Removal of one of the indepented variables because it is statistically non significant is not necessary and some points should be consider when decide to remove it:
Test wether there is collinearity between that variable and others, so the removal of this variable is encourged
If that variable is important factor in the research or strongly relevent to your research question, you should not ignore it even if it is non statistically significant
If R square value is enhanced after removal this variable, you may remove it.
You can conduct stepwise regression procedures to can evaluate the importance of this variable when be associated with other independent variables.
Why are you doing this regression? If you want a predictive equation, there's no harm in extra variables. But if you want to interpret the coefficients, you need to do more analysis and thinking. Look at partial regression (some call them added variable) plots for all the variables. Those will help you decide if the predictor in question simply has no relationship with the response variable or if (as some have suggested) you have a collinearity problem. Also check for outliers and non-linear relationships (which might call for re-expressing some of your variables).
In my research, there are hypotheses, to test them I used multiple linear regression, and after that I want to write a prediction equation in the dependent variable. According to this equation: Y=A+B1X1+B2X2+,........
(Before starting to test the hypotheses, I conducted a test for outliers , a normal distribution, and multicollinearity, and all the results were good.)