integrate(log(1+(1+x**2)**0.5), x) -> incorrect result
Subject: integrate(log(1+(1+x**2)**0.5), x) -> incorrect result
From: Stavros Macrakis
Date: Fri, 28 Nov 2008 22:28:50 -0500
On Fri, Nov 28, 2008 at 10:10 PM, Raymond Toy <toy.raymond at gmail.com> wrote:
> > integrate(log(1+(1+x**2)**(1/2)), x)
>
> Maxima can't compute the integral completely by itself, unfortunately,
> but it does get close:
>
Clever, Ray! Just to make this easy for new users, here's the explicit
transcript:
(%i1) integrate(log(1+(1+x**2)**(1/2)), x);
(%o1) 'integrate(x^2/((x^2+1)^(3/2)+x^2+1),x)+x*log(sqrt(x^2+1)+1)+atan(x)-x
(%i2) ratsimp(%),algebraic;
(%o2)
x*log(sqrt(x^2+1)+1)+'integrate((sqrt(x^2+1)-1)/(x^2+1),x)+atan(x)-x
<<< rationalizes denominators
(%i3) %,integrate;
(%o3) x*log(sqrt(x^2+1)+1)+asinh(x)-x <<< correct
integral
Now check it:
(%i4) diff(%o3,x);
(%o4) log(sqrt(x^2+1)+1)+x^2/(sqrt(x^2+1)*(sqrt(x^2+1)+1))+1/sqrt(x^2+1)-1
(%i5) ratsimp(%);
(%o5) log(sqrt(x^2+1)+1)