How to integrate $\int e^{2x} \sec{3x} \,dx$

Question

$\int e^{2x} \sec{3x} \,dx$

I have tried every elementary (e.g. methods taught in first course of integration) way to integrate this function . But couldn't make it. Please someone solve it and mention the concepts you have used to solve it.

Answer 1

As said in comments, there is no closed form solution and the only possible result is a gaussian hypergeometric function ... which corresponds to an infinite summation.

So, as @ person suggested, use $$\sec(t)=\sum_{n=0}^\infty \frac{\left|E_{2 n}\right|}{(2 n)!}t^{2n}$$ making $$I=\int e^{2x} \sec(3x) \,dx=\sum_{n=0}^\infty 9^n \int \frac{\left|E_{2 n}\right|}{(2 n)!}x^{2n}\,e^{2x}\,dx$$ that is to say $$I=\sum_{n=0}^\infty \frac{ 9^n \left|E_{2 n}\right| }{2^{2 n+1}\,(2 n)!}\,\Gamma (2 n+1,-2 x)$$ $$J=\int_0^t e^{2x} \sec(3x) \,dx=\sum_{n=0}^\infty \frac{ 9^n \left|E_{2 n}\right| }{2^{2 n+1}\,(2 n)!}\,\Big[\Gamma (2 n+1,-2 t)-\Gamma (2 n+1)\Big]$$

Expanded as a Taylor series $$J=\sum_{n=1}^\infty \frac{a_n}{n!} t^n$$ coefficients $a_n$ make the sequence $$\{1,2,13,62,637,4802,70993,742142,14537497,195123842,\cdots\}$$ which is not found in $OEIS$.