Rewrite as

sin(x)sin^2(x)cos^2(x)= sin(x)(1-cos^2(x))cos^2(x)

Let u = cos(x) du = -sin(x)

you obtain -integral((u^2-u^4)du)

Typically when you have sin^n(x)cos^m(x) if n is odd separate out the sin(x) and convert to cosines . If m is odd separate out a cos(x) and convert the remaining cosines to sines