1 | initial version |

Mathematica is correct. You can see that easily by plotting the integrand from 0 to 2Pi. Clearly the area is not zero. It looks like about 8 is correct.

Here is a workaround in sagemath,

Write the integrand using trig function by converting it using Euler's formula. This will give sqrt(2-2*cos(t)) and now integrate this instead of the complex exponential. But make sure to use "fricas" as algorithm

```
sage: var("t")
sage: g = sqrt(2 - 2 * cos(t))
sage: integrate(g, t, 0, 2 * pi, algorithm="fricas")
8
```

2 | No.2 Revision |

Mathematica is correct. You can see that easily by plotting the integrand from 0 to 2Pi. Clearly the area is not zero. It looks like about 8 is correct.

Here is a workaround in sagemath,

Write the integrand using trig function by converting it using Euler's formula. This will give sqrt(2-2*cos(t)) and now integrate this instead of the complex exponential. But make sure to use "fricas" as algorithm

~~sage: ~~var("t")
~~sage: ~~f = sqrt((e^(I * t) - 1) * (e^(-I*t) - 1))
g = simplify(expand(sageobj(f._maxima_().demoivre())))
sqrt(cos(t)^2 + sin(t)^2 - 2*cos(t) + 1)

The above is `sqrt(2 - 2 * `

but for some reason simplify does not know that ~~cos(t))
~~cos(t))`cos(t)^2+sin(t)^2=1`

? strange.

Now integrate the above, but use "fricas" to get correct answer

```
sage: integrate(g, t, 0, 2 * pi, algorithm="fricas")
8
```

3 | No.3 Revision |

Mathematica is correct. You can see that easily by plotting the integrand from 0 to 2Pi. Clearly the area is not zero. It looks like about 8 is correct.

Here is a workaround in sagemath,

Write the integrand using trig function by converting it using Euler's formula. This will give ~~sqrt(2-2*cos(t)) ~~`sqrt(2-2*cos(t))`

and now integrate this instead of the complex exponential. But make sure to use "fricas" as ~~algorithm~~algorithm. The default algorithm and others give zero which is wrong.

```
var("t")
f = sqrt((e^(I * t) - 1) * (e^(-I*t) - 1))
g = simplify(expand(sageobj(f._maxima_().demoivre())))
sqrt(cos(t)^2 + sin(t)^2 - 2*cos(t) + 1)
```

The above is `sqrt(2 - 2 * cos(t))`

but for some reason simplify does not know that `cos(t)^2+sin(t)^2=1`

? strange.

Now integrate the above, but use "fricas" to get correct answer

```
sage: integrate(g, t, 0, 2 * pi, algorithm="fricas")
8
```

4 | No.4 Revision |

Here is a workaround in sagemath,

Write the integrand using trig function by converting it using Euler's formula. This will give `sqrt(2-2*cos(t))`

and now integrate this instead of the complex exponential. But make sure to use "fricas" as algorithm. The default algorithm and others give zero which is wrong.

```
var("t")
f = sqrt((e^(I * t) - 1) * (e^(-I*t) - 1))
g =
```~~simplify(expand(sageobj(f._maxima_().demoivre())))
~~expand(sageobj(f._maxima_().demoivre())).trig_simplify()
~~ sqrt(cos(t)^2 ~~sqrt(-2*cos(t) + ~~sin(t)^2 - 2*cos(t) + 1)
~~2)

The above is `sqrt(2 - 2 * cos(t))`

but for some reason simplify does not know that `cos(t)^2+sin(t)^2=1`

? strange.

Now integrate the above, but use "fricas" to get correct answer

```
sage: integrate(g, t, 0, 2 * pi, algorithm="fricas")
8
```

5 | No.5 Revision |

Here is a workaround in sagemath,

Write the integrand using trig function by converting it using Euler's formula. This will give `sqrt(2-2*cos(t))`

and now integrate this instead of the complex exponential. But make sure to use "fricas" as algorithm. The default algorithm and others give zero which is wrong.

```
var("t")
f = sqrt((e^(I * t) - 1) * (e^(-I*t) - 1))
g = expand(sageobj(f._maxima_().demoivre())).trig_simplify()
sqrt(-2*cos(t) + 2)
```

Now integrate the above, but use "fricas" to get correct answer

```
sage: integrate(g, t, 0, 2 * pi, algorithm="fricas")
8
```

Compare to

```
sage: integrate(g, t, 0, 2 * pi)
0
```

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