Document default integration method behavior in docs

Add clear explanation that integrate() tries Risch first, then falls
back to RuleBasedMethod. This matches the README but was missing from
the main documentation page.

Changes:
- Added "Default Behavior" paragraph explaining Risch-first strategy
- Added example showing automatic fallback from Risch to Rule based
- Added examples of explicit method selection

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <[email protected]>

Author: ChrisRackauckas

docs/src/index.md

@@ -39,7 +39,11 @@ integrate(f, x, RischMethod(use_algebraic_closure=true))  # With options
 
 ## Integration Methods
 
-SymbolicIntegration.jl provides two integration algorithms: Rule based and Risch method. Here is a quick table to see what they can integrate:
+SymbolicIntegration.jl provides two integration algorithms: Risch method and Rule based method.
+
+**Default Behavior:** When no method is explicitly specified, `integrate()` will first attempt the **Risch method**. If the Risch method fails (e.g., due to unsupported expressions like `sqrt(x)` or `abs(x)`), it will automatically fall back to the **Rule based method**. This ensures maximum coverage while prioritizing the theoretically complete Risch algorithm when applicable.
+
+Here is a quick table to see what each method can integrate:
 
 feature | Risch | Rule based
 --------|-------|-----------
@@ -55,6 +59,29 @@ multiple symbols | ❌ | ✅
 
 [→ See complete methods documentation](methods/overview.md)
 
+### Example: Automatic Fallback Behavior
+
+When no method is specified, the integration will try Risch first, then fall back to Rule based if needed:
+
+```julia
+# This uses sqrt which is not supported by Risch, so it falls back to RuleBasedMethod
+integrate(sqrt(x))
+# ┌ Warning: NotImplementedError: integrand contains unsupported expression sqrt(x)
+# └ @ SymbolicIntegration
+#
+#  > RischMethod failed returning ∫(sqrt(x), x)
+#  > Trying with RuleBasedMethod...
+#
+# (2//3)*(x^(3//2))
+```
+
+You can also explicitly specify which method to use:
+
+```julia
+integrate(sqrt(x), x, RuleBasedMethod())  # Skip Risch, use Rule based directly
+integrate(x^2 + 1, x, RischMethod())      # Use only Risch
+```
+
 ### RischMethod
 This method is based on the algorithms from the book: