[updated]
It is not nice. But here it is.
Well, maybe you should look in a calculus 2 textbook and you will see that there's a whole section dedicated to "trigonometric integration" with most, if not all, of the solutions being as, or more, "difficult", as you would say, than this one. How did you get title of "MHF Expert" and make a completely idiotic remark like that? That seems like it is the very opposite of "expert."
Perhaps you are referring to trigonometric substitution? I can assure you though that the integral you posted is considerably difficult, much more advanced than Calculus 2. Don't forget that even an extra or missing x can complicate an expression unimaginably, maybe you missed one?
What an extremely rude and disrespectful comment to make. I have asked you a pertinent question, seeing as you are unlikely to have heard of the Fresnel C and Fresnel F integrals. It is quite possible you did not write the correct integrand, as a similar integrand could be integrated exactly using substitution.
"Trigonometric integration" simply refers to the integration of trigonometric functions, NOT a METHOD of integration, which is what you were asking for.