Finding the "right" technique for a given integral can be difficult, it requires a strategy. Let's start by sorting out the different techniques.
First, start by simplifying the integrand as much as possible (using simple algebraic manipulations or basic trigonometric identities). Not necessarily a simpler form but more a form that we know how to integrate.
Next, look for obvious substitution (a function whose derivative also occurs), one that will get you an integral that is easy to do.
If you can’t solve the integral using simplification or substitution, try to classify the integrand into one of the following: product of trig powers, rational functions, radicals, or a product of a polynomial and a transcendental function.
Rational functions: use partial fractions if the degree of the numerator is less than the degree of the denominator, otherwise use long division.
Product of a polynomial and a transcendental function: use Integration by parts.
Radicals: use trig substitution if the integral contains sqrt(a^2+x^2) or sqrt(x^2-a^2), for (ax+b)^1/n try simple substitution.
If none of the above techniques work, you should take some more aggressive measures; advanced algebraic manipulations, trig identities, integration by parts with no product (assume 1 as a multiplier). In some cases, you need to use multiple techniques.
The best strategy is to assume easy until easy doesn’t work, always try the simplest techniques first, and remember there is more than one way to solve an integral.