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If you consider that you work on the real field, Maple answers a complex number, which is just weird. If you consider that you work on the complex field, your function is not well defined since there are 3 different choices for the cubical root, leading to different integrals.

If you want the cube root to be the real one, your integral will just go to -Infinity as x approaches -2.

Let us specify the real cube root as follows for the negative numbers:

sage: cuberoot(x) = -((abs(x))^(1/3))

Then,

sage: h(x) = 1/(cuberoot((x + 1)) + 1)
sage: integral_numerical(h, -6, -2)
(-inf, nan)

If you consider that you work on the real field, Maple answers a complex number, which is just weird. Sage's Symbolic Ring does the same artificial choice since:

sage: h(-3.3)
0.408623955144076 - 0.281398327127290*I

If you consider that you work on the complex field, your function is not well defined since there are 3 different choices for the cubical root, leading to different integrals.

If you want the cube root to be the real one, your integral will just go to -Infinity as x approaches -2.

Let us specify the real cube root as follows for the negative numbers:numbers, so that you will get the real cube root:

sage: cuberoot(x) = -((abs(x))^(1/3))

Then,

sage: h(x) = 1/(cuberoot((x + 1)) + 1)
sage: integral_numerical(h, -6, -2)
(-inf, nan)

If you consider that you work on the real field, Maple answers a complex number, which is just weird. Sage's Symbolic Ring does the same artificial choice since:

sage: h(-3.3)
0.408623955144076 - 0.281398327127290*I

If you consider that you work on the complex field, your function is not well defined since there are 3 different choices for the cubical root, leading to different integrals.

If you want the cube root to be the real one, your integral will just go to -Infinity as x approaches -2.

Let us specify the real cube root as follows for the negative numbers, so that you will get the real cube root:

sage: cuberoot(x) = -((abs(x))^(1/3))

Then,

sage: h(x) = 1/(cuberoot((x + 1)) + 1)
sage: integral_numerical(h, -6, -2)
(-inf, nan)

If you consider that you work on the real field, Maple answers a complex number, which is just weird. Sage's Symbolic Ring does the same artificial choice since:

sage: h(-3.3)
0.408623955144076 - 0.281398327127290*I

So, i agree that there is a kind of bug here, see also this question.

If you consider that you work on the complex field, your function is not well defined since there are 3 different choices for the cubical root, leading to different integrals.

If you want the cube root to be the real one, your integral will just go to -Infinity as x approaches -2.

. And Sage can see this. Let us specify the real cube root as follows for the negative numbers, so that you will get the real cube root:

sage: cuberoot(x) = -((abs(x))^(1/3))

Then,

sage: h(x) = 1/(cuberoot((x + 1)) + 1)
sage: integral_numerical(h, -6, -2)
(-inf, nan)

If you consider that you work on the real field, Maple answers a complex number, which is just weird. Sage's Symbolic Ring does the same artificial choice since:

sage: h(-3.3)
0.408623955144076 - 0.281398327127290*I

So, i agree that there is a kind of bug here, see also this question.

If you consider that you work on the complex field, your function is not well defined since there are 3 different choices for the cubical root, leading to different integrals.

If you want the cube root to be the real one, your integral will go to -Infinity as x approaches -2. And Sage can see this. Let us specify the real cube root as follows for the negative numbers, so that you will get the real cube root:root (not a complex one):

sage: cuberoot(x) = -((abs(x))^(1/3))

Then,

sage: h(x) = 1/(cuberoot((x + 1)) + 1)
sage: integral_numerical(h, -6, -2)
(-inf, nan)