WIP #1064 Added error message for not converging integral

- idiv - Integral of `1` does not converge on `2`.
- JUnit test for this issue `testXReciprocalIssue1064()`
- `NIntegrate` with `LegendreGauss` method was called as default, other
`NIntegrate` methods don't return a result

symja_android_library/matheclipse-core/src/main/java/org/matheclipse/core/eval/Errors.java

@@ -132,6 +132,7 @@ public static void initGeneralMessages() {
       "herm", "The matrix `1` is not hermitian or real and symmetric.", //
       "idim", "`1` and `2` must have the same length.", //
       "idir", "Direction vector `1` has zero magnitude.", //
+      "idiv", "Integral of `1` does not converge on `2`.", //
       "ifun", "Inverse functions are being used. Values may be lost for multivalued inverses.", //
       "ilsmn",
       "Single or list of non-negative machine-sized integers expected at position `1` of `2`.", //

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

@@ -308,7 +308,7 @@ public IExpr evaluate(IAST holdallAST, final int argSize, final IExpr[] option,
           IAST copy = holdallAST.setAtCopy(2, xList.arg1());
           IExpr temp = engine.evaluate(copy);
           if (temp.isFreeAST(S.Integrate)) {
-            return definiteIntegral(temp, xList, engine);
+            return definiteIntegral(temp, xList, holdallAST, engine);
           }
           return F.NIL;
         }
@@ -633,7 +633,8 @@ private static IExpr integrateAbs(IExpr function, final IExpr x) {
    * @param engine the evaluation engine
    * @return
    */
-  private static IExpr definiteIntegral(IExpr function, IAST xValueList, EvalEngine engine) {
+  private static IExpr definiteIntegral(IExpr function, IAST xValueList, IAST originalAST,
+      EvalEngine engine) {
     // see Rubi rule for definite integrals
     IExpr x = xValueList.arg1();
     IExpr lower = xValueList.arg2();
@@ -653,11 +654,21 @@ private static IExpr definiteIntegral(IExpr function, IAST xValueList, EvalEngin
     }
     IExpr lowerLimit = engine.evaluate(F.Limit(function, F.Rule(x, lower), lowerDirection));
     if (!lowerLimit.isFree(S.DirectedInfinity, true) || !lowerLimit.isFree(S.Indeterminate, true)) {
+      if (lowerLimit.isDirectedInfinity() || lowerLimit.isIndeterminate()) {
+        // Integral of `1` does not converge on `2`.
+        return Errors.printMessage(S.Integrate, "idiv", F.List(originalAST.arg1(), xValueList),
+            engine);
+      }
       LOGGER.log(engine.getLogLevel(), "Not integrable: {} for limit {} -> {}", function, x, lower);
       return F.NIL;
     }
     IExpr upperLimit = engine.evaluate(F.Limit(function, F.Rule(x, upper), upperDirection));
     if (!upperLimit.isFree(S.DirectedInfinity, true) || !upperLimit.isFree(S.Indeterminate, true)) {
+      if (upperLimit.isDirectedInfinity() || upperLimit.isIndeterminate()) {
+        // Integral of `1` does not converge on `2`.
+        return Errors.printMessage(S.Integrate, "idiv", F.List(originalAST.arg1(), xValueList),
+            engine);
+      }
       LOGGER.log(engine.getLogLevel(), "Not integrable: {} for limit {} -> {}", function, x, upper);
       return F.NIL;
     }

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

@@ -27,7 +27,6 @@
 import org.matheclipse.core.generic.UnaryNumerical;
 import org.matheclipse.core.interfaces.IAST;
 import org.matheclipse.core.interfaces.IExpr;
-import org.matheclipse.core.interfaces.IReal;
 import org.matheclipse.core.interfaces.ISymbol;
 import org.matheclipse.core.numerics.integral.ClenshawCurtis;
 import org.matheclipse.core.numerics.integral.GaussLobatto;
@@ -109,8 +108,6 @@ public class NIntegrate extends AbstractFunctionEvaluator {
    */
   public static double integrate(String method, IAST list, final double min, final double max,
       IExpr function, int maxPoints, int maxIterations) throws MathIllegalStateException {
-    GaussIntegratorFactory factory = new GaussIntegratorFactory();
-
     if (!list.arg1().isSymbol()) {
       // `1` is not a valid variable.
       String str = Errors.getMessage("ivar", F.list(list.arg1()), EvalEngine.get());
@@ -173,7 +170,9 @@ public static double integrate(String method, IAST list, final double min, final
       if (min == Double.NEGATIVE_INFINITY || max == Double.POSITIVE_INFINITY) {
         return gausKronrodRule(maxIterations, f, min, max);
       }
+
       // default: LegendreGauss
+      GaussIntegratorFactory factory = new GaussIntegratorFactory();
       GaussIntegrator integ = factory.legendre(maxPoints, min, max);
       return integ.integrate(f);
     }
@@ -242,7 +241,7 @@ public IExpr evaluate(final IAST ast, EvalEngine engine) {
       }
       option = options.getOption(S.MaxPoints);
       if (option.isReal()) {
-        maxPoints = ((IReal) option).toIntDefault(-1);
+        maxPoints = option.toIntDefault(-1);
         if (maxPoints <= 0) {
           // Inappropriate parameter: `1`.
           return Errors.printMessage(ast.topHead(), "par", F.List(S.MaxPoints), engine);
@@ -258,7 +257,7 @@ public IExpr evaluate(final IAST ast, EvalEngine engine) {
       }
       option = options.getOption(S.PrecisionGoal);
       if (option.isReal()) {
-        precisionGoal = ((IReal) option).toIntDefault(-1);
+        precisionGoal = option.toIntDefault(-1);
         if (precisionGoal <= 0) {
           // Inappropriate parameter: `1`.
           return Errors.printMessage(ast.topHead(), "par", F.List(S.PrecisionGoal), engine);

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

@@ -602,4 +602,19 @@ public void testN_Integrate() {
         "6.36777");
   }
 
+  @Test
+  public void testXReciprocalIssue1064() {
+    // message - NIntegrate: maximal count (10,000) exceeded.
+    check("NIntegrate(1/x, {x,0,1},Method->\"GaussKronrod\")", //
+        "NIntegrate(1/x,{x,0,1},Method->GaussKronrod)");
+    check("NIntegrate(1/x, {x,0,1},Method->\"LegendreGauss\")", //
+        "10.37476");
+    check("N(Integrate(1/x, {x,0,1}))", //
+        "10.37476");
+
+    // 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})");
+  }
+
 }