In[]:=
k=10;​​Table[​​Rest[ResourceFunction["IntegerCompositions"][n+2,2]],​​{n,1,k}];​​partProd[tuple2_List]:=Times@@(((1+#1u^2+#2s^2)&@@#)&/@tuple2);​​Table[s^(2n)-u^(2n),{n,1,k}]*1/partProd/@%%;​​NIntegrate[%,{s,0,1},{u,0,∞}]//Quiet;​​Grid[{{"Integrand","Approx. value"}}~Join~Transpose[{%%,%}]]
Out[]=
Integrand
Approx. value
2
s
-
2
u
(1+2
2
s
+
2
u
)(1+
2
s
+2
2
u
)(1+3
2
u
)
3.27823×
-18
10
4
s
-
4
u
(1+3
2
s
+
2
u
)(1+2
2
s
+2
2
u
)(1+
2
s
+3
2
u
)(1+4
2
u
)
-5.37459×
-19
10
6
s
-
6
u
(1+4
2
s
+
2
u
)(1+3
2
s
+2
2
u
)(1+2
2
s
+3
2
u
)(1+
2
s
+4
2
u
)(1+5
2
u
)
3.64754×
-20
10
8
s
-
8
u
(1+5
2
s
+
2
u
)(1+4
2
s
+2
2
u
)(1+3
2
s
+3
2
u
)(1+2
2
s
+4
2
u
)(1+
2
s
+5
2
u
)(1+6
2
u
)
7.17331×
-21
10
10
s
-
10
u
(1+6
2
s
+
2
u
)(1+5
2
s
+2
2
u
)(1+4
2
s
+3
2
u
)(1+3
2
s
+4
2
u
)(1+2
2
s
+5
2
u
)(1+
2
s
+6
2
u
)(1+7
2
u
)
-1.76986×
-15
10
12
s
-
12
u
(1+7
2
s
+
2
u
)(1+6
2
s
+2
2
u
)(1+5
2
s
+3
2
u
)(1+4
2
s
+4
2
u
)(1+3
2
s
+5
2
u
)(1+2
2
s
+6
2
u
)(1+
2
s
+7
2
u
)(1+8
2
u
)
7.86314×
-23
10
14
s
-
14
u
(1+8
2
s
+
2
u
)(1+7
2
s
+2
2
u
)(1+6
2
s
+3
2
u
)(1+5
2
s
+4
2
u
)(1+4
2
s
+5
2
u
)(1+3
2
s
+6
2
u
)(1+2
2
s
+7
2
u
)(1+
2
s
+8
2
u
)(1+9
2
u
)
-5.84196×
-24
10
16
s
-
16
u
(1+9
2
s
+
2
u
)(1+8
2
s
+2
2
u
)(1+7
2
s
+3
2
u
)(1+6
2
s
+4
2
u
)(1+5
2
s
+5
2
u
)(1+4
2
s
+6
2
u
)(1+3
2
s
+7
2
u
)(1+2
2
s
+8
2
u
)(1+
2
s
+9
2
u
)(1+10
2
u
)
-2.22434×
-22
10
18
s
-
18
u
(1+10
2
s
+
2
u
)(1+9
2
s
+2
2
u
)(1+8
2
s
+3
2
u
)(1+7
2
s
+4
2
u
)(1+6
2
s
+5
2
u
)(1+5
2
s
+6
2
u
)(1+4
2
s
+7
2
u
)(1+3
2
s
+8
2
u
)(1+2
2
s
+9
2
u
)(1+
2
s
+10
2
u
)(1+11
2
u
)
8.5551×
-26
10
20
s
-
20
u
(1+11
2
s
+
2
u
)(1+10
2
s
+2
2
u
)(1+9
2
s
+3
2
u
)(1+8
2
s
+4
2
u
)(1+7
2
s
+5
2
u
)(1+6
2
s
+6
2
u
)(1+5
2
s
+7
2
u
)(1+4
2
s
+8
2
u
)(1+3
2
s
+9
2
u
)(1+2
2
s
+10
2
u
)(1+
2
s
+11
2
u
)(1+12
2
u
)
-2.91351×
-27
10