Integrate $\int x\sqrt{1+\sec^4x}dx$

Question

I've been solving some homework in Area of a Surface of Revolution and this integration popped up, I couldn't solve it and can't find anything online about it, plus integral calculator can't solve it either? $$ \int x \, \sqrt{1+\sec^4x} \, dx $$

Here is the problem in the book, and no its the integral is supposed to be from 0 to $\pi/3$

Q1: Set up an integral for the area of the surface obtained by rotating the curve about the $y$-axis:

a. $y= \tan(x)$

I posted it without limits because in my exam calculators are prohibited thus I cant use one and I have to find the integral first then substitute.

Answer 1

Since there is no much hope for a solution, use a series solution $$\sqrt{1+\sec ^4(x)}=\sqrt 2\sum_{n=0}^\infty \frac {a_n}{(2n)!} x^{2n}$$ the first $a_n$ making the unknown sequence $$\{1,1,11,211,5921,237121,14592731,1371063331,164080073441,\cdots\}$$

Using only the terms given above, this leads to a value of $1.26193$ while numerical integration gives $1.26294$. Using twice more terms, it would be $1.2629384$ instead of $1.2629409$