What values for x in​​g(x)=
∞
∫
0
Im(
x
1+It
(1+It)
)csch(πt)t=CMRB?

​
In[]:=
m=NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},​​Method"AlternatingSigns",WorkingPrecision20]
Out[]=
0.1878596424602145778
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[10]},WorkingPrecision20]]
Out[]=
{0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[11,20]},WorkingPrecision20]]
Out[]=
{0.99999999999153599881,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599879,0.99999999999153599880}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[21,30]},WorkingPrecision20]]
Out[]=
{0.99999999999153599879,0.99999999999153599879,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[31,40]},WorkingPrecision20]]
Out[]=
{25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[41,50]},WorkingPrecision20]]
Out[]=
{25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[51,60]},WorkingPrecision20]]
Out[]=
{25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[61,70]},WorkingPrecision20]]
Out[]=
{25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[71,80]},WorkingPrecision20]]
Out[]=
{25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[81,90]},WorkingPrecision20]]
Out[]=
{25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
Quiet[​​N[x]/.FindRoot[​​m-NIntegrate[​​Csch[Pit]Im[((1+It)^(x/(1+It)))],{t,0,Infinity},​​WorkingPrecision20,AccuracyGoal20,​​Method"Trapezoidal"],{x,Range[91,100]},WorkingPrecision20]]
Out[]=
{25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358,25.656654035105872358}
In[]:=
N[E^Pi]
Out[]=
23.1407