# Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Another integral similar to my previous question: $$\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$$ Could you suggets how to evaluate it? Is there a closed form?

• I think it is about time to create closedform.stackexchange.com. – Julien Nov 23 '13 at 23:03
• This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. – Carl Mummert Nov 25 '13 at 3:57
• In particular, please try to explain where you encountered this integral and why it is interesting to evaluate. – Carl Mummert Nov 25 '13 at 3:58
• On another forum, a very similar integral has been evaluated: See here .Your integral can be calculated using the same technique. – Shobhit Bhatnagar Feb 3 '14 at 11:30

## 1 Answer

Yes, there is a closed form: $$\frac{\pi^2}3-\ln^22-4\,G,$$ where $G$ is the Catalan constant: $$G=-\int_0^1\frac{\ln x}{x^2+1}dx.$$