Integration shows RootSum and really long expressions

Question

I am doing this integration and when I use $C_alpha(T)$ as just constant numbers(Subscript[C, [Alpha]][T_]= Piecewise[{{1.2, T < 100}, {8, T > 1000}, {5, 100 <= T <= 1000}}]), I am able to plot the graph nicely. But once I use a straight line equation in that Piecewise function, integration shows weird answers. What am I doing wrong here? Remove["Global`*"]; f[p_, T_] := 1/(Exp[p/T] + 1) Needs["Rubi`"] [Theta] = (7*10^-11)^(1/2) Subscript[M, 1] = 2*10^-3 Tf =100 Subscript[C, [Alpha]][T_] = Piecewise[{{1.2, T < 100}, {8, T > 1000}, {8 + (900/6.8)*(T - 1000), 100 <= T <= 1000}}] [CapitalGamma][p_, T_] = Subscript[C, [Alpha]][T]*10^-11 * p* T^4 // PiecewiseExpand [CapitalDelta][p_] := (Subscript[M, 1]^2 /(2*p)) h[p_, T_] = (1/ 8) ([CapitalGamma][p, T]* [CapitalDelta][p]^2 * Sin[2*[Theta]]^2)/([CapitalDelta][ p]^2 + ([CapitalGamma][p, T]/2)^2) // FullSimplify // PiecewiseExpand fDW[p_, t_] = f[p*t/Tf, t] Integrate[h[p*t/1000, t], t] // PiecewiseExpand x[a_, b_] := a/b xf = Table[{x[p, T], x[p, T]^2*fDW[p, T]}, {p, 1, 10^3}, {T, 100, 500, 100}] // N plot1 = ListLogLogPlot[xf, AxesLabel -> {x, Subscript[x.b2f, DW]}, Ticks -> {Automatic, Automatic}, PlotLabel -> Style["!(*SubscriptBox[(M), (1)])=7keV,[Theta].b2 = 7x10⁻¹¹"]] ```

Bob Hanlon · Accepted Answer

Use exact values until the Table for xf is calculated. $Version (* "12.2.0 for Mac OS X x86 (64-bit) (December 12, 2020)" *) Clear["Global`*"] f[p_, T_] := 1/(Exp[p/T] + 1) Use of Rubi is not necessary (* Needs["Rubi`"] *) θ = (7*10^-11)^(1/2); Subscript[M, 1] = 2*10^-3; Tf = 100; Subscript[C, α][T_] = Piecewise[ {{6/5, T < 100}, {8, T > 1000}, {8 + (2250/17)*(T - 1000), 100 <= T <= 1000}}]; Γ[p_, T_] = Subscript[C, α][T]*10^-11*p*T^4 // PiecewiseExpand; Δ[p_] := (Subscript[M, 1]^2/(2*p)) h[p_, T_] = (1/ 8) (Γ[p, T]*Δ[p]^2* Sin[2*θ]^2)/(Δ[ p]^2 + (Γ[p, T]/2)^2) // FullSimplify // PiecewiseExpand; fDW[p_, t_] = f[p*t/Tf, t] Integrate[h[p*t/1000, t], t] // PiecewiseExpand; x[a_, b_] := a/b EDIT: To eliminate complex artifacts (imaginary parts negligible compared to real part) use high precision and Chop[#, 10^-30]&. Also note that the iterators in the Table are done in the opposite order to group by T values . xf = Table[{x[p, T], x[p, T]^2*fDW[p, T]}, {T, 100, 500, 100}, {p, 1, 10^3}] // N[#, 20] & // Chop[#, 10^-30] &; plot1 = ListLogLogPlot[xf, Joined -> True, AxesLabel -> {x, Subscript[x.b2f, DW]}, Ticks -> {Automatic, Automatic}, PlotLabel -> Style["!(*SubscriptBox[(M), (1)])=7keV,[Theta].b2 = 7x10⁻¹¹"], PlotLegends -> LineLegend[Range[100, 500, 100], LegendLabel -> "T"]]

Integration shows RootSum and really long expressions

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