<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] Since integral to function $f(x)$ is such function $F(x)$ so that $F'(x) = f(x) + C$ for $C \in \mathbb{R}$ you can find derivative of your result and compare it to the original task.
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