I get this other error message:

Error, (in numapprox:-chebpade) cannot determine if this expression is true or false: abs(Int(sin(u^3)*tan(u),u = 0. .. 1.)+Int(sin(u^3)*tan(u),u = 0. .. -1.)+2.0*Int(sin(u^3)*tan(u),u = 0. .. .50)+2.0*Int(sin(u^3)*tan(u),u = 0. .. -.50)-2.0*Int(sin(u^3)*tan(u),u = 0. .. .8660254037845)-2.0*Int(sin(u^3)*tan(u),u = 0. .. -.8660254037845)-2.0*Int(sin(u^3)*tan(u),u = 0. .. 0.)-2.0*Int(sin(u^3)*tan(u...
 

It may be a hint.

I get this other error message:

Error, (in numapprox:-chebpade) cannot determine if this expression is true or false: abs(Int(sin(u^3)*tan(u),u = 0. .. 1.)+Int(sin(u^3)*tan(u),u = 0. .. -1.)+2.0*Int(sin(u^3)*tan(u),u = 0. .. .50)+2.0*Int(sin(u^3)*tan(u),u = 0. .. -.50)-2.0*Int(sin(u^3)*tan(u),u = 0. .. .8660254037845)-2.0*Int(sin(u^3)*tan(u),u = 0. .. -.8660254037845)-2.0*Int(sin(u^3)*tan(u),u = 0. .. 0.)-2.0*Int(sin(u^3)*tan(u...
 

It may be a hint.

The examples in ?numapprox[pade] and ?numapprox[pade] are "academic" in the sense that for numerical evaluation it does not seem to provide an advantage to evaluate exp(x) using a Pade approximant.

I think that "real life" applications include functions that are "costly" to evaluate, eg a function involving a numerical integral that takes many seconds or minutes at each evaluation point. If it has to be used many times (eg to find an extremum), it may be advantageous to calculate an approximant and then use it instead  of the numerical integral.

In fact, this would be particularly useful for functions of several variables. I wonder about Pade approximants for them. 

The examples in ?numapprox[pade] and ?numapprox[pade] are "academic" in the sense that for numerical evaluation it does not seem to provide an advantage to evaluate exp(x) using a Pade approximant.

I think that "real life" applications include functions that are "costly" to evaluate, eg a function involving a numerical integral that takes many seconds or minutes at each evaluation point. If it has to be used many times (eg to find an extremum), it may be advantageous to calculate an approximant and then use it instead  of the numerical integral.

In fact, this would be particularly useful for functions of several variables. I wonder about Pade approximants for them. 

You may play with tricks like this to improve visibility of the dot:

animate( DEplot,[ SYS, [x(t),y(t)], t=0..10, x=-3..3,y=-3..3 ,
labels=[`#mi("x",fontweight = "bold")`,
`#mover(mi("x",fontweight = "bold"),mi(".",
fontweight = "bold")))`]], epsilon=0..1, frames=20 );

You may play with tricks like this to improve visibility of the dot:

animate( DEplot,[ SYS, [x(t),y(t)], t=0..10, x=-3..3,y=-3..3 ,
labels=[`#mi("x",fontweight = "bold")`,
`#mover(mi("x",fontweight = "bold"),mi(".",
fontweight = "bold")))`]], epsilon=0..1, frames=20 );

I am afraid that I do not understand whether "substitute arccos(x) with 2x" means the change of variables (like that made by Jacques):

with(IntegrationTools):
J:=Int(arccos(x)/(1+x^4),x):

Jt:=(simplify@Change)(J,x=cos(2*t)) assuming 0<t,t<
Pi/4;

Not sure either whether this is your result: sqrt(2)/4*arctanh(1/2*((arccos(x)^2/4+1)*sqrt(2))/sqrt(arccos(x)^2/4+1))-arctanh(1/2*((arccos(x)^2/4-1)*sqrt(2))/sqrt(arccos(x)^2/4+1)); But clearly it is different from value(Jt), which has sums of dilogs. So, could you give the detail of your steps?

The integrand was

arccos(x)/(1+x^4)

not

arctan(x)/(1+x^4)

And as you know,

diff(arctan(x),x);

                                                1
                                              ------
                                                   2
                                              1 + x

So, by the chain rule

arctan(arctan(x^2)):
diff(%,x);


           2 x
--------------------------
      4               2 2
(1 + x ) (1 + arctan(x ) )

which is a different function.

On the other hand, In Mathematica the input should be: 

Integrate[ArcTan[x]/(1 + x^4), x]

ie, with square brackets.

is relevant here, I think.  Eg. this article on Wikipedia has a small section on this issue. For sure, uncorrelated measurements with different uncertainties are quite frequent in Physics and elsewhere.

As far as I know, the algorithms used by Origin are not clearly documented. So, I have doubts about using it. But it is a very popular black box.

is relevant here, I think.  Eg. this article on Wikipedia has a small section on this issue. For sure, uncorrelated measurements with different uncertainties are quite frequent in Physics and elsewhere.

As far as I know, the algorithms used by Origin are not clearly documented. So, I have doubts about using it. But it is a very popular black box.

I save thread pages regularly (currently under Win XP with Firefox 2.0.0.13). I would very much prefer to be able to save the threads in plain text format though.

The Aladjev library discused in this thread provides a command with a different evaluation of floats:

Evalf(3/4);
                                 0.75

This is for Maple 10.06.

The Aladjev library discused in this thread provides a command with a different evaluation of floats:

Evalf(3/4);
                                 0.75

This is for Maple 10.06.

that the default is to assume that 0.75 is an approximation to 2 digits and not the exact decimal representation of 3/4. It is the user who has to decide in which of these two contexts the interpretation is to be made.

that the default is to assume that 0.75 is an approximation to 2 digits and not the exact decimal representation of 3/4. It is the user who has to decide in which of these two contexts the interpretation is to be made.