WIP #1064 implement for determining default numerical integral method

- If the expression has `Abs` function, use the `LegendreGauss` method

  Example input: `Integrate[Abs(x^2-2x), {x, -10, 10}] // N`
  Result should be `669.3282335875249`

- If the expression contains a variable that occurs in the exponent of
`Power` function, use the `GaussKronrod` method
  Example input: `x^x`, `3^(2x)`, `E^(-Sin(t))`

- Otherwise, use the `Romberg` method, since it is the optimized version
of trapezoid and Simpson methods

symja_android_library/matheclipse-core/src/main/java/org/matheclipse/core/reflection/system/NIntegrate.java

@@ -229,7 +229,7 @@ public NIntegrate() {
    */
   @Override
   public IExpr evaluate(final IAST ast, EvalEngine engine) {
-    String method = "LegendreGauss";
+    String method = "";
     int maxPoints = DEFAULT_MAX_POINTS;
     int maxIterations = DEFAULT_MAX_ITERATIONS;
     int precisionGoal = 16; // automatic scale value
@@ -270,13 +270,23 @@ public IExpr evaluate(final IAST ast, EvalEngine engine) {
       IAST list = (IAST) ast.arg2();
       IExpr function = ast.arg1();
       if (list.isAST3() && list.arg1().isSymbol()) {
+        IExpr x = list.arg1();
         double min = list.arg2().evalf();
         double max = list.arg3().evalf();
-        // if (min != null && max != null) {
         if (function.isEqual()) {
           IAST equalAST = (IAST) function;
           function = F.Plus(equalAST.arg1(), F.Negate(equalAST.arg2()));
         }
+        if (method.isEmpty()) {
+          method = "Romberg";
+          if (list.arg2().isDirectedInfinity() || list.arg3().isDirectedInfinity()) {
+            method = "LegendreGauss";
+          } else if (!function.isFree(a -> a == S.Abs || a == S.RealAbs, true)) {
+            method = "LegendreGauss";
+          } else if (!function.isFree(a -> a.isPower() && a.exponent().isFree(x), false)) {
+            method = "GaussKronrod";
+          }
+        }
         try {
           double result = integrate(method, list, min, max, function, maxPoints, maxIterations);
           result = Precision.round(result, precisionGoal);

symja_android_library/matheclipse-core/src/test/java/org/matheclipse/core/system/IntegrateTest.java

@@ -262,10 +262,10 @@ public void testIntegrate() {
     // see github #120
     check("Integrate(Ln(x)^2, {x,0,2})", //
         "4-4*Log(2)+2*Log(2)^2");
-    check("Integrate(Ln(x)^2, {x,0,2}) // N", //
-        "2.18832");
-    check("NIntegrate(Ln(x)^2, {x,0,2}) // N", //
-        "2.1857");
+    checkNumeric("Integrate(Ln(x)^2, {x,0,2}) // N", //
+        "2.1883173055966214");
+    checkNumeric("NIntegrate(Ln(x)^2, {x,0,2}) // N", //
+        "2.188317305596199");
 
     // see github #116
     // should give (2*ArcTan((1 + 2*x)/Sqrt(3)))/Sqrt(3)
@@ -390,16 +390,21 @@ public void testNIntegrate() {
     checkNumeric("NIntegrate(Cos(x), {x, 0, Pi})", //
         "1.0E-16");
     checkNumeric("NIntegrate(1/Sin(Sqrt(x)), {x, 0, 1}, PrecisionGoal->10)", //
-        "2.1108620052");
+        "2.1195255867");
   }
 
   @Test
   public void testSystem171() {
-    check("N(Integrate(Sin(x),x))", "-Cos(x)");
-    check("N(Sin(x))", "Sin(x)");
-    check("Cancel((1*x+1/2*2)^((-1)*2)*1^(-1)^(-1))", "1/(1+x)^2");
-    check("Integrate(1/(a+b*x),x)", "Log(a+b*x)/b");
-    check("Integrate((a+b*x)^(1/3),x)", "3/4*(a+b*x)^(4/3)/b");
+    check("N(Integrate(Sin(x),x))", //
+        "-Cos(x)");
+    check("N(Sin(x))", //
+        "Sin(x)");
+    check("Cancel((1*x+1/2*2)^((-1)*2)*1^(-1)^(-1))", //
+        "1/(1+x)^2");
+    check("Integrate(1/(a+b*x),x)", //
+        "Log(a+b*x)/b");
+    check("Integrate((a+b*x)^(1/3),x)", //
+        "3/4*(a+b*x)^(4/3)/b");
   }
 
   @Test
@@ -610,11 +615,15 @@ public void testXReciprocalIssue1064() {
     check("NIntegrate(1/x, {x,0,1},Method->\"LegendreGauss\")", //
         "10.37476");
     check("N(Integrate(1/x, {x,0,1}))", //
-        "10.37476");
+        "Integrate(1/x,{x,0,1})");
 
     // message - Integrate: Integral of 1/x does not converge on {x,0,1}.
     check("Integrate(1/x, {x,0,1})", //
         "Integrate(1/x,{x,0,1})");
+
+    checkNumeric("Integrate(Abs(x^2-2*x), {x, -10, 10}) // N", //
+        "669.3282335875249");
+
   }
 
 }