A slight complement to Nasser's answer :

Contemplate :

```
sage: graphics_array(([f(t).demoivre(force=True).simplify_full().integrate(t, algorithm=u).plot((0,2*pi), title=u) for u in ("maxima", "giac", "fricas", "mathematica_free")]),ncols=2)
```

(`sympy`

fails to integrate the expression).

`maxima`

problems with various trig integrals have been documented for a long time... `giac`

, `fricas`

and `mathematica`

give different expressions of similar functions (up to a constant), but proving the identities isn't trivlal (long algebrico/trigonometric massage ahead...).

But note :

```
sage: table([[u, f(t).demoivre(force=True).simplify_full().integrate(t, algorithm=u)] for u in ("maxima", "giac", "fricas", "mathematica_free")], header_row=["Algorithm", "Antiderivative"])
Algorithm Antiderivative
+------------------+---------------------------------------------------+
maxima -4/sqrt(sin(t)^2/(cos(t) + 1)^2 + 1)
giac -4*cos(1/2*t)*sgn(sin(1/2*t)) + 4*sgn(sin(1/2*t))
fricas -2*(cos(t) + 1)*sqrt(-2*cos(t) + 2)/sin(t)
mathematica_free -2*sqrt(-2*cos(t) + 2)*cot(1/2*t)
```

*ALL* those "CAS primitives" are periodic of period $2\pi$, whereas the function $\displaystyle\int_0^t f(t)\,dt$ is **not** ($f$ is positive except for a null-measure set ($t=2k\pi,\ k\in\mathbb{Z}$) where it is zero, therefore its integral is increasing). Computing the integral by taking limits of the primitive at bounds is wrong.

This happens to be plotted in the above figure for the `fricas`

and `mathematica`

cases. This (and the fact that it is *not* plotted for `giac`

) is pure numerical happenstance...

Morality : **never** trust a CAS result without checking it...

HTH,

See also https://trac.sagemath.org/ticket/17183.