Here's my steps of figuring out how to solve integrals so far (I'm still learning calculus!)
(1.) Do whatever the hell feels right and following my hunch for what method will work. Or just guessing.
(2.) If my mind didn't grasp any particular method, I'll just "dip my toes in the water" and see what works (simplifying expressions, dividing the fraction inside the integral, using the simple integral formula, etc.).
(3.) If that doesn't work, then look for specific clues for specific formulas (derivatives and integrals for arcsin and arctan, ln (x) in an expression with the denominator having a higher-ordered variable than the numerator or so, memorized integrals or substitutions of trigonometric expressions, etc.)
(4.) If that still doesn't work, then maybe do U-substitution.
(5.) Integration by parts, as the last option (because I don't encounter it too often in my calculus course).
Of course, it's different with everyone but the above steps have worked for me.