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FWIW, the question Is floor((%i+1)/(2*%pi)) positive, negative or zero? is nonsensical : $\displaystyle{\frac{i+1}{2\pi}}$ is not real and therefore has no sign. Maxima was out for a short beer, but can be hand-guided.

BTW :

sage: cos(1+I*t).trig_expand().integrate(t,0,1) # Hand-guide Maxima
-I*cosh(1)*sin(1) + cos(1)*sinh(1) + I*sin(1)
sage: cos(1+I*t).integrate(t,0,1, algorithm="giac")
-I*sin(I + 1) + I*sin(1)
sage: cos(1+I*t).integrate(t,0,1, algorithm="fricas")
1/2*((e^(2*I) - 1)*e^(-I + 1) + e^(-I + 2) - e^I)*e^(-1) # Exponentialized...
sage: cos(1+I*t).integrate(t,0,1, algorithm="sympy")
1/2*(e - e^(2*I - 1))*e^(-I) + 1/2*(e^(2*I) - 1)*e^(-I) # Same as fricas
sage: cos(1+I*t).integrate(t,0,1, algorithm="mathematica_free")
0.634964000000000 - 0.456987000000000*I # Numerical result, correct but nonsensical here...
sage: cos(1+I*t)._mathematica_().Integrate((t, 0, 1)).sage()
-I*sin(I + 1) + I*sin(1) # Same as giac
sage: cos(1+I*t)._mathematica_().Integrate((t, 0, 1)).sage().trig_expand()
-I*cosh(1)*sin(1) + cos(1)*sinh(1) + I*sin(1) # Same as hand-guided Maxima

And, BTW :

sage: bool(cos(1+I*t).integrate(t,0,1, algorithm="fricas")==cos(1+I*t).trig_expand().integrate(t,0,1).exponentialize())
True

All these expressions are therefore equal.

HTH,