numerical integration - How to speed up plotting of Sophomore's Dream function

Let $$\textbf{Sphd}(\alpha;x)=\int_{0}^xt^{\alpha t}dt$$

I want make a graphic like this by using Mathematica

I tried:

Plot[Table[NIntegrate[t^(a t), {t, 0, x}], {a, -1, 10, 1}], {x, 1, 4}]

but it is very slow..

Answer

Reformulation as a ParametricNDSolve[] problem is very fast:

ff = 10;
pfun = ParametricNDSolveValue[{y'[t] == (t + $MachineEpsilon)^(a t), 
   y[0] == 1}, y, {t, 0, ff}, {a}]

LogPlot[Evaluate[
  Table[pfun[a][
    x], {a, {-50, -30, -20, -10, -5, -1, 0, 0.5, 1, 2, 3, 5, 
     10}}]], {x, 0, 10}, PlotRange -> {1, 10^10}]

Comments

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