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SandBoxSymbolicInverseTrig

Edit detail for SandBoxSymbolicInverseTrig revision 1 of 2

	1 2
Editor: Bill Page Time: 2009/10/21 18:31:21 GMT-7
Note: clean up

changed:
-
How does Axiom compute inverse trigonometric functions symbolically?

\begin{axiom}
v:EXPR INT:= x+a * y
t:=asec v
simplify t
s:=atan(sqrt(v^2-1))
(s=t)::Boolean
\end{axiom}

\begin{axiom}
subst(t,a=%i)
w:COMPLEX EXPR INT:=x+%i*y
\end{axiom}

The following command generates a huge expression that is
too large for MathAction to handle properly::

  !\begin{axiom}
  asec(w)
  \end{axiom}


\begin{axiom}
)set output algebra on
)set output tex off
\end{axiom}

\begin{axiom}
asec(w)
\end{axiom}

\begin{axiom}
asec( %i )
\end{axiom}

\begin{axiom}
simplify asec( %i )
\end{axiom}

I guess this one also causes a problem with MathAction::

  !\begin{axiom}
  simplify(sec(asec(w)))
  \end{axiom}

Not coherent ?

\begin{axiom}
)set output algebra off
)set output tex on
\end{axiom}

\begin{axiom}
[atan(tan(x)), atan(tan(%pi))]
\end{axiom}

\begin{reduce}
atan(tan(x));
\end{reduce}

How does Axiom compute inverse trigonometric functions symbolically?

fricas

v:EXPR INT:= x+a * y

$\label{eq1}{a \ y}+ x$

(1)

Type: Expression(Integer)

fricas

t:=asec v

$\label{eq2}asec \left({{a \ y}+ x}\right)$

(2)

Type: Expression(Integer)

fricas

simplify t

$\label{eq3}asec \left({{a \ y}+ x}\right)$

(3)

Type: Expression(Integer)

fricas

s:=atan(sqrt(v^2-1))

$\label{eq4}\arctan \left({\sqrt{{{{a}^{2}}\ {{y}^{2}}}+{2 \ a \ x \ y}+{{x}^{2}}- 1}}\right)$

(4)

Type: Expression(Integer)

fricas

(s=t)::Boolean

$\label{eq5} \mbox{\rm false}$

(5)

Type: Boolean

fricas

subst(t,a=%i)

$\label{eq6}asec \left({{i \ y}+ x}\right)$

(6)

Type: Expression(Complex(Integer))

fricas

w:COMPLEX EXPR INT:=x+%i*y

$\label{eq7}x +{y \ i}$

(7)

Type: Complex(Expression(Integer))

The following command generates a huge expression that is too large for MathAction? to handle properly:

  \begin{axiom}
  asec(w)
  \end{axiom}

fricas

)set output algebra on

fricas

)set output tex off

fricas

asec(w)

   (8)
     -
          2
       *
          atan
                                        +-------------------------------+
                                        | 4      2      2    4     2
                       5     2 3    4   |y  + (2x  + 2)y  + x  - 2x  + 1     5
                     (y  + 2x y  + x y) |-------------------------------  + y
                                        |         4     2 2    4
                                       \|        y  + 2x y  + x
                   + 
                        2      3     4    2
                     (2x  + 1)y  + (x  - x )y
                *
                   ROOT
                               8      2      6      4    2  4      6    4  2
                              y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                            + 
                               8    6
                              x  - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           8      2      6      4     2      4
                          y  + (4x  + 2)y  + (6x  + 2x  + 1)y
                        + 
                             6     4     2  2    8     6    4
                          (4x  - 2x  + 2x )y  + x  - 2x  + x
                     /
                               8      2      6      4    2  4      6    4  2
                              y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                            + 
                               8    6
                              x  - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           8      2      6      4     2      4      6     4  2
                          y  + (4x  + 2)y  + (6x  + 2x  + 1)y  + (4x  - 2x )y
                        + 
                           8     6    4
                          x  - 2x  + x
               + 
                           6     2 4     4 2    6
                       (- y  - 3x y  - 3x y  - x )
                    *
                        +-------------------------------+
                        | 4      2      2    4     2
                        |y  + (2x  + 2)y  + x  - 2x  + 1
                        |-------------------------------
                        |         4     2 2    4
                       \|        y  + 2x y  + x
                   + 
                        6        2      4     4 2    6    4
                     - y  + (- 3x  - 1)y  - 3x y  - x  + x
                *
                    +-------------------------------+
                    | 4      2      2    4     2
                    |y  + (2x  + 2)y  + x  - 2x  + 1
                    |-------------------------------
                   4|         4     2 2    4
                   \|        y  + 2x y  + x
            /
                                               +-------------------------------+
                                               | 4      2      2    4     2
                       6     2 4     4 2    6  |y  + (2x  + 2)y  + x  - 2x  + 1
                     (y  + 3x y  + 3x y  + x ) |-------------------------------
                                               |         4     2 2    4
                                              \|        y  + 2x y  + x
                   + 
                      6      2      4     4 2    6    4
                     y  + (3x  + 1)y  + 3x y  + x  - x
                *
                   ROOT
                               8      2      6      4    2  4      6    4  2
                              y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                            + 
                               8    6
                              x  - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           8      2      6      4     2      4
                          y  + (4x  + 2)y  + (6x  + 2x  + 1)y
                        + 
                             6     4     2  2    8     6    4
                          (4x  - 2x  + 2x )y  + x  - 2x  + x
                     /
                               8      2      6      4    2  4      6    4  2
                              y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                            + 
                               8    6
                              x  - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           8      2      6      4     2      4      6     4  2
                          y  + (4x  + 2)y  + (6x  + 2x  + 1)y  + (4x  - 2x )y
                        + 
                           8     6    4
                          x  - 2x  + x
                *
                   ROOT
                                      21         2       19
                                  - 2y   + (- 20x  - 10)y
                                + 
                                        4      2       17
                                  (- 90x  - 72x  - 20)y
                                + 
                                         6       4      2       15
                                  (- 240x  - 216x  - 98x  - 20)y
                                + 
                                         8       6       4      2       13
                                  (- 420x  - 336x  - 180x  - 62x  - 10)y
                                + 
                                           10       8       6      4      2
                                    (- 504x   - 252x  - 142x  - 62x  - 18x  - 2)
                                 *
                                     11
                                    y
                                + 
                                         12      8      6     4     2  9
                                  (- 420x   - 40x  - 20x  - 8x  - 2x )y
                                + 
                                         14       12      10  7
                                  (- 240x   + 168x   - 30x  )y
                                + 
                                        16       14      12      10  5
                                  (- 90x   + 144x   - 68x   + 10x  )y
                                + 
                                        18      16      14      12     10  3
                                  (- 20x   + 54x   - 50x   + 18x   - 2x  )y
                                + 
                                       20     18      16     14     12
                                  (- 2x   + 8x   - 12x   + 8x   - 2x  )y
                             *
                                 +-------------------------------+
                                 | 4      2      2    4     2
                                 |y  + (2x  + 2)y  + x  - 2x  + 1
                                 |-------------------------------
                                 |         4     2 2    4
                                \|        y  + 2x y  + x
                            + 
                                  21         2       19
                              - 2y   + (- 20x  - 12)y
                            + 
                                    4      2       17
                              (- 90x  - 86x  - 30)y
                            + 
                                     6       4       2       15
                              (- 240x  - 256x  - 144x  - 40)y
                            + 
                                     8       6       4       2       13
                              (- 420x  - 392x  - 254x  - 116x  - 30)y
                            + 
                                     10       8       6       4      2       11
                              (- 504x   - 280x  - 180x  - 100x  - 44x  - 12)y
                            + 
                                          12      10      8      6      4     2
                                    - 420x   + 28x   - 30x  - 20x  - 12x  - 6x
                                  + 
                                    - 2
                             *
                                 9
                                y
                            + 
                                     14       12      10  7
                              (- 240x   + 224x   - 40x  )y
                            + 
                                    16       14       12      10  5
                              (- 90x   + 184x   - 114x   + 20x  )y
                            + 
                                    18      16      14      12     10  3
                              (- 20x   + 68x   - 84x   + 44x   - 8x  )y
                            + 
                                   20      18      16      14      12     10
                              (- 2x   + 10x   - 20x   + 20x   - 10x   + 2x  )y
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                            4|         4     2 2    4
                            \|        y  + 2x y  + x
                         *
                            ROOT
                                        8      2      6      4    2  4
                                       y  + (4x  + 1)y  + (6x  + x )y
                                     + 
                                          6    4  2    8    6
                                       (4x  - x )y  + x  - x
                                  *
                                      +-------------------------------+
                                      | 4      2      2    4     2
                                      |y  + (2x  + 2)y  + x  - 2x  + 1
                                      |-------------------------------
                                      |         4     2 2    4
                                     \|        y  + 2x y  + x
                                 + 
                                    8      2      6      4     2      4
                                   y  + (4x  + 2)y  + (6x  + 2x  + 1)y
                                 + 
                                      6     4     2  2    8     6    4
                                   (4x  - 2x  + 2x )y  + x  - 2x  + x
                              /
                                        8      2      6      4    2  4
                                       y  + (4x  + 1)y  + (6x  + x )y
                                     + 
                                          6    4  2    8    6
                                       (4x  - x )y  + x  - x
                                  *
                                      +-------------------------------+
                                      | 4      2      2    4     2
                                      |y  + (2x  + 2)y  + x  - 2x  + 1
                                      |-------------------------------
                                      |         4     2 2    4
                                     \|        y  + 2x y  + x
                                 + 
                                    8      2      6      4     2      4
                                   y  + (4x  + 2)y  + (6x  + 2x  + 1)y
                                 + 
                                      6     4  2    8     6    4
                                   (4x  - 2x )y  + x  - 2x  + x
                        + 
                                   22       2      20       4      2       18
                                  y   + (11x  + 5)y   + (55x  + 40x  + 10)y
                                + 
                                       6       4      2       16
                                  (165x  + 135x  + 56x  + 10)y
                                + 
                                       8       6       4      2      14
                                  (330x  + 240x  + 122x  + 38x  + 5)y
                                + 
                                       10       8       6      4      2      12
                                  (462x   + 210x  + 126x  + 52x  + 13x  + 1)y
                                + 
                                       12      8      6      4     2  10
                                  (462x   + 70x  + 30x  + 11x  + 2x )y
                                + 
                                       14       12      10     6    4  8
                                  (330x   - 210x   + 70x   + 3x  + x )y
                                + 
                                       16       14       12      10     8  6
                                  (165x   - 240x   + 126x   - 30x   + 3x )y
                                + 
                                           18       16       14      12      10
                                        55x   - 135x   + 122x   - 52x   + 11x
                                      + 
                                           8
                                        - x
                                 *
                                     4
                                    y
                                + 
                                           20      18      16      14      12
                                        11x   - 40x   + 56x   - 38x   + 13x
                                      + 
                                            10
                                        - 2x
                                 *
                                     2
                                    y
                                + 
                                   22     20      18      16     14    12
                                  x   - 5x   + 10x   - 10x   + 5x   - x
                             *
                                 +-------------------------------+
                                 | 4      2      2    4     2
                                 |y  + (2x  + 2)y  + x  - 2x  + 1
                                 |-------------------------------
                                 |         4     2 2    4
                                \|        y  + 2x y  + x
                            + 
                               22       2      20       4      2       18
                              y   + (11x  + 6)y   + (55x  + 48x  + 15)y
                            + 
                                   6       4      2       16
                              (165x  + 162x  + 83x  + 20)y
                            + 
                                   8       6       4      2       14
                              (330x  + 288x  + 176x  + 72x  + 15)y
                            + 
                                   10       8       6      4      2      12
                              (462x   + 252x  + 168x  + 88x  + 33x  + 6)y
                            + 
                                   12      8      6      4     2      10
                              (462x   + 70x  + 40x  + 21x  + 8x  + 1)y
                            + 
                                   14       12      10     6     4    2  8
                              (330x   - 252x   + 70x   + 3x  + 2x  + x )y
                            + 
                                   16       14       12      10     8  6
                              (165x   - 288x   + 168x   - 40x   + 3x )y
                            + 
                                  18       16       14      12      10     8  4
                              (55x   - 162x   + 176x   - 88x   + 21x   - 2x )y
                            + 
                                     20      18      16      14      12     10
                                  11x   - 48x   + 83x   - 72x   + 33x   - 8x
                                + 
                                   8
                                  x
                             *
                                 2
                                y
                            + 
                               22     20      18      16      14     12    10
                              x   - 6x   + 15x   - 20x   + 15x   - 6x   + x
                         *
                             +-------------------------------+2
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                            4|         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                               20       2      18       4      2       16
                              y   + (10x  + 5)y   + (45x  + 35x  + 10)y
                            + 
                                   6       4      2       14
                              (120x  + 100x  + 46x  + 10)y
                            + 
                                   8       6      4      2      12
                              (210x  + 140x  + 76x  + 28x  + 5)y
                            + 
                                   10      8      6      4     2      10
                              (252x   + 70x  + 50x  + 24x  + 8x  + 1)y
                            + 
                                   12      10      8     6     4    2  8
                              (210x   - 70x   + 20x  + 6x  + 3x  + x )y
                            + 
                                   14       12      10     8  6
                              (120x   - 140x   + 50x   - 6x )y
                            + 
                                  16       14      12      10     8  4
                              (45x   - 100x   + 76x   - 24x   + 3x )y
                            + 
                                  18      16      14      12     10    8  2
                              (10x   - 35x   + 46x   - 28x   + 8x   - x )y
                            + 
                               20     18      16      14     12    10
                              x   - 5x   + 10x   - 10x   + 5x   - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           20       2      18       4      2       16
                          y   + (10x  + 6)y   + (45x  + 42x  + 15)y
                        + 
                               6       4      2       14
                          (120x  + 120x  + 68x  + 20)y
                        + 
                               8       6       4      2       12
                          (210x  + 168x  + 108x  + 52x  + 15)y
                        + 
                               10      8      6      4      2      10
                          (252x   + 84x  + 60x  + 36x  + 18x  + 6)y
                        + 
                               12      10      8     6     4     2      8
                          (210x   - 84x   + 10x  + 4x  + 3x  + 2x  + 1)y
                        + 
                               14       12      10     8  6
                          (120x   - 168x   + 60x   - 4x )y
                        + 
                              16       14       12      10     8  4
                          (45x   - 120x   + 108x   - 36x   + 3x )y
                        + 
                              18      16      14      12      10     8  2    20
                          (10x   - 42x   + 68x   - 52x   + 18x   - 2x )y  + x
                        + 
                              18      16      14      12     10    8
                          - 6x   + 15x   - 20x   + 15x   - 6x   + x
                     /
                               22       2      20       4      2       18
                              y   + (11x  + 5)y   + (55x  + 40x  + 10)y
                            + 
                                   6       4      2       16
                              (165x  + 135x  + 56x  + 10)y
                            + 
                                   8       6       4      2      14
                              (330x  + 240x  + 122x  + 38x  + 5)y
                            + 
                                   10       8       6      4      2      12
                              (462x   + 210x  + 126x  + 52x  + 13x  + 1)y
                            + 
                                   12      8      6      4     2  10
                              (462x   + 70x  + 30x  + 11x  + 2x )y
                            + 
                                   14       12      10     6    4  8
                              (330x   - 210x   + 70x   + 3x  + x )y
                            + 
                                   16       14       12      10     8  6
                              (165x   - 240x   + 126x   - 30x   + 3x )y
                            + 
                                  18       16       14      12      10    8  4
                              (55x   - 135x   + 122x   - 52x   + 11x   - x )y
                            + 
                                  20      18      16      14      12     10  2
                              (11x   - 40x   + 56x   - 38x   + 13x   - 2x  )y
                            + 
                               22     20      18      16     14    12
                              x   - 5x   + 10x   - 10x   + 5x   - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           22       2      20       4      2       18
                          y   + (11x  + 6)y   + (55x  + 48x  + 15)y
                        + 
                               6       4      2       16
                          (165x  + 162x  + 83x  + 20)y
                        + 
                               8       6       4      2       14
                          (330x  + 288x  + 176x  + 72x  + 15)y
                        + 
                               10       8       6      4      2      12
                          (462x   + 252x  + 168x  + 88x  + 33x  + 6)y
                        + 
                               12      8      6      4     2      10
                          (462x   + 70x  + 40x  + 21x  + 8x  + 1)y
                        + 
                               14       12      10     6     4    2  8
                          (330x   - 252x   + 70x   + 3x  + 2x  + x )y
                        + 
                               16       14       12      10     8  6
                          (165x   - 288x   + 168x   - 40x   + 3x )y
                        + 
                              18       16       14      12      10     8  4
                          (55x   - 162x   + 176x   - 88x   + 21x   - 2x )y
                        + 
                              20      18      16      14      12     10    8  2
                          (11x   - 48x   + 83x   - 72x   + 33x   - 8x   + x )y
                        + 
                           22     20      18      16      14     12    10
                          x   - 6x   + 15x   - 20x   + 15x   - 6x   + x
               + 
                                         +-------------------------------+
                                         | 4      2      2    4     2
                         4     3 2    5  |y  + (2x  + 2)y  + x  - 2x  + 1
                     (x y  + 2x y  + x ) |-------------------------------
                                         |         4     2 2    4
                                        \|        y  + 2x y  + x
                   + 
                        4      3      2    5    3
                     x y  + (2x  + x)y  + x  - x
                *
                   ROOT
                               8      2      6      4    2  4      6    4  2
                              y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                            + 
                               8    6
                              x  - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           8      2      6      4     2      4
                          y  + (4x  + 2)y  + (6x  + 2x  + 1)y
                        + 
                             6     4     2  2    8     6    4
                          (4x  - 2x  + 2x )y  + x  - 2x  + x
                     /
                               8      2      6      4    2  4      6    4  2
                              y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                            + 
                               8    6
                              x  - x
                         *
                             +-------------------------------+
                             | 4      2      2    4     2
                             |y  + (2x  + 2)y  + x  - 2x  + 1
                             |-------------------------------
                             |         4     2 2    4
                            \|        y  + 2x y  + x
                        + 
                           8      2      6      4     2      4      6     4  2
                          y  + (4x  + 2)y  + (6x  + 2x  + 1)y  + (4x  - 2x )y
                        + 
                           8     6    4
                          x  - 2x  + x
               + 
                                   +-------------------------------+
                                   | 4      2      2    4     2
                        3     3    |y  + (2x  + 2)y  + x  - 2x  + 1
                 (- 2x y  - 2x y)  |-------------------------------
                                  4|         4     2 2    4
                                  \|        y  + 2x y  + x
   + 
     -
            log
                               21         2       19         4      2       17
                           - 2y   + (- 20x  - 10)y   + (- 90x  - 72x  - 20)y
                         + 
                                  6       4      2       15
                           (- 240x  - 216x  - 98x  - 20)y
                         + 
                                  8       6       4      2       13
                           (- 420x  - 336x  - 180x  - 62x  - 10)y
                         + 
                                  10       8       6      4      2      11
                           (- 504x   - 252x  - 142x  - 62x  - 18x  - 2)y
                         + 
                                  12      8      6     4     2  9
                           (- 420x   - 40x  - 20x  - 8x  - 2x )y
                         + 
                                  14       12      10  7
                           (- 240x   + 168x   - 30x  )y
                         + 
                                 16       14      12      10  5
                           (- 90x   + 144x   - 68x   + 10x  )y
                         + 
                                 18      16      14      12     10  3
                           (- 20x   + 54x   - 50x   + 18x   - 2x  )y
                         + 
                                20     18      16     14     12
                           (- 2x   + 8x   - 12x   + 8x   - 2x  )y
                      *
                          +-------------------------------+
                          | 4      2      2    4     2
                          |y  + (2x  + 2)y  + x  - 2x  + 1
                          |-------------------------------
                          |         4     2 2    4
                         \|        y  + 2x y  + x
                     + 
                           21         2       19         4      2       17
                       - 2y   + (- 20x  - 12)y   + (- 90x  - 86x  - 30)y
                     + 
                              6       4       2       15
                       (- 240x  - 256x  - 144x  - 40)y
                     + 
                              8       6       4       2       13
                       (- 420x  - 392x  - 254x  - 116x  - 30)y
                     + 
                              10       8       6       4      2       11
                       (- 504x   - 280x  - 180x  - 100x  - 44x  - 12)y
                     + 
                              12      10      8      6      4     2      9
                       (- 420x   + 28x   - 30x  - 20x  - 12x  - 6x  - 2)y
                     + 
                              14       12      10  7
                       (- 240x   + 224x   - 40x  )y
                     + 
                             16       14       12      10  5
                       (- 90x   + 184x   - 114x   + 20x  )y
                     + 
                             18      16      14      12     10  3
                       (- 20x   + 68x   - 84x   + 44x   - 8x  )y
                     + 
                            20      18      16      14      12     10
                       (- 2x   + 10x   - 20x   + 20x   - 10x   + 2x  )y
                  *
                      +-------------------------------+
                      | 4      2      2    4     2
                      |y  + (2x  + 2)y  + x  - 2x  + 1
                      |-------------------------------
                     4|         4     2 2    4
                     \|        y  + 2x y  + x
                  *
                     ROOT
                                 8      2      6      4    2  4      6    4  2
                                y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                              + 
                                 8    6
                                x  - x
                           *
                               +-------------------------------+
                               | 4      2      2    4     2
                               |y  + (2x  + 2)y  + x  - 2x  + 1
                               |-------------------------------
                               |         4     2 2    4
                              \|        y  + 2x y  + x
                          + 
                             8      2      6      4     2      4
                            y  + (4x  + 2)y  + (6x  + 2x  + 1)y
                          + 
                               6     4     2  2    8     6    4
                            (4x  - 2x  + 2x )y  + x  - 2x  + x
                       /
                                 8      2      6      4    2  4      6    4  2
                                y  + (4x  + 1)y  + (6x  + x )y  + (4x  - x )y
                              + 
                                 8    6
                                x  - x
                           *
                               +-------------------------------+
                               | 4      2      2    4     2
                               |y  + (2x  + 2)y  + x  - 2x  + 1
                               |-------------------------------
                               |         4     2 2    4
                              \|        y  + 2x y  + x
                          + 
                             8      2      6      4     2      4      6     4  2
                            y  + (4x  + 2)y  + (6x  + 2x  + 1)y  + (4x  - 2x )y
                          + 
                             8     6    4
                            x  - 2x  + x
                 + 
                            22       2      20       4      2       18
                           y   + (11x  + 5)y   + (55x  + 40x  + 10)y
                         + 
                                6       4      2       16
                           (165x  + 135x  + 56x  + 10)y
                         + 
                                8       6       4      2      14
                           (330x  + 240x  + 122x  + 38x  + 5)y
                         + 
                                10       8       6      4      2      12
                           (462x   + 210x  + 126x  + 52x  + 13x  + 1)y
                         + 
                                12      8      6      4     2  10
                           (462x   + 70x  + 30x  + 11x  + 2x )y
                         + 
                                14       12      10     6    4  8
                           (330x   - 210x   + 70x   + 3x  + x )y
                         + 
                                16       14       12      10     8  6
                           (165x   - 240x   + 126x   - 30x   + 3x )y
                         + 
                               18       16       14      12      10    8  4
                           (55x   - 135x   + 122x   - 52x   + 11x   - x )y
                         + 
                               20      18      16      14      12     10  2
                           (11x   - 40x   + 56x   - 38x   + 13x   - 2x  )y
                         + 
                            22     20      18      16     14    12
                           x   - 5x   + 10x   - 10x   + 5x   - x
                      *
                          +-------------------------------+
                          | 4      2      2    4     2
                          |y  + (2x  + 2)y  + x  - 2x  + 1
                          |-------------------------------
                          |         4     2 2    4
                         \|        y  + 2x y  + x
                     + 
                        22       2      20       4      2       18
                       y   + (11x  + 6)y   + (55x  + 48x  + 15)y
                     + 
                            6       4      2       16
                       (165x  + 162x  + 83x  + 20)y
                     + 
                            8       6       4      2       14
                       (330x  + 288x  + 176x  + 72x  + 15)y
                     + 
                            10       8       6      4      2      12
                       (462x   + 252x  + 168x  + 88x  + 33x  + 6)y
                     + 
                            12      8      6      4     2      10
                       (462x   + 70x  + 40x  + 21x  + 8x  + 1)y
                     + 
                            14       12      10     6     4    2  8
                       (330x   - 252x   + 70x   + 3x  + 2x  + x )y
                     + 
                            16       14       12      10     8  6
                       (165x   - 288x   + 168x   - 40x   + 3x )y
                     + 
                           18       16       14      12      10     8  4
                       (55x   - 162x   + 176x   - 88x   + 21x   - 2x )y
                     + 
                           20      18      16      14      12     10    8  2
                       (11x   - 48x   + 83x   - 72x   + 33x   - 8x   + x )y
                     + 
                        22     20      18      16      14     12    10
                       x   - 6x   + 15x   - 20x   + 15x   - 6x   + x
                  *
                      +-------------------------------+2
                      | 4      2      2    4     2
                      |y  + (2x  + 2)y  + x  - 2x  + 1
                      |-------------------------------
                     4|         4     2 2    4
                     \|        y  + 2x y  + x
                 + 
                        20       2      18       4      2       16
                       y   + (10x  + 5)y   + (45x  + 35x  + 10)y
                     + 
                            6       4      2       14
                       (120x  + 100x  + 46x  + 10)y
                     + 
                            8       6      4      2      12
                       (210x  + 140x  + 76x  + 28x  + 5)y
                     + 
                            10      8      6      4     2      10
                       (252x   + 70x  + 50x  + 24x  + 8x  + 1)y
                     + 
                            12      10      8     6     4    2  8
                       (210x   - 70x   + 20x  + 6x  + 3x  + x )y
                     + 
                            14       12      10     8  6
                       (120x   - 140x   + 50x   - 6x )y
                     + 
                           16       14      12      10     8  4
                       (45x   - 100x   + 76x   - 24x   + 3x )y
                     + 
                           18      16      14      12     10    8  2    20
                       (10x   - 35x   + 46x   - 28x   + 8x   - x )y  + x
                     + 
                           18      16      14     12    10
                       - 5x   + 10x   - 10x   + 5x   - x
                  *
                      +-------------------------------+
                      | 4      2      2    4     2
                      |y  + (2x  + 2)y  + x  - 2x  + 1
                      |-------------------------------
                      |         4     2 2    4
                     \|        y  + 2x y  + x
                 + 
                    20       2      18       4      2       16
                   y   + (10x  + 6)y   + (45x  + 42x  + 15)y
                 + 
                        6       4      2       14
                   (120x  + 120x  + 68x  + 20)y
                 + 
                        8       6       4      2       12
                   (210x  + 168x  + 108x  + 52x  + 15)y
                 + 
                        10      8      6      4      2      10
                   (252x   + 84x  + 60x  + 36x  + 18x  + 6)y
                 + 
                        12      10      8     6     4     2      8
                   (210x   - 84x   + 10x  + 4x  + 3x  + 2x  + 1)y
                 + 
                        14       12      10     8  6
                   (120x   - 168x   + 60x   - 4x )y
                 + 
                       16       14       12      10     8  4
                   (45x   - 120x   + 108x   - 36x   + 3x )y
                 + 
                       18      16      14      12      10     8  2    20     18
                   (10x   - 42x   + 68x   - 52x   + 18x   - 2x )y  + x   - 6x
                 + 
                      16      14      12     10    8
                   15x   - 20x   + 15x   - 6x   + x
              /
                        22       2      20       4      2       18
                       y   + (11x  + 5)y   + (55x  + 40x  + 10)y
                     + 
                            6       4      2       16
                       (165x  + 135x  + 56x  + 10)y
                     + 
                            8       6       4      2      14
                       (330x  + 240x  + 122x  + 38x  + 5)y
                     + 
                            10       8       6      4      2      12
                       (462x   + 210x  + 126x  + 52x  + 13x  + 1)y
                     + 
                            12      8      6      4     2  10
                       (462x   + 70x  + 30x  + 11x  + 2x )y
                     + 
                            14       12      10     6    4  8
                       (330x   - 210x   + 70x   + 3x  + x )y
                     + 
                            16       14       12      10     8  6
                       (165x   - 240x   + 126x   - 30x   + 3x )y
                     + 
                           18       16       14      12      10    8  4
                       (55x   - 135x   + 122x   - 52x   + 11x   - x )y
                     + 
                           20      18      16      14      12     10  2    22
                       (11x   - 40x   + 56x   - 38x   + 13x   - 2x  )y  + x
                     + 
                           20      18      16     14    12
                       - 5x   + 10x   - 10x   + 5x   - x
                  *
                      +-------------------------------+
                      | 4      2      2    4     2
                      |y  + (2x  + 2)y  + x  - 2x  + 1
                      |-------------------------------
                      |         4     2 2    4
                     \|        y  + 2x y  + x
                 + 
                    22       2      20       4      2       18
                   y   + (11x  + 6)y   + (55x  + 48x  + 15)y
                 + 
                        6       4      2       16
                   (165x  + 162x  + 83x  + 20)y
                 + 
                        8       6       4      2       14
                   (330x  + 288x  + 176x  + 72x  + 15)y
                 + 
                        10       8       6      4      2      12
                   (462x   + 252x  + 168x  + 88x  + 33x  + 6)y
                 + 
                        12      8      6      4     2      10
                   (462x   + 70x  + 40x  + 21x  + 8x  + 1)y
                 + 
                        14       12      10     6     4    2  8
                   (330x   - 252x   + 70x   + 3x  + 2x  + x )y
                 + 
                        16       14       12      10     8  6
                   (165x   - 288x   + 168x   - 40x   + 3x )y
                 + 
                       18       16       14      12      10     8  4
                   (55x   - 162x   + 176x   - 88x   + 21x   - 2x )y
                 + 
                       20      18      16      14      12     10    8  2    22
                   (11x   - 48x   + 83x   - 72x   + 33x   - 8x   + x )y  + x
                 + 
                       20      18      16      14     12    10
                   - 6x   + 15x   - 20x   + 15x   - 6x   + x
         /
            2
       *
          %i

Type: Complex(Expression(Integer))

fricas

asec( %i )

   (9)  asec(%i)

Type: Expression(Complex(Integer))

fricas

simplify asec( %i )

   (10)  asec(%i)

Type: Expression(Complex(Integer))

I guess this one also causes a problem with MathAction?:

  \begin{axiom}
  simplify(sec(asec(w)))
  \end{axiom}

Not coherent ?

fricas

)set output algebra off

fricas

)set output tex on

fricas

[atan(tan(x)), atan(tan(%pi))]

$\label{eq8}\left[{\arctan \left({\tan \left({x}\right)}\right)}, \: 0 \right]$

(8)

Type: List(Expression(Integer))

atan(tan(x));

reduce

$\displaylines{\qdd \atan \(\tan \(x$