The responses here cover a good part of the technical issues, so I won't repeat their advice.  When it comes to what you report, however, the issue shifts from technical to theoretical, as follows:

1) You had a reason (I have to assume) built from existing literature as to why those five variables would be related to your outcome.

2) Presuming you have tested the things that typically screw up the estimate of significance in multiple regression (i.e., you have a large enough sample size to support 5 IV {which is a minimum of 150 cases in typical social science research btw}, you have tested for colinearity between IV's, you examined the residuals for problems, etc), you are left with the question of how and why your sample is not behaving in the way that your theory predicted. 

3) Using basic logic, there are two options:

  - your sample is flawed (not bad, but substantively different from that on which your theory was built)

  - your theory is flawed (again, not bad, but substantively different from what plays out in the real world.

    Either way, you have something interesting to report, because "Theory says A, here is empirical support of A" is kinda boring.

As a reader, I would like to know what the model looks like with and without the fifth variable in it, so I could see what happens to the other coefficients when it gets removed.  As a reviewer, I would ask for a "revise and resubmit" to include that information if you didn't include it.  But ultimately, since it is a paper, it is up to your adviser what you do with the troublesome variable.

 Dear Levan, 

Check this paper out and see how I am describing the linear regression analyses in tables where variables (significant or not) are included in the significant models.  A non-significant variable and also its position can contribute to the whole model. The variance of the whole model is given in R2 but the squared part correlation gives the % with which a significant independent variable per se contributes to the variance. Nota bene they do not add up to the R2. 

Article Student nurses and the general population in Sweden: Trends ...

Levan -

p-values are often misinterpreted and you would likely understand better by looking at the estimated coefficients and their estimated standard errors. This is all sample size dependent however.  Further, for multiple regression, regressors may interact heavily with one another, and thus these results may be much further misleading.

If you want to know what happens if you were to drop that regressor, often this may decrease variance, but increase bias.  Still, the estimated variance of the prediction error can be an important measure.  (If, for example, SAS PROC REG were to cover the kind of regression you want, I know that STDI is the square root of that measure.)  If you have test data, you could compare model results. You might even want to research "cross validation."  (Generally, I think that you might be better off without a variable that you don't really need, but that can possibly create substantial bias.) 

You should, however, consider what your subject matter theory says.  If there is reason to believe that the regressor you may drop is related in an important way to your dependent variable, then you might want to do a separate simple regression to look into that.  (Don't forget heteroscedasticity.  Also, if y should be zero when x is zero, you could consider the classical ratio estimator.  This can all apply to multiple regression as well.)  However, even if there is a strong relationship between that x and your y, it may still not be needed in your overall multiple regression model, as it may be covered by the other regressors due to collinearity or some other reason or reasons.  This may be apparent from the subject matter theory. 

Scatterplots can help greatly on all of this.  They can help you understand the relationship between y and your predicted y for each model.  They can help you understand relationships between possible regressors.  (I am assuming continuous data.)  They can help greatly in residual analyses. 

You asked "...should I use second model's results (Beta, p values, etc...) (with 5th variable removed) in my paper?"  You should report the estimated coefficients for the model that you decide to use.  But you could also note any exploratory model results, perhaps in an appendix.  As for p-values, I know many people still use (and misuse) them, but I think that more and more journals and researchers are realizing that a p-value is of very limited use, and "effect size" is very important, from a practical point of view.  (You could research "effect size.")  Instead of reporting p-values, you could report the estimated standard error for each estimated coefficient.  (This is still sample size dependent.  A bigger sample reduces your estimated standard error.) 

Regarding p-values, you might consider the following, in general:

 https://www.researchgate.net/publication/262971440_Practical_Interpretation_of_Hypothesis_Tests_-_letter_to_the_editor_-_TAS

If you note that the same regressor in two different models (say, with and without the fifth regressor) show very different coefficient to standard error ratios, that is because of the regressor interactions within each model, and the changed number of regressors.

Note that R-square (r-square in the 'simple'case) is redefined for each model. There are papers by Willett and Singer on that, I think in The American Statistician.  Anyway, R-square can be volatile and misleading.   

I hope some of this helps, and that I did not misunderstand your situation.

Best wishes and have fun with your project.

Cheers - Jim

Article Practical Interpretation of Hypothesis Tests - letter to the...

High p-values accompanying the estimated coefficients simply indicate that you could not reject the null hypothesis. I assume that the null hypothesis is that the estimated coefficient equal zero against the alternative that it is different from zero. For example a p-value of 0.734 implies that the coefficient of interest is insignificantly different from zero.   Accordingly, the independent variable of concern (i.e. of that coefficient) is not important in explaining variation in the dependent variable.

Before going further, I assume that you have checked to make sure that the model should fit. A non-significant outcome could be cause by a poor model. So you have looked at residuals, and maybe you have tried a few transformations to get a better fit. However, the variable is still not significant.

1) Remove the non-significant variable from the model, and reanalyze the data without the non-significant variable. This assumes that there are no significant interaction terms that include this variable. You might say something like variable X was initially entered in the model but was removed because it was not significant.

2) Rethink the problem and gather more data. If others have shown that the variable is significant and there is sound theoretical basis indicating that it should be a significant factor, then the fact that it is not significant in your model indicates that a mistake was made at some point. Of course, there is always a small possibility that the accepted model is the one that is wrong.

The multiple regression is a sensitive issue. There are 5 assumptions that should be met to good a reliable model. unfortunately these assumptions are  mostly overlooked, then the model is not that good to continue with making inference on the population where that sample it taken from. High p-value is an evidence that the variable (5) is not important in controlling the response, just say it is not significant and as one colleague wrote earlier one of the assumptions is the collinearity means the correlation between two or more independent variable might lead to insignificant variable in the model and many other issues 

The responses here cover a good part of the technical issues, so I won't repeat their advice.  When it comes to what you report, however, the issue shifts from technical to theoretical, as follows:

1) You had a reason (I have to assume) built from existing literature as to why those five variables would be related to your outcome.

2) Presuming you have tested the things that typically screw up the estimate of significance in multiple regression (i.e., you have a large enough sample size to support 5 IV {which is a minimum of 150 cases in typical social science research btw}, you have tested for colinearity between IV's, you examined the residuals for problems, etc), you are left with the question of how and why your sample is not behaving in the way that your theory predicted. 

3) Using basic logic, there are two options:

  - your sample is flawed (not bad, but substantively different from that on which your theory was built)

  - your theory is flawed (again, not bad, but substantively different from what plays out in the real world.

    Either way, you have something interesting to report, because "Theory says A, here is empirical support of A" is kinda boring.

As a reader, I would like to know what the model looks like with and without the fifth variable in it, so I could see what happens to the other coefficients when it gets removed.  As a reviewer, I would ask for a "revise and resubmit" to include that information if you didn't include it.  But ultimately, since it is a paper, it is up to your adviser what you do with the troublesome variable.

I sort of agree with Julia's suggestion as a reviewer. I think it depends on what the overall manuscript is like. If this is a simple manuscript then I would agree with Julia's idea. However, reviewers may get annoyed if you present several "failed" models and reviewers are now on page 45. Maybe consider putting the detailed analysis in supplemental material.

My view is that I don't really want to wade through models that have problems. I would like to know what you think the data are telling us. I want enough methods to get a sense that you did the basic steps and that you seriously considered other possibilities. If I am really lucky I would like access to the raw data as a supplement. That way I can reanalyze the data if I am seriously interested in your experiment. I would combine it with other data to try and tell a larger story.

You should strongly use the theory on the subject matter as a guide to build the model

Examine the potential problems of multicollinearity and even the quality of the sample

If you delete the variable with no significant contribution and the other coefficients remain roughly equal, use the simplified model and add a supplement to comment your procedure. Otherwise, there are more complex problems to be solved