Integrate $\tan(x)\sqrt{1+\sec^4(x)}$

Question

I am trying to integrate this function: $\tan(x) \sqrt{1+\sec^4(x)}$. Please help me with this. Additionally, how do I confront all these problem in the future? Thank you.

Answer 1

Hint:

Just substitute $u = \sec^4 x +1$ to get $\frac {du}{dx} = 4\sec^4 x \tan x $. Then, we have, $$I= \int \tan x \sqrt {1+\sec^4 x}\mathrm {d}x $$ $$= \frac {1}{4} \int \frac {\sqrt {u}}{u-1} \mathrm {d}u $$

Hope you can take it from here.