In[]:=
$Assumptions=0≤
x
1
≤a&&0≤
x
2
≤a&&
x
2
>
x
1
&&0≤
y
1
≤b&&0≤
y
2
≤b&&
y
2
>
y
1
&&a>0&&b>0&&a>b;
In[]:=
K1=
1
2
a
2
b
b
∫
0
b
∫
0
a
∫
0
a
∫
0
1
2
Log[
2
(
x
2
-
x
1
)
+
2
(
y
2
-
y
1
)
]
x
1

x
2

y
1

y
2
1
12
2
a
2
b
a
2
b
(-25a+8bπ)-8a
3
b
ArcTan
a
b
+8ab(
2
a
-2
2
b
)ArcTan
b
a
+2
4
a
Log[a]+2
4
b
Log[b]-(
4
a
-6
2
a
2
b
+
4
b
)Log[
2
a
+
2
b
]
In[]:=
a=5;b=2;
In[]:=
K1//N
0.448187
In[]:=
a=.;b=.;
In[]:=
NU=Numerator[K1]//Expand
Out[]=
-25
2
a
2
b
+8a
3
b
π-8a
3
b
ArcTan
a
b
+8
3
a
bArcTan
b
a
-16a
3
b
ArcTan
b
a
+2
4
a
Log[a]+2
4
b
Log[b]-
4
a
Log[
2
a
+
2
b
]+6
2
a
2
b
Log[
2
a
+
2
b
]-
4
b
Log[
2
a
+
2
b
]
In[]:=
NU=NU/.2
4
a
Log[a]-
4
a
Log[
2
a
+
2
b
]->-
4
a
Log1+
2
b
2
a
;
In[]:=
NU=NU/.2
4
b
Log[b]-
4
b
Log[
2
a
+
2
b
]->-
4
b
Log1+
2
a
2
b
;
In[]:=
K1=
NU
Denominator[K1]
Out[]=
1
12
2
a
2
b
-25
2
a
2
b
+8a
3
b
π-8a
3
b
ArcTan
a
b
+8
3
a
bArcTan
b
a
-16a
3
b
ArcTan
b
a
-
4
b
Log1+
2
a
2
b
+6
2
a
2
b
Log[
2
a
+
2
b
]-
4
a
Log1+
2
b
2
a

In[]:=
a=5;b=2;
In[]:=
K1//N
Out[]=
0.448187
In[]:=
a=.;b=.;
In[]:=
Limit[K1,b0]//Expand
Out[]=
-
3
2
+Log[a]
In[]:=
K=K1/.ba//FullSimplify
Out[]=
1
12
(-25+4π+Log[16])+Log[a]
In[]:=
K//N
Out[]=
-0.805087+Log[a]