​​WiththehelpofWolframMathematicaIhaveobtainedaclosedformforthedoubleintegral:
J=Integrate
Log[1+xy]
1+x^2+y^2
,{x,0,1},{y,0,1}=(*y->
t
x
*)Integrate
xLog[1+t]
x^2(1+x^2)+t^2
,{t,0,x},{x,0,1}=Integrate
xLog[1+t]
x^2(1+x^2)+t^2
,{x,0,t},{t,0,1}=Integrate
Log[1+t]
1-4
2
t
ArcTanh
1-4
2
t
(1-t^2)
1+5
2
t
,{t,0,1}=(*t->
u
1+
2
u
*)
Integrate-
ArcTanh
-1+
6
u
1+8
2
u
+8
4
u
+
6
u
Log1+
u
1+
2
u

1+
2
u
,u,0,
1
2
1-
3
.
takingintoaccount
ArcTanh[z]->1/2Log
1+z
1-z
itturnsout:
ArcTanh
-1+
6
u
1+8
2
u
+8
4
u
+
6
u

1
2
Log
2
u
2
(2+
2
u
)
2
(1+2
2
u
)
->Log
u(2+
2
u
)
(1+2
2
u
)
->Log[u]+Log
2+
2
u
1+2
2
u

J=Integrate
Log[u]Log
1+
2
u
1+u+
2
u

1+
2
u
+
Log
2+
2
u
1+2
2
u
Log
1+
2
u
1+u+
2
u

1+
2
u
,u,0,
1
2
1-
3
;
beJ1=Integrate
Log[u]Log
1+
2
u
1+u+
2
u

1+
2
u
,u,0,
1
2
1-
3
;
andJ2=Integrate
Log
2+
2
u
1+2
2
u
Log
1+
2
u
1+u+
2
u

1+
2
u
,u,0,
1
2
1-
3
;therefore​​J=J1+J2
andinturnJ1=J11-J12,being
J11=Integrate
Log[u]Log[1+
2
u
]
1+
2
u
,u,0,
1
2
1-
3
=(*u->Tan[w]*)Integrate-2Log[Cos[w]]Log[Tan[w]],w,0,ArcTan
1
2
1-
3
//Simplify;
Short[Expand[J11],3]
-

5
π
8(1)
-
3
5
π
8(1)
+267+
3
2
π
PolyLog3,
(2+)+
3
2+2
3
2+
3

2
π
-1-91+
3
Log[4]+1-(2+)+
3
Log[+
3
]
J12=Integrate
Log[u]Log[1+u+
2
u
]
1+
2
u
,u,0,
1
2
1-
3
=Integrate-
Log[u]Log
1
2
-

3
2
+u
2(-+u)
,u,0,
1
2
1-
3
+Integrate
Log[u]Log
1
2
-

3
2
+u
2(+u)
,u,0,
1
2
1-
3
-Integrate
Log[u]Log
1
2
+

3
2
+u
2(-+u)
,u,0,
1
2
1-
3
+Integrate
Log[u]Log
1
2
+

3
2
+u
2(+u)
,u,0,
1
2
1-
3
;
Short[J12,2]
-
1
324
4+273Log
-+
3
+
3

2
Log
2
(1)11

+28+61+61+1+1+
1
144
(1)
J2=Integrate
Log[1+
2
u
]Log[2+
2
u
]
1+
2
u
,u,0,
1
2
1-
3
-Integrate
Log[2+
2
u
]Log[1+u+
2
u
]
1+
2
u
,u,0,
1
2
1-
3
-Integrate
Log[1+
2
u
]Log[1+2
2
u
]
1+
2
u
,u,0,
1
2
1-
3
+Integrate
Log[1+u+
2
u
]Log[1+2
2
u
]
1+
2
u
,u,0,
1
2
1-
3
;
Short[J2,4]
-
3
π
12
+
1
6
2
π
ArcTan
1
2
(1-
3
)-
5
12

2
π
Log[2]+2059
​
J=J11-J12+J2;
Short[J,4]
-
3
π
12
+2069+
1
144
-2
2
π
Log[8]-9Log-
1
(-2+)+
3
-6Log[(-2+)+
3
]+Log-
+
3
(-2+)+
3
-2Log1+
7+4
3
+Log-4-4
7+4
3
-Log
1
2
+
7+4
3
-12π(1)+3(1)
​​
numericalvalue=NIntegrate
Log[1+t]
1-4
2
t
ArcTanh
1-4
2
t
(1-t^2)
1+5
2
t
,{t,0,1},WorkingPrecision->49,AccuracyGoal->30,PrecisionGoal->30
0.1111215849008960477686265005659304994890975121227
Chop[N[J11-J12+J2,48]]
0.1111215849008960477686265005659304994890975121227
​​
Chop[N[J11-J12+J2,48]]===numericalvalue
True