# Indefinite integral is incorrect

`indefinite_integral(sqrt(1+cos(x)**2), x).full_simplify()`

gives `1/6*sin(x)^3`

, which is incorrect.

Indefinite integral is incorrect

`indefinite_integral(sqrt(1+cos(x)**2), x).full_simplify()`

gives `1/6*sin(x)^3`

, which is incorrect.

3

This is a bug. Furthermore, it is *not* a `maxima`

bug, as it is often the case.
Here, Sage truly screws things up itself:

Maxima doesn't give a false answer:

`sage: maxima.integrate(sqrt(1+cos(x)^2),x).sage() integrate(sqrt(cos(x)^2 + 1), x)`

When one tries to "ease" the problem,

maxima doesn't recognize the "obvious", but does not give a false answer:

`sage: maxima.integrate(sqrt(1-m*sin(x)^2),x).sage() integrate(sqrt(-m*sin(x)^2 + 1), x)`

Sage does:

`sage: integrate(sqrt(1-m*sin(x)^2),x) 1/4*m*x - 1/8*m*sin(2*x)`

BTW: what is expected:

```
sage: elliptic_e(x,1/2).diff(x)
sqrt(-1/2*sin(x)^2 + 1)
```

One can easily check that `sympy`

, `giac`

and `fricas`

all fail to integrate,
but that none of them gives misleading "answers".

This one does not seem to be related to existing indefinite integral bugs, and is an original, genuine, Sage-specific one. Reported as Trac #26563.

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Asked: ** 2018-10-26 14:08:37 +0200 **

Seen: **312 times**

Last updated: **Oct 27 '18**

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Where did you get the

`indefinite_integral`

function ?in 8.5.b0:

@tmonteli, I use sage 8.1

Using integral(sqrt(1+cos(x)**2), x).full_simplify() as suggested still results in an answer which is incorrect, doesn't it? This is a nonelementary integral. See here or here