On 02/18/2011 03:59 PM, Stavros Macrakis wrote:
> Hmm. Mathematica
> (integrals.wolfram.com<http://integrals.wolfram.com/index.jsp?expr=+sqrt(x^2%2B4)/(x^2%2B1)>)
> gets
> asin(x/2)+sqrt(3)*atan((sqrt(3)*x)/sqrt(4+x^2)), but that is not correct
> according to Maxima:
>
> ex0: sqrt(x^2+4)/(x^2+1)$
> ex1: asin(x/2)+sqrt(3)*atan(sqrt(3)*x/sqrt(4+x^2))$
>
> The difference d: diff(ex1,x)-ex0 is messy and hard to simplify to something
> simple.
>
> Plotting d shows that it is non-constant and taylor agrees that the
> difference is non-constant:
>
> taylor(diff(ex1,x)-ex0,x,0,10);
> => x^2/8+5*x^6/1024+63*x^10/262144
>
> Have I made some blunder, or is the Mathematica result incorrect?
I have
(%i1) display2d : false;
(%o1) false
(%i2) diff(sqrt(p-1)*atan(sqrt(p-1)*x/sqrt(x^2+p)) + asinh(x/sqrt(p)),x);
(%o2) sqrt(p-1)*(sqrt(p-1)/sqrt(x^2+p)-sqrt(p-1)*x^2/(x^2+p)^(3/2))
/((p-1)*x^2/(x^2+p)+1)
+1/(sqrt(p)*sqrt(x^2/p+1))
(%i3) radcan(%);
(%o3) sqrt(x^2+p)/(x^2+1)
you have a typo asin should read asinh
Andre