I am trying to make a table, in a range where $\alpha$ = 0.01, 0.1,.....1.5....2.0, and $\mu$=0.01, 0.1, 0.2,....1.0, 1.1, ....10 for the following "Minimize" function, and compare it also to "NMinimize" with different methods: NMinimize[ $\frac{\frac{64 b^5 \left(-16 b^5+4 b^3+\frac{3 \pi \sqrt{\frac{1}{b^2}} b}{\sqrt{1-b^2}}-\frac{6 \sin ^{-1}(b)}{\sqrt{1-b^2}}+6 b\right)}{\pi \left(96 b^5-96 b^7\right)}-\frac{4\, \alpha\, b^3}{(2 b+\mu )^2}}{b^2+1}$] What's the recommended set of commands to do this? I also tried D[ $\frac{\frac{64 b^5 \left(-16 b^5+4 b^3+\frac{3 \pi \sqrt{\frac{1}{b^2}} b}{\sqrt{1-b^2}}-\frac{6 \sin ^{-1}(b)}{\sqrt{1-b^2}}+6 b\right)}{\pi \left(96 b^5-96 b^7\right)}-\frac{4\, \alpha\, b^3}{(2 b+\mu )^2}}{b^2+1},\,b$], and the using the result with Solve[result==0,b], but the evaluation doesn't complete even after 10 minutes on an i7 machine with 16 GB RAM. Also doesn't seem to work even with ND[ ].

Thank you

Many thanks Jim! I am now trying to plot the list of triplets of the form `{0.1,0.01,{1.99839,{b->0.0478935}}}`. Being a newbie, I am transferring all of the values in another editor, substitute the `->` and then plotting a 3D graph. Is there a better way to combine Plot2D and Plot3D in the same (or sequential) declarative command? "Table" command also appears to be working, but I have to check it thoroughly. I don't know how to combine Plot in either the list from the first method or from the Table directly.

Sorry, I should have pointed out that this was a conditional expression `Im[b^2] != 0 \|\| Re[b^2] >= 0]`, and Mathematica calculated this from `Integrate[(p^2 (p^2 + 1)^(3/2))/(b^2 + p^2)^4, {p, 0, Infinity}]`. I am unsure how to do a loop in Mathematica to construct a table of the value of the the minimum expression, and the value of "b" where the minimum occurs, either using NMinimize or Minimize i.e. vary alpha from 0.01,0.02,...0.1,0.2,0.3....1.0,1.1...2.0, and vary [Mu] from 0.01,0.02,..0.1,0.2,0.3....1.0,1.1,....10. It's an uneven grid, so alpha values go from 0.01 in increments of 0.01, then from 0.9 to 2.0 in increments of 0.1 etc. NMinimize[{(-((4 alpha b^3)/(2 b + \[Mu])^2) + ( 64 b^5 (6 b + 4 b^3 - 16 b^5 + (3 Sqrt[1/b^2] b \[Pi])/Sqrt[ 1 - b^2] - (6 ArcSin[b])/Sqrt[1 - b^2]))/((96 b^5 - 96 b^7) \[Pi]))/(1 + b^2), 0 <= b <= 0.9999999}, {b}] An aside is the problem that evaluating `D(-((4 alpha b^3)/(2 b + \[Mu])^2) + ( 64 b^5 (6 b + 4 b^3 - 16 b^5 + (3 Sqrt[1/b^2] b \[Pi])/Sqrt[ 1 - b^2] - (6 ArcSin[b])/Sqrt[1 - b^2]))/((96 b^5 - 96 b^7) \[Pi]))/(1 + b^2), b]`, then equating the result to == 0, and solving it -- the calculation doesn't complete even after 15 minutes. Same is the case if numerical differentiation is performed and the result equated to 0.

constructing a table with Minimize and NMinimize