Does not work because Integrate can't you give a Antiderivative.
Integrate[Sqrt[2*m*l^2/(H + m*g*l*Cos[p[t]])], p[t]]
(* No way *)
We can solve like this:
ClearAll["`*"]; Remove["`*"];
l = 1;
b = 0;
m = 1;
g = 9.81;
ps = 0;
dps = 1;
H = 1/2*m*l^2*dps^2 - m*g*l*Cos[ps];
EQ = D[b + t ==
1/2*Integrate[Sqrt[2*m*l^2/(H + m*g*l*Cos[p[t]])], p[t]], t]
s = DSolve[{EQ, p[0] == 0}, p[t], {t, 0, 100}] //
Quiet(*I assume a Initial condition !!! *)
Plot[p[t] /. s, {t, 0, 100}, WorkingPrecision -> 20,
PlotPoints -> 100]
Plot[p[t] /. s, {t, 0, 2}, PlotLabel -> "Analytically"]
Plot[p[t] /. s, {t, 0, 1/2}, PlotLabel -> "Analytically"]
s1 = NDSolve[{EQ, p[0] == 0}, p, {t, 0, 1/2},
Method ->
"LinearlyImplicitEuler"];(*I assume a Initial condition !!! *)
Plot[(p[t] /. s1), {t, 0, 1/2}, PlotLabel -> "Numerically"]