Question

What is `int tan(2x^3)-1/cos(1-3x)`?

Answer 1

The first term of this integral has no known integration. The second is supplied.

Explanation:

I assume that there ought to be a `dx` at the end of the expression?

This integral has two terms, which may be treated separately. The second term is messy but solvable in terms of elementary functions. The first term is not. I will not go through all the possible gory details here as the question simply doesn't have an analytic answer in terms of known functions.

`1/cosx=secx`, we know from definition.

We know (from e.g. https://www.math.ubc.ca/~feldman/m121/secx.pdf)
`int secxdx=log|secx+tanx|+C`
and we see easily from this that
`int sec(1-3x)dx=-1/3log|sec(1-3x)+tan(1-3x)|+C`

So the second term integrates as
`int -1/cos(1-3x)dx=1/3log|sec(1-3x)+tan(1-3x)|+C`

The first term has no analytic integration in terms of elementary functions. The online Wolfram integrator (integrals.wolfram.com) doesn't even find one in terms of the extended set of transcendental functions. Nor does it find one for the simpler function `int tan(x^2)dx`.