c=
2
t
+1;d=
1
2
log(c);​​CMRB=
∞
∫
0
d+t
-1
tan
(t)
c

sin
-1
tan
(t)-dt
c
sinh(πt)
t
Out[]=
$Aborted
In[]:=
cmrbInt=
∞
∫
0
integrand(t)t
In[]:=
(mrb=N[NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->110],100])
Out[]=
0.1878596424620671202485179340542732300559030949001387861720046840894772315646602137032966544331074969
In[]:=
ClearAll[t,integrand,cmrbInt,prec];​​​​prec=100;​​​​c=
2
t
+1;d=
1
2
log(c);integrand(t_):=
exp
d+t
-1
tan
(t)
c
sin
-1
tan
(t)-td
c
sinh(πt)
;​​​​cmrbInt=NIntegrate[integrand[t],{t,0,72},WorkingPrecision->prec+20,Method->"DoubleExponential"];​​​​N[cmrbInt,prec]​​​​N[mrb-cmrbInt,prec]
Out[]=
0.1878596424620671202485179340542732300559030949001387861720046840894772315646602137032966544331074971
Out[]=
-2.×
-100
10
(E^((d+t*ArcTan[t])/c)*​​Sin[(ArcTan[t]-t*d)/c])/Sinh[Pi*t]
In[]:=
integrand(t)
Out[]=
tArcTan[t]+
1
2
Log[1+
2
t
]
1+
2
t

Csch[πt]Sin
ArcTan[t]-
1
2
tLog[1+
2
t
]
1+
2
t

In[]:=
c=t^2+1;d=(1/2)*Log[c];;Plot(E^((d+t*ArcTan[t])/c)*​​Sin[(ArcTan[t]-t*d)/c])Sinh[Pi*t],{t,0,10},Filling->Axis,PlotRange->Full,PlotLabel->
tArcTan[t]+
1
2
Log[1+
2
t
]
1+
2
t

Csch[πt]Sin
ArcTan[t]-
1
2
tLog[1+
2
t
]
1+
2
t
""
Out[]=