integrate(log(1+(1+x**2)**0.5), x) -> incorrect result

Joost Witteveen wrote:
> When I ask maxima to:
>   integrate(log(1+(1+x**2)**0.5), x)
> it comes with the answer:
>       x*log((x**2+1)**0.5+1)  -  (x**2+1)**1.5 / (3*(x**2+1)+3*x+3)
> but that is wrong (differentiating it again to x gives the wrong result).
>
> I tried it with maxima 5.13.0 (from debian-testing (lenny)), and
> 5.10.0 (debian etch).
> I attempted to build the cvs version as well, but gcl segfaults during
> ./configure.
>
> I notice that if I change the expression to:
>   integrate(log(1+(1+x**2)**(1/2)), x)
> that now at least maxima notices it cannot integrate it.
Using 0.5 instead of the exact 1/2 must confuse some part of the integrator.

Maxima can't compute the integral completely by itself, unfortunately,
but it does get close:

'integrate(x^2/((x^2+1)^(3/2)+x^2+1),x)+x*log(sqrt(x^2+1)+1)+atan(x)-x

If we factor the integrand, we get

x^2/(x^2+1)/(1+sqrt(x^2+1)).

Rationalize the denominator to get rid of the sqrt to get

(sqrt(x^2+1)-1)/(x^2+1)

Maxima can integrate this to get asinh(x)-atan(x).  Combining this with
the previous result give the final answer as

x*log(sqrt(x^2+1)+1)+asinh(x)-x

which matches Mma answer.

It's unfortunate that maxima can't do this by itself.

Ray



>
> For what it's worth, Wolfram does integrate it, and claims:
>
> /Integrate[Log[1 + (1 + x^2)^(1/2)], x] ==/
> -x + ArcSinh[x] + x*Log[1 + Sqrt[1 + x^2]]
>
> (this seems to be correct)
>
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