Opened 5 years ago

# exponential integral of arccos

Reported by: Owned by: mforets major sage-8.1 calculus integration, maxima Reported upstream. No feedback yet.

Maxima interface returns a wrong result of this definite integral:

```sage: integrate(exp(arccos(x)),x,0,1, algorithm='maxima')
-1/2*e^(-1/2*pi) + 1/2
```

while it should be:

```sage: integrate(exp(arccos(x)),x,0,1, algorithm='sympy')
1/2*e^(1/2*pi) + 1/2
```

This was reported in ask.sage: an integral with arccos wrong

### comment:1 Changed 5 years ago by mforets

• Description modified (diff)

### comment:2 Changed 5 years ago by rws

This is a Maxima problem because

```(%i10) integrate(exp(acos(x)),x,0,1);
- %pi/2
1   %e
(%o10)                           - - ---------
2       2
```

### comment:3 Changed 5 years ago by mforets

• Report Upstream changed from Not yet reported upstream; Will do shortly. to Reported upstream. No feedback yet.

Ok, thank you!

i've sent an upstream report

### comment:4 Changed 12 months ago by chapoton

There is something strange going on:

```maxima: f:exp(acos(x));
%e^acos(x)
maxima: integrate(f,x,0,1);
1/2-%e^-(%pi/2)/2
maxima: integrate(1/f,x,0,1);
1/2-%e^-(%pi/2)/2
```

How would one try to debug that ?

### comment:5 Changed 12 months ago by chapoton

The problem is also there (after change of variable) in

```(%i8) integrate(exp(acos(cos(y)))*sin(y),y,0,%pi/2);
- %pi/2
1   %e
(%o8)                            - - ---------
2       2
(%i9) integrate(exp(y)*sin(y),y,0,%pi/2);
%pi/2
%e        1
(%o9)                             ------- + -
2      2

(%i12) integrate(exp(-y)*sin(y),y,0,%pi/2);
- %pi/2
1   %e
(%o12)                           - - ---------
2       2
```
Last edited 12 months ago by chapoton (previous) (diff)

### comment:6 Changed 12 months ago by chapoton

so this depends on the allowed trig simplifications:

```(%i50) triginverses:all;
(%o50)                                all
(%i51) integrate(exp(acos(cos(y)))*sin(y),y,0,%pi/2);
%pi/2
%e        1
(%o51)                            ------- + -
2      2
(%i52) triginverses:true;
(%o52)                               true
(%i53) integrate(exp(acos(cos(y)))*sin(y),y,0,%pi/2);
- %pi/2
1   %e
(%o53)                           - - ---------
2       2
```
Last edited 12 months ago by chapoton (previous) (diff)
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