<s>[INST] How to integrate $\ln \big( b + \sqrt{b^2 + c^2 + x^2}\,\big)$? I am looking to demonstrate the following result. Any ideas are much appreciated. 
$$
\begin{align}\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx  = &\;x \ln \left( b + \sqrt{b^2 +c^2 +x^2}\right) +b \ln \left(2x + 2\sqrt{ b^2 +c^2 +x^2} \right)\\&\;  - c \arctan \left(\frac{ b x} { c  \sqrt{b^2 +c^2 +x^2}}\right) + c \arctan  \left(\frac{x}{c}\right) -x  \end{align}$$
 [/INST] From the first term at the right let's try integrations by parts :
\begin{align}
I(b,c)&:=\int \ln \left( b + \sqrt{b^2 + c^2 + x^2}\right) dx-x \ln \left( b + \sqrt{b^2 +c^2 +x^2}\right) \\
&= -\int x\frac{1}{b + \sqrt{b^2 +c^2 +x^2}}\frac{2\,x}{2\sqrt{b^2 +c^2 +x^2}}\,dx\\
&= \int \frac{{b - \sqrt{b^2 +c^2 +x^2}}}{c^2 +x^2}\frac{x^2}{\sqrt{b^2 +c^2 +x^2}}\,dx\\
&= b\int \frac{c^2+x^2-c^2}{(c^2 +x^2)\sqrt{b^2 +c^2 +x^2}}\,dx-\int\frac{x^2}{c^2 +x^2}\,dx\\
&= b\int \frac{dx}{\sqrt{b^2 +c^2 +x^2}}-\int \frac{b\,c^2}{(c^2 +x^2)\sqrt{b^2 +c^2 +x^2}}\,dx+\int\frac{c^2-(c^2+x^2)}{c^2 +x^2}\,dx\\
&= b\,\ln \left(2x + 2\sqrt{ b^2 +c^2 +x^2} \right)  - c\,\arctan \frac{ b x} { c  \sqrt{b^2 +c^2 +x^2}} + c\, \arctan \frac{x}{c} -x\\
\end{align}
The three last ones should be more direct !
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