In[]:=
Module[​​{a=RandomReal[{0.99999,1}]},​​​​{​​NIntegrate[(Sqrt[x]ArcTan[x]Log[1+x^2])/(a^2+x^2),{x,0,∞}],​​    ​​π/(2Sqrt[2])(NIntegrate[(8Log[a]+8Log[1-x^2]-4Sqrt[a]x(1+ax^2)Log[1-ax^2])/(1-a^2x^4),{x,0,1}]+(8Catalan+π^2-Log[2]^2)),​​​​(*perQuýNhân'scomment*)​​π/(2Sqrt[2])(NIntegrate[(8-4x(1+x^2))/(1-x^4)Log[1-x^2],{x,0,1}]+8NIntegrate[(Log[1-x^2]-Log[1-ax^2])/(1-a^2x^4),{x,0,1}]+(8Catalan+π^2-Log[2]^2)),​​​​Sqrt[2]π(Catalan+π^2/12+π/4Log[2]+3/4Log[2]^2)//N​​​​}//Column​​​​]
Out[]=
11.7433
11.7432
11.7433
11.7433
In[]:=
π/(2Sqrt[2])(NIntegrate[(8-4x(1+x^2))/(1-x^4)Log[1-x^2],{x,0,1}]+8Catalan+π^2-Log[2]^2)
Out[]=
15.3974