<s>[INST] How to find $\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15}}}{1+x^{2+\sqrt{3}}}\right)}{\left(1+x^2\right)\log x}\mathrm dx$ I was challenged to prove this identity
$$\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15\vphantom{\large A}}}}{1+x^{2+\sqrt{3\vphantom{\large A}}}}\right)}{\left(1+x^2\right)\log x}\mathrm dx=\frac{\pi}{4}\left(2+\sqrt{6}\sqrt{3-\sqrt{5}}\right).$$
I was not successful, so I want to ask for your help. Can it be somehow related to integrals listed in that question?
 [/INST] I do not know how to answer your question. However, in order you to challenge your challenger, I give you a few amazing results (obtained using a CAS) for $$f(n)=\int_0^\infty\frac{\log\left(\frac{1+x^a}{1+x^b}\right)}{\left(1+x^2\right)\log x}dx$$ in which $a=2n+\sqrt{4 n^2-1}$ and $b=n+\sqrt{n^2-1}$. 
$$f(1)=\frac{1}{4} \left(1+\sqrt{3}\right) \pi$$
$$f(2)=\frac{1}{4} \left(2-\sqrt{3}+\sqrt{15}\right) \pi$$
$$f(3)=\frac{1}{4} \left(3-2 \sqrt{2}+\sqrt{35}\right) \pi$$
$$f(4)=\frac{1}{4} \left(4+3 \sqrt{7}-\sqrt{15}\right) \pi$$
$$f(5)=\frac{1}{4} \left(5-2 \sqrt{6}+3 \sqrt{11}\right) \pi$$
$$f(6)=\frac{1}{4} \left(6-\sqrt{35}+\sqrt{143}\right) \pi$$
$$f(7)=\frac{1}{4} \left(7-4 \sqrt{3}+\sqrt{195}\right) \pi$$
$$f(8)=\frac{1}{4} \left(8-3 \sqrt{7}+\sqrt{255}\right) \pi$$
$$f(9)=\frac{1}{4} \left(9-4 \sqrt{5}+\sqrt{323}\right) \pi$$
$$f(10)=\frac{1}{4} \left(10-3 \sqrt{11}+\sqrt{399}\right) \pi$$
which are exactly what Pranav Arora answered (what I missed) $$f(n)=\frac \pi 4(a-b)$$
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