sqrt2}+\frac1{2\sqrt2}\ln\left|\frac{x+x^{-1}-\sqrt2}{x+x^{-1}+\sqrt2}\right|\right)+k$