# [Maxima] integrate(log(1+(1+x**2)**0.5), x) -> incorrect result

Stavros Macrakis macrakis at alum.mit.edu
Fri Nov 28 21:28:50 CST 2008

```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)
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