# 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.

Please start posting anonymously - your entry will be published after you log in or create a new account.

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

Seen: **262 times**

Last updated: **Oct 27 '18**

Plotting an integral with a variable as a limit

integral() failing with "segmentation fault"

integral from sin at plus minus infinity seems to be bad

How do I understand the result of symbolic integrals

Strange behviour when trying to integrate gaussian function. bug?

How can I speed up symbolic function evaluation?

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.

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