This bug is more subtle than the other "falsely claim unintegrable" bug.
Because the integral is not reproducible in a fresh session, but after
some other computation.
A minimal reproducible example I found is:
====
integrate(0 - sin(1/sin(x))*cos(x)*cos(cos(1/sin(x)))/sin(x)^2, x)
integrate((x^2*sin(x)*cos(cos(x))/sin(cos(x))^2 +
2*x/sin(cos(x)))*sin(x^2/sin(cos(x))) + 0, x)
g0 := tan(sin((x*tan(x) + x)/2));
f := D(g0, x);
g := integrate(f,x)
====
If you integrate 'f' in a fresh session, you can get correct result.
But after the previous 2 integrals, it returns as unintegrable.
Tracing shows the computation diverge at "extendedEuclidean" in
"normalHermiteIntegrate" of TranscendentalHermiteIntegration in
intrf.spad.
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