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## Another simple integral

As it has appeared recently, in some examples Maple can't find int(1+f(x), x) in some cases where it can find int(f(x), x). For example, for f(x) = (x^2)^(1/3).

Now, reading Axel Vogt's post, I noticed that in some examples Maple can't do such a simple change of variables as changing the sign.

```J:=Int(exp(-eta^2+eta)/(1+exp(eta)),eta = -infinity .. infinity);
infinity
/                 2
|          exp(-eta  + eta)
J :=  |          ---------------- deta
|            1 + exp(eta)
/
-infinity

value(J);

infinity
/                 2
|          exp(-eta  + eta)
|          ---------------- deta
|            1 + exp(eta)
/
-infinity
```

Changing the sign manually also has appeared to be not such a simple operation as it may seem,

```J1:=IntegrationTools:-Change(J,eta=-theta);
infinity
/
|            exp(-theta (theta + 1))
J1 := - |          - ----------------------- dtheta
|                1 + exp(-theta)
/
-infinity
value(J1);
infinity
/
|            exp(-theta (theta + 1))
- |          - ----------------------- dtheta
|                1 + exp(-theta)
/
-infinity
```

Even double minus signs didn't get cancelled! It took me some time to find the correct simplification steps,

```J1:=simplify(IntegrationTools:-Expand(J1));
infinity
/                    2
|           exp(-theta )
J1 :=  |          -------------- dtheta
|          exp(theta) + 1
/
-infinity

value(J1);

1/2
Pi
-----
2
```

Shouldn't it be more simple than that?

Alec

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