How to quickly calculate the limit of integral with parameters

Question

I want to quickly calculate the limit value of $\lim _{x \rightarrow 0} \frac{\int_{0}^{x} t \ln (1+t \sin t) d t}{1-\cos x^{2}}$.

But using the code below I need to take 40 seconds to get the result:

Limit[Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}]/(1 - Cos[x^2]), 
 x -> 0]

I get an error message if I use the numerical method to solve:

Needs["NumericalCalculus`"]
NumericalCalculus`NLimit[
 Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}]/(1 - Cos[x^2]), x -> 0]

What should I do to get the correct limit value quickly?

Answer 1

$Version

(* "12.1.1 for Mac OS X x86 (64-bit) (June 19, 2020)" *)

Clear["Global`*"]

f[x_?NumericQ] := 
  NIntegrate[t*Log[1 + t*Sin[t]], {t, 0, x}, 
    WorkingPrecision -> 25]/(1 - Cos[x^2]);

Plot[f[x], {x, 10^-9, 1/100},
  PlotPoints -> 100,
  MaxRecursion -> 5,
  PlotRange -> All,
  WorkingPrecision -> 25] // Quiet

The order of the next two operations affects the relative times

Series[Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}]/(1 - Cos[x^2]), {x, 0, 1}] //
   Normal // AbsoluteTiming

(* {24.254, 1/2} *)

Limit[Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}]/(1 - Cos[x^2]), 
  x -> 0] // AbsoluteTiming

(* {54.4468, 1/2} *)

Starting with a fresh kernel and reversing the order

Clear["Global`*"]

Limit[Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}]/(1 - Cos[x^2]), 
  x -> 0] // AbsoluteTiming

(* {72.6784, 1/2} *)

Series[Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}]/(1 - Cos[x^2]), {x, 0, 1}] //
   Normal // AbsoluteTiming

(* {5.99999, 1/2} *)

Answer 2

L'Hôpital's rule needs ~10% of the calculation time

Normal[Series[Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}]  , {x, 0, 4}]]/Normal[Series[(1 - Cos[x^2])  , {x, 0, 4}]]
(* 1/2*)