$\int_0^\infty e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right) dx$

f[x_] = E^(I*Pi*x)*(1 - (x + 1)^(1/(x + 1))); 
g[x_] = x^(1/x); u := t/(1 - t); 

sub = Im[NIntegrate[(f[(-I t)] - f[( I t)])/(Exp[2 Pi t] - 1), {t,
          0, Infinity}, WorkingPrecision -> 100]]

0.1170836031505383167089899122239912286901483986967757585888318959258587743002\
7817712246477316693025869

m = NSum[f[( t)] , {t, 0, Infinity}, WorkingPrecision -> 100, 
  Method -> "AlternatingSigns"]

0.1878596424620671202485179340542732300559030949001387861720046840894772315646\
6021370329665443217278

m - sub

0.0707760393115288035395280218302820013657546962033630275831727881636184572643\
8203658083188126524252

Is the same as 


{Re[NIntegrate[f[t], {t, 0, Infinity}]], and, 
 Re[NIntegrate[f[t], {t, 0, Infinity I}, WorkingPrecision -> 100]]}

NIntegrate::deodiv: DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.

NIntegrate::deodiv: DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.

{0.070776, and, \
0.0707760393115288035395280218302820013657546962033630275831727881636184572643\
8203658083188126617723821}

To be continued.