Oh dear! In FriCAS 1.3.9 try:

radicalSolve(64*z^4 + 64*z^3 + 32*z^2 - 8*z + 1)

and then try:

ii := rootOf(ii^2+1)
factor(64*z^4 + 64*z^3 + 32*z^2 - 8*z + 1, [ii])

to see what is wrong with radicalSolve(). (The integer quartic splits
into gaussian-integer quadratics.)

Also, unparse() in FriCAS refuses to convert the result from
radicalSolve() of the type List(Equation(Expression(Integer))) back
to InputForm. Why shouldn't this be be allowed?!? (The prettyprinted
ASCII monster is appended below.)

Derive 6.10 factors the integer polynomial right away as:

FACTOR(64*z^4 + 64*z^3 + 32*z^2 - 8*z + 1, Complex)
= 64*(z - SQRT(2)/4 + 1/4 + #i*(1/4 - SQRT(2)/4))
*(z + SQRT(2)/4 + 1/4 + #i*(SQRT(2)/4 + 1/4))
*(z + SQRT(2)/4 + 1/4 - #i*(SQRT(2)/4 + 1/4))
*(z - SQRT(2)/4 + 1/4 + #i*(SQRT(2)/4 - 1/4))

This polynomial arises in integral 5.66 (#401) from the Timofeev test
suite:

INT(TAN(x)/(SQRT(TAN(x)) - 1)^2, x)
= - x/2 + 1/(1 - SQRT(TAN(x))) + 1/2*LN(COS(x))
+ LN(1 - SQRT(TAN(x)))
+ 1/SQRT(2)*LN((1 + TAN(x) + SQRT(2)*SQRT(TAN(x)))/SEC(x))
- 1/SQRT(2)*ATAN(1 + SQRT(2)*SQRT(TAN(x)))
+ 1/SQRT(2)*ATAN(1 - SQRT(2)*SQRT(TAN(x)))
= - x/2 + 1/(1 - SQRT(TAN(x))) + 1/2*LN(COS(x))
+ LN(1 - SQRT(TAN(x)))
+ 1/SQRT(2)*ATANH((1 + TAN(x))/(SQRT(2)*SQRT(TAN(x))))
- 1/SQRT(2)*ATAN(1 + SQRT(2)*SQRT(TAN(x)))
+ 1/SQRT(2)*ATAN(1 - SQRT(2)*SQRT(TAN(x)))

which was discussed in the <sci.math.symbolic> thread
"Boooooooooooooooooooooooooo" from April to June 2016. Note that the
one and only radical extension needed to express the antiderivative is
SQRT(2).

No doubt, radicalSolve() in FriCAS needs to be overhauled.

Martin.

PS: The result from radicalSolve(64*z^4 + 64*z^3 + 32*z^2 - 8*z + 1) in
FriCAS 1.3.9 is:

[
z
=
-
+--+
\|13
*
ROOT
+------------------+2
+-+ +---+ 3| +---+ +-+
(68 \|3 - 18 \|- 1 )\|18 \|- 1 \|3 + 35
+
+------------------+
+-+ +---+ 3| +---+
+-+ +-+
(208 \|3 - 234 \|- 1 )\|18 \|- 1 \|3 + 35 -
676 \|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
+------------------+2
+---+ +-+ 3| +---+ +-+
(702 \|- 1 \|3 - 1365)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
- 6591 \|18 \|- 1 \|3 + 35 - 13182
+
+-+
13 \|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
- 507
/
2028
,

z
=
+--+
\|13
*
ROOT
+------------------+2
+-+ +---+ 3| +---+ +-+
(68 \|3 - 18 \|- 1 )\|18 \|- 1 \|3 + 35
+
+------------------+
+-+ +---+ 3| +---+
+-+ +-+
(208 \|3 - 234 \|- 1 )\|18 \|- 1 \|3 + 35 - 676
\|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
+------------------+2
+---+ +-+ 3| +---+ +-+
(702 \|- 1 \|3 - 1365)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
- 6591 \|18 \|- 1 \|3 + 35 - 13182
+
+-+
13 \|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
- 507
/
2028
,

z
=
-
+--+
\|13
*
ROOT
+------------------+2
+-+ +---+ 3| +---+ +-+
(- 68 \|3 + 18 \|- 1 )\|18 \|- 1 \|3 + 35
+
+------------------+
+-+ +---+ 3| +---+
+-+ +-+
(- 208 \|3 + 234 \|- 1 )\|18 \|- 1 \|3 + 35 +
676 \|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
+------------------+2
+---+ +-+ 3| +---+ +-+
(702 \|- 1 \|3 - 1365)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
- 6591 \|18 \|- 1 \|3 + 35 - 13182
+
-
+-+
13 \|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
- 507
/
2028
,

z
=
+--+
\|13
*
ROOT
+------------------+2
+-+ +---+ 3| +---+ +-+
(- 68 \|3 + 18 \|- 1 )\|18 \|- 1 \|3 + 35
+
+------------------+
+-+ +---+ 3| +---+
+-+ +-+
(- 208 \|3 + 234 \|- 1 )\|18 \|- 1 \|3 + 35 +
676 \|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
+------------------+2
+---+ +-+ 3| +---+ +-+
(702 \|- 1 \|3 - 1365)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
- 6591 \|18 \|- 1 \|3 + 35 - 13182
+
-
+-+
13 \|3
*
ROOT
+------------------+2
+---+ +-+ 3| +---+ +-+
(- 18 \|- 1 \|3 + 35)\|18 \|- 1 \|3 + 35
+
+------------------+
3| +---+ +-+
169 \|18 \|- 1 \|3 + 35 - 169
+
- 507
/
2028
]

Type: List(Equation(Expression(Integer)))