<s>[INST] Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$...Definite Integral Calculate
$$
I_1:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx, \ p \geq 1.
$$
I am trying to solve this integral $I_1$.  I know how to solve a related integral $I_2$
$$
I_2:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^2} dx=\frac{\pi}{4}\bigg(2\ln 2 +3\ln \pi-4\ln\Gamma\big(\frac{1}{4}\big) \bigg)
$$
but I am not sure how to use that result here. In this case I just use the substitution $x=e^{-\xi}$ and than use a series expansion.  The result is
$$
I_2=\int_0^\infty \frac{\xi^s e^{-\xi}}{1+e^{-2\xi}} d\xi=\sum_{n=0}^\infty (-1)^n \frac{\Gamma(s+1)}{(2n+1)^{s+1}}=\Gamma(s+1)L(s+1,\chi_4)
$$
where L is the Dirichlet L-Function where $\chi_4$ is the unique non-principal character.  This result is further simplified but takes some work.  I am interested in the general case above, $I_1$ Thanks
 [/INST] Just for your information, I used a CAS without any success for the general case. However, I obtained some formulas.   
For $p=2$,
$$\frac{1}{8} \left(-\gamma _1\left(\frac{1}{8}\right)+\gamma
   _1\left(\frac{5}{8}\right)-\sqrt{2} (\gamma +\log (8)) \left(\pi +2 \log
   \left(\cot \left(\frac{\pi }{8}\right)\right)\right)\right)$$
For $p=3$,
$$\frac{1}{36} \left(-2 \gamma _1\left(\frac{1}{12}\right)+\gamma
   _1\left(\frac{5}{12}\right)+2 \gamma _1\left(\frac{7}{12}\right)-\gamma
   _1\left(\frac{11}{12}\right)+12 \sqrt{3} \log (2) \log \left(\sqrt{3}-1\right)+6
   \sqrt{3} \log (3) \log \left(\sqrt{3}-1\right)-12 \sqrt{3} \log (2) \log
   \left(1+\sqrt{3}\right)-6 \sqrt{3} \log (3) \log \left(1+\sqrt{3}\right)-2 \gamma 
   \left(\pi +3 \sqrt{3} \left(\log \left(1+\sqrt{3}\right)-\log
   \left(\sqrt{3}-1\right)\right)\right)+\pi  \left(-3 \log (3)+\log (16)+12 \log
   (\pi )-16 \log \left(\Gamma \left(\frac{1}{4}\right)\right)\right)\right)$$ For $p=4$,$$\frac{1}{16} \left(-\gamma _1\left(\frac{1}{16}\right)+\gamma
   _1\left(\frac{9}{16}\right)+16 \log (2) \sin \left(\frac{\pi }{8}\right) \log
   \left(\sin \left(\frac{3 \pi }{16}\right)\right)-4 \pi  \log (2) \csc
   \left(\frac{\pi }{8}\right)-16 \log (2) \sin \left(\frac{\pi }{8}\right) \log
   \left(\cos \left(\frac{3 \pi }{16}\right)\right)+16 \log (2) \cos \left(\frac{\pi
   }{8}\right) \log \left(\tan \left(\frac{\pi }{16}\right)\right)-\gamma  \left(\pi 
   \csc \left(\frac{\pi }{8}\right)+4 \left(\sin \left(\frac{\pi }{8}\right)
   \left(\log \left(\cos \left(\frac{3 \pi }{16}\right)\right)-\log \left(\sin
   \left(\frac{3 \pi }{16}\right)\right)\right)+\cos \left(\frac{\pi }{8}\right) \log
   \left(\cot \left(\frac{\pi }{16}\right)\right)\right)\right)\right)$$ In these formulas, $\gamma$ is the Euler constant and $\gamma_1$ is the  Stieltjes constant.
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