What is the value of this integral?

Question

Let us consider the definite integral $$\int\limits_0^1\frac{1}{-\sin \left(\frac{355 x}{113}\right)-\sin (\pi x)+2}\,dx.$$

The integrand has the sharp maximum at approximately x==1/2 as

Plot[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 0, 1},  PlotRange -> {0, 1000}]

and

1/(2 - Sin[Pi*x] - Sin[355/113*x]) /. x -> 0.5

1.1259*10^14

show. Let us calculate its numerical value by

NIntegrate[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 0, 1}, WorkingPrecision -> 500]

1.2362169441996136851883195633922114040481857268750706513973706856485258255036921164094489402293477582881130975970830174783445922472199291683034580223409700064733593412912404064803381979555313504509727777905495203840976429627566636757997635903087515772977188679568523976731890867113508843031431576691732124347894884567712227795096153264965184100880948635131023672835134263978050533965702731340161896810515594850298388648619221758337473963896532597731360204009495075235122659383639365026092262109738888*10^7

and

NIntegrate[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 0, 1}, WorkingPrecision -> 1000]

1.001238555284578042264617144192400882420859884217313673973393029717575862379157756534310218994101070236160802301957339871913913756328227472506781430533481037902156433981036539993869651405529876630723938976081293563905374365419140155264038994215106030388424098155233913474224499823849830641075764497571015128909503298524419408682320418929661264138488676605033434972671072242011026479745573132128871944031984928168529268107459810009286814370366477980818868498042156604771607758181001778158976146808678042800024409528816766163438147284276803597703017272639164188183298370503946710095132978382457347421323244213392134426628339851044899564836060073374226540743034549243746615692589205033957730838390841727891835162993345859514680681840739765635659862420614154264942957376804156215178774043250223996004039988875402248125243337150092116281087960046032952525230784220279493253698303801199053339187333072097391177874201614455353679697657089606621124448308475019101617295155680442626137130223538965822513429241*10^7

As we see, the results differ. Up to Maple, this value equals 1.499451605234141071490295*10^7.

The question arises: what is the true value of this integral?

Answer 1

You can get a result close to Maple's using the method "LocalAdaptive":

NIntegrate[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 0, 1}, 
  MinRecursion -> 3, MaxRecursion -> 100, Method -> "LocalAdaptive", 
  PrecisionGoal -> 10] // InputForm

(* 1.4994515633482287*^7 *)

---

I have not looked into Maple's numerical integration algorithms for more than 15 years, but I remember that at the time their default (and only) integration strategy was similar to "LocalAdaptive".

---

Using "GlobalAdaptive" with larger recursion start- and end values:

NIntegrate[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 0, 1}, 
 MinRecursion -> 6, MaxRecursion -> 100, 
 Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 10^8}, 
 PrecisionGoal -> 10, WorkingPrecision -> 30]

(* 1.49945160523414228214390320871*10^7 *)

---

NIntegrate allows custom breakdown of the integration ranges.

For example, let us try to "isolate" the max point:

Reduce[{-((-(355/113) Cos[(355 x)/
       113] - \[Pi] Cos[\[Pi] x])/(2 - Sin[(355 x)/113] - 
      Sin[\[Pi] x])^2) == 0, 0 <= x <= 1}, x]

(* x == Root[{
  355 Cos[Rational[355, 113] #] + 113 Pi Cos[Pi #]& , 
   0.49999997877158033540739364320327348112`20.331736988545668}] *)

N[ToRules[%], 100]

(* {x -> 0.4999999787715803354073608859722141098496639023446210807132737788751325902685677891429692202993799552} *)

Block[{c = 10^(-7)},
 Plot[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 1/2 - c, 1/2 + c}, 
  PlotRange -> All, MaxRecursion -> 15]
 ]

Then we can integrate like this:

Block[{c = 10^(-7)},
 NIntegrate[
  1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 0, 1/2 - c, 1/2 + c, 1}, 
  MaxRecursion -> 100, WorkingPrecision -> 30, PrecisionGoal -> 10]
 ]

(* 1.49945160523414228208295107976*10^7 *)

Answer 2

You may help MMA a bit by splitting the integration range into 3 pieces. The first one the easy piece from 0 up to near the peak, the second over the peak, where you also specify "Method->"LocalAdaptive"". And the third from after the peak up to 1

NIntegrate[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 0, 1/2 - 0.001}] + 
 NIntegrate[
  1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 1/2 - 0.001, 1/2 + 0.001}, 
  Method -> "LocalAdaptive"] +
 NIntegrate[1/(2 - Sin[Pi*x] - Sin[355/113*x]), {x, 1/2 + 0.001, 1}]

1.49888*10^7