For the first part of your question, it seems that there is
a very basic mistake of definition between
int(cos(t^2),t) and int(cos(%pi*t^2/2),t)
diff --git a/src/algebra/limitps.spad b/src/algebra/limitps.spad
index 02a85a8e..1ee8682a 100644
--- a/src/algebra/limitps.spad
+++ b/src/algebra/limitps.spad
@@ -298,8 +298,8 @@
is?(ker, 'acsc) => 0
is?(ker, 'acsch) => 0
is?(ker, 'acoth) => 0
- is?(ker, 'fresnelS) => sqrt(pi()/(8 :: FE))
- is?(ker, 'fresnelC) => sqrt(pi()/(8 :: FE))
+ is?(ker, 'fresnelS) => 1/(2 :: FE)
+ is?(ker, 'fresnelC) => 1/(2 :: FE)
is?(ker, 'Si) => pi()/(2 :: FE)
is?(ker, 'Ci) => 0
is?(ker, 'airyAi) => 0
@@ -323,8 +323,8 @@
is?(ker, 'acsc) => -pi()
is?(ker, 'acsch) => 0
is?(ker, 'acoth) => 0
- is?(ker, 'fresnelS) => -sqrt(pi()/(8 :: FE))
- is?(ker, 'fresnelC) => -sqrt(pi()/(8 :: FE))
+ is?(ker, 'fresnelS) => -1/(2 :: FE)
+ is?(ker, 'fresnelC) => -1/(2 :: FE)
is?(ker, 'Si) => -pi()/(2 :: FE)
is?(ker, 'airyAi) => 0
is?(ker, 'airyBi) => 0
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