Enhance trigonometric function tests to verify mathematical correctness

Updated tests to verify that integrals produce mathematically correct results:
- ∫sin(x)dx derivative should equal sin(x)
- ∫cos(x)dx derivative should equal cos(x)
- ∫tan(x)dx derivative should equal tan(x)

This ensures the Risch method not only runs without errors but produces
the correct mathematical results that calculus expects.

Author: ChrisRackauckas

test/methods/risch/test_trigonometric_functions.jl

@@ -12,7 +12,7 @@ using Symbolics
         @test_nowarn integrate(sin(x), x, RischMethod())
         @test_nowarn integrate(cos(x), x, RischMethod()) 
 
-        # Test that the result is computed (not just that it doesn't error)
+        # Test that the results are mathematically correct
         sin_result = integrate(sin(x), x, RischMethod())
         cos_result = integrate(cos(x), x, RischMethod())
 
@@ -21,5 +21,23 @@ using Symbolics
         @test cos_result !== nothing
         @test sin_result isa Union{Number, SymbolicUtils.BasicSymbolic}
         @test cos_result isa Union{Number, SymbolicUtils.BasicSymbolic}
+        
+        # Test mathematical correctness by verifying derivatives
+        # ∫sin(x)dx should give -cos(x) (up to constants), so d/dx of result should be sin(x)
+        @test isequal(Symbolics.derivative(sin_result, x), sin(x))
+        
+        # ∫cos(x)dx should give sin(x) (up to constants), so d/dx of result should be cos(x)  
+        @test isequal(Symbolics.derivative(cos_result, x), cos(x))
+    end
+    
+    @testset "Additional trigonometric functions" begin
+        # Test more trigonometric functions if the basic ones work
+        @test_nowarn integrate(tan(x), x, RischMethod())
+        
+        tan_result = integrate(tan(x), x, RischMethod())
+        @test tan_result !== nothing
+        
+        # Verify derivative: d/dx[∫tan(x)dx] = tan(x)
+        @test isequal(Symbolics.derivative(tan_result, x), tan(x))
     end
 end