https://math.stackexchange.com/questions/4943865/how-to-integrate-over-this-multi-function-surface

​

numerical integration :
In[]:=
NIntegrate[1/(Sqrt[u^2+v^2](1+u^2+v^2)),{u,0,1},{v,0,1}]
Out[]=
1.31941
{evaluation time, exact value} :
In[]:=
Integrate[1/(Sqrt[u^2+v^2](1+u^2+v^2)),{u,0,1},{v,0,1}]//AbsoluteTiming
Out[]=
46.1995,
1
12
7
2
π
+3πArcSin
2+
2
3/4
2
+2ArcSinh[1]-3ArcSinh
2-
2
3/4
2
-Log[4]+Log[(1-)+
2
]+Log[(1+)+
2
]+Log[((-1+)+
2
)]+Log[-2+
2
]+Log[2+
2
]+Log[((1+)+
2
)]-Log[1-(-1+
2
)
ArcSinh[1]

]-Log[1+(-1+
2
)
ArcSinh[1]

]-Log[1-(1+
2
)
ArcSinh[1]

]-Log[1+(1+
2
)
ArcSinh[1]

]-32
2
ArcSinh[1]
-8ArcSinh
2-
2
3/4
2
ArcTan
(2+
2
)Cot
1
4
(π+2ArcSinh[1])
2
+4ArcSinh
2-
2
3/4
2
Log[((-1+)+
2
)]-4ArcSinh
2-
2
3/4
2
Log[((1+)+
2
)]-4ArcSinh
2-
2
3/4
2
Log[1-(-1+
2
)
ArcSinh[1]

]+4ArcSinh
2-
2
3/4
2
Log[1-(1+
2
)
ArcSinh[1]

]+ArcSin
2+
2
3/4
2
-8ArcTan
(-2+
2
)Cot
1
4
(π+2ArcSinh[1])
2
+4Log[(1-)+
2
]-Log[(1+)+
2
]-Log[1+(-1+
2
)
ArcSinh[1]

]+Log[1+(1+
2
)
ArcSinh[1]

]+ArcSinh[1]-Log[9]+2-2Log[1+
2
]+Log[1-(-1+
2
)
ArcSinh[1]

]+Log[1+(-1+
2
)
ArcSinh[1]

]+Log[1-(1+
2
)
ArcSinh[1]

]+Log[1+(1+
2
)
ArcSinh[1]

]-2Log[-1+
2ArcSinh[1]

]+2PolyLog2,-(-1+
2
)+2PolyLog2,(-1+
2
)-PolyLog2,-3+2
2
+2PolyLog2,
-2ArcSinh[1]

-2PolyLog2,-(-1+
2
)
-ArcSinh[1]

-2PolyLog2,

2
-ArcSinh[1]

2+
2
+2PolyLog2,-(-1+
2
)
ArcSinh[1]

+2PolyLog2,(-1+
2
)
ArcSinh[1]



integral over [-1, 1]

In[]:=
Integrate[ArcSin[t]/(1+t^2),{t,1/Sqrt[2],1}]
Out[]=
1
16
2
2
π
+
2
Log[2]
-4
2
Log[1+
2
]
-4Log[2]Log[2+
2
]+4
2
Log[2+
2
]
-πLog[2]-2Log--
1/4
(-1)
+Log(2+)-
1/4
(-1)
-Log-+
1/4
(-1)
-Log(2+)+
1/4
(-1)
+2Log[(1+)-
2
]-2Log[(1-)+
2
]+2Log[(1+)+
2
]-Log[1+
2
]-2Log[-((1+)+
2
)]+Log[2+
2
]-8PolyLog2,(1-)-
1-
2
+8PolyLog2,(-1-)+
1+
2
-8PolyLog2,
1/4
(-1)
(-1+
2
)+8PolyLog2,
3/4
(-1)
(-1+
2
)
In[]:=
NIntegrate[1/(Sqrt[u^2+v^2](1+u^2+v^2)),{u,v}∈Rectangle[{-1,-1},{1,1}]]​​π^2-2πArcTan[Sqrt[2]]+8NIntegrate[ArcSin[t]/(1+t^2),{t,1/Sqrt[2],1}]​​π^2+8Re[PolyLog[2,(Sqrt[2]-1)E^(3Iπ/4)]-PolyLog[2,(Sqrt[2]-1)E^(Iπ/4)]]//N
Out[]=
5.27764
Out[]=
5.27764
Out[]=
5.27764