diff --git a/funsor/integrate.py b/funsor/integrate.py
index ab6f98ad..2be76294 100644
--- a/funsor/integrate.py
+++ b/funsor/integrate.py
@@ -177,13 +177,23 @@ def eager_integrate(delta, integrand, reduced_vars):
     if reduced_vars.isdisjoint(delta_fresh):
         return None
     reduced_names = frozenset(v.name for v in reduced_vars)
+    new_integrand = integrand
+    new_log_measure = delta
+
+    # reduced_vars that are in integrand.inputs are substituted in delta and integrand
     subs = tuple(
         (name, point)
         for name, (point, log_density) in delta.terms
-        if name in reduced_names
+        if name in reduced_names and name in integrand.inputs
     )
-    new_integrand = Subs(integrand, subs)
-    new_log_measure = Subs(delta, subs)
+    if subs:
+        new_integrand = Subs(new_integrand, subs)
+        new_log_measure = Subs(new_log_measure, subs)
+    # reduced vars that are not in integrand.inputs are reduced over in delta
+    reduced_names = reduced_names.difference(integrand.inputs)
+    if reduced_names:
+        new_log_measure = new_log_measure.reduce(ops.logaddexp, reduced_names)
+
     result = Integrate(new_log_measure, new_integrand, reduced_vars - delta_fresh)
     return result