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How does FriCAS 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
   




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