Sr Examen

Otras calculadoras

Resolución de integrales/ 4-x^2/ 2-y^2/ x^2-y/ x+sqrt(4-x^2-y^2)+y

Integral de x+sqrt(4-x^2-y^2)+y dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                              
  /                              
 |                               
 |  /       _____________    \   
 |  |      /      2    2     |   
 |  \x + \/  4 - x  - y   + y/ dx
 |                               
/                                
0
∫01(y+(x+−y2+(4−x2))) dx\int\limits_{0}^{1} \left(y + \left(x + \sqrt{- y^{2} + \left(4 - x^{2}\right)}\right)\right)\, dx 
Integral(x + sqrt(4 - x^2 - y^2) + y, (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                  //                                                                                                                                                                                                                                                                                                                                                                                                                  _________________________                                                        _________________________                                                           \
                                                  ||                            _________________________                                                                                                                                                                                                                  _________________________                                                                                                                 /               2                                                                /               2                                                                    |
                                                  ||                           /               2                                                                                                                                                                                                                          /               2                                                                                                                         /               x           /     2\      /           x           \        2     /               x           /     2\      /           x           \                   |
                                                  ||                          /               x           /     2\                                                                                                                                                                                                 2     /               x           /     2\                                                                                               8*I*   /   -1 + ------------------ *\4 - y /*acosh|-----------------------|   2*I*y *   /   -1 + ------------------ *\4 - y /*acosh|-----------------------|                   |
                                                  ||                 4*pi*   /   -1 + ------------------ *\4 - y /                                                   ____________________                                                         ____________________                                         pi*y *   /   -1 + ------------------ *\4 - y /                                                   ____________________                              /                   /     2\                |   ____________________|            /                   /     2\                |   ____________________|                   |
                                                  ||                        /                   /     2\                                                        3   /           /     2\                                                         /           /     2\                                                  /                   /     2\                                                        2   /           /     2\                             \/          polar_lift\4 - y /                |  /           /     2\ |          \/          polar_lift\4 - y /                |  /           /     2\ |      |    2  |    |
                                                  ||                      \/          polar_lift\4 - y /                                                   2*I*x *\/  polar_lift\4 - y /                                                 8*I*x*\/  polar_lift\4 - y /                                                \/          polar_lift\4 - y /                                                 2*I*x*y *\/  polar_lift\4 - y /                                                                           \\/  polar_lift\4 - y / /                                                        \\/  polar_lift\4 - y / /      |   x   |    |
                                                  ||- --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + --------------------------------------------------------------------------- + --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + --------------------------------------------------------------------------- - ------------------------------------------------------------------------------  for |-------| > 1|
                                                  ||             _________________________              _________________________              _________________________              _________________________              _________________________              _________________________              _________________________              _________________________              _________________________              _________________________              _________________________              _________________________               _________________________              _________________________        |      2|    |
                                                  ||            /               2                      /               2                      /               2                      /               2                      /               2                      /               2                      /               2                      /               2                      /               2                      /               2                      /               2                      /               2                       /               2                      /               2                 |-4 + y |    |
  /                                               ||           /               x                2     /               x                      /               x                2     /               x                      /               x                2     /               x                      /               x                2     /               x                      /               x                2     /               x                      /               x                2     /               x                       /               x                2     /               x                               |
 |                                                ||  - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------     - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------                      |
 | /       _____________    \           2         ||         /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\            /                   /     2\           /                   /     2\                      |
 | |      /      2    2     |          x          ||       \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /          \/          polar_lift\4 - y /         \/          polar_lift\4 - y /                      |
 | \x + \/  4 - x  - y   + y/ dx = C + -- + x*y + |<                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
 |                                     2          ||                                                                                                                                                                                                                                                                                                                                 ________________________                                                    ________________________                                                                                                                                                  |
/                                                 ||                                                                                                                                                                                                                                                                                                                                /              2                                                            /              2                                                                                                                                                           |
                                                  ||                                                                                                                                                                                                                                                                                                                               /              x           /     2\     /           x           \     2     /              x           /     2\     /           x           \                                                                                                           |
                                                  ||                                                                                                                                                                                                                                                                                                                         4*   /   1 - ------------------ *\4 - y /*asin|-----------------------|    y *   /   1 - ------------------ *\4 - y /*asin|-----------------------|                                                                                                           |
                                                  ||                                                                                                                     ____________________                                                       ____________________                                                        ____________________                             /                  /     2\               |   ____________________|         /                  /     2\               |   ____________________|                                                                                                           |
                                                  ||                                                                                                                3   /           /     2\                                                       /           /     2\                                                    2   /           /     2\                            \/         polar_lift\4 - y /               |  /           /     2\ |       \/         polar_lift\4 - y /               |  /           /     2\ |                                                                                                           |
                                                  ||                                                                                                               x *\/  polar_lift\4 - y /                                                 4*x*\/  polar_lift\4 - y /                                                 x*y *\/  polar_lift\4 - y /                                                                        \\/  polar_lift\4 - y / /                                                   \\/  polar_lift\4 - y / /                                                                                                           |
                                                  ||                                                                                        ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - ------------------------------------------------------------------------ + ------------------------------------------------------------------------                                                                                              otherwise    |
                                                  ||                                                                                                  ________________________              ________________________             ________________________              ________________________             ________________________              ________________________             ________________________              ________________________             ________________________              ________________________                                                                                                           |
                                                  ||                                                                                                 /              2                      /              2                     /              2                      /              2                     /              2                      /              2                     /              2                      /              2                     /              2                      /              2                                                                                                                    |
                                                  ||                                                                                                /              x                2     /              x                     /              x                2     /              x                     /              x                2     /              x                     /              x                2     /              x                     /              x                2     /              x                                                                                                                     |
                                                  ||                                                                                        - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------                                                                                                            |
                                                  ||                                                                                              /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\                                                                                                            |
                                                  \\                                                                                            \/         polar_lift\4 - y /         \/         polar_lift\4 - y /        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /                                                                                                            /
∫(y+(x+−y2+(4−x2))) dx=C+x22+xy+{−2ix3polar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1−2ixy2polar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1+8ixpolar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1−2iy2(4−y2)x2polar_lift⁡(4−y2)−1acosh⁡(xpolar_lift⁡(4−y2))4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1+πy2(4−y2)x2polar_lift⁡(4−y2)−14y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1+8i(4−y2)x2polar_lift⁡(4−y2)−1acosh⁡(xpolar_lift⁡(4−y2))4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1−4π(4−y2)x2polar_lift⁡(4−y2)−14y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1for ∣x2y2−4∣>1x3polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1+xy2polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1−4xpolar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1+y2(4−y2)−x2polar_lift⁡(4−y2)+1asin⁡(xpolar_lift⁡(4−y2))2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1−4(4−y2)−x2polar_lift⁡(4−y2)+1asin⁡(xpolar_lift⁡(4−y2))2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1otherwise\int \left(y + \left(x + \sqrt{- y^{2} + \left(4 - x^{2}\right)}\right)\right)\, dx = C + \frac{x^{2}}{2} + x y + \begin{cases} - \frac{2 i x^{3} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i x \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{\pi y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{4 \pi \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2} - 4}}\right| > 1 \\\frac{x^{3} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} + \frac{x y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 x \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} + \frac{y^{2} \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
          1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
          /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  /                                                                                                                                                                                                                                                                                                                                                             _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               _________________________                                                                                                                                                                                                                                                                                                                                                                                                                _________________________                                                                                                                                                                                                                                                                                                     _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                              
         |  |                                                                                                                                                                                                                                                                                                                                                            /               2                   /                                                                                2                      \             ____________________ /                                                                                2                      \                                                                                                                    ____________________ /                                                                                2                      \              /               2                   /                                                                                2                      \                                                                                                                                                  ____________________ /                                                                                2                      \            /               2                   /                                                                                2                      \                                                                                                                                                                                 /               2                   /                                                                                2                      \                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                                                           /               x           /     2\ |                       16*x                                                4*x*y                       |        3   /           /     2\  |                       16*x                                                4*x*y                       |                                                                                                                   /           /     2\  |                       16*x                                                4*x*y                       |       2     /               x           /     2\ |                       16*x                                                4*x*y                       |                                                                                                                                             2   /           /     2\  |                       16*x                                                4*x*y                       |           /               x           /     2\ |                       16*x                                                4*x*y                       |      /           x           \                                                                                                                                           2     /               x           /     2\ |                       16*x                                                4*x*y                       |      /           x           \                                                                                                                                                                                                                                                                                            
         |  |                                                                                                                                                                                                                                                                                                                                                  4*pi*   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|   2*I*x *\/  polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                           8*I*x*\/  polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------|   pi*y *   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                      2*I*x*y *\/  polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------|   8*I*   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                      2*I*y *   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                            
         |  |                                                                                                                                                                                                                                                                                                                                                         /                   /     2\           |      _________________________                            _________________________                   |                                  |      _________________________                            _________________________                   |                                                                                                                                         |      _________________________                            _________________________                   |           /                   /     2\           |      _________________________                            _________________________                   |                                                                                                                                                                       |      _________________________                            _________________________                   |         /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                               /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                                                                                                                                                                            
         |  |                                                                                                                                                                                                                                                                                                                                                       \/          polar_lift\4 - y /           |     /               2                                    /               2                            |                                  |     /               2                                    /               2                            |                                                                                                                                         |     /               2                                    /               2                            |         \/          polar_lift\4 - y /           |     /               2                                    /               2                            |                                                                                                                                                                       |     /               2                                    /               2                            |       \/          polar_lift\4 - y /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                                                                                                             \/          polar_lift\4 - y /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                  /     2\      /           x           \                                                                                            2 /     2\      /           x           \                                                              
         |  |                                                                                                                                                                                                                                                                                                                                                                                                |    /               x                     /     2\       /               x                     /     2\|                                  |    /               x                     /     2\       /               x                     /     2\|                                                                                                                                         |    /               x                     /     2\       /               x                     /     2\|                                                  |    /               x                     /     2\       /               x                     /     2\|                                                                                                                                                                       |    /               x                     /     2\       /               x                     /     2\|                                                |    /               x                     /     2\       /               x                     /     2\|      \\/  polar_lift\4 - y / /                                                                                                                                                                                      |    /               x                     /     2\       /               x                     /     2\|      \\/  polar_lift\4 - y / /                                            8*I*x*\4 - y /*acosh|-----------------------|                                                                                     2*I*x*y *\4 - y /*acosh|-----------------------|                                                              
         |  |                               ____________________                                                            ____________________                                                          ____________________                                                                                                                                                                               |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                  |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                                                                         |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                  |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                                                                                                       |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                                                                                                                                                     |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                               |   ____________________|                                                                                                            |   ____________________|                                                              
         |  |                              /           /     2\                                                        2   /           /     2\                                                      2   /           /     2\                                                                           /     2\                                                                                             |  /                   /     2\                         /                   /     2\                    |                                  |  /                   /     2\                         /                   /     2\                    |                                                   2 /     2\                                                                            |  /                   /     2\                         /                   /     2\                    |                                                  |  /                   /     2\                         /                   /     2\                    |                                                                  /     2\                                                                                             |  /                   /     2\                         /                   /     2\                    |                                                |  /                   /     2\                         /                   /     2\                    |                                                                                                2 /     2\                                                                                                           |  /                   /     2\                         /                   /     2\                    |                                                                                               |  /           /     2\ |                                                                                                            |  /           /     2\ |                                                   2          
         |  |                        8*I*\/  polar_lift\4 - y /                                                   6*I*x *\/  polar_lift\4 - y /                                                 2*I*y *\/  polar_lift\4 - y /                                                                       8*I*\4 - y /                                                                                             \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                  \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                              2*I*y *\4 - y /                                                                            \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                  \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                           4*pi*x*\4 - y /                                                                                             \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                                                          pi*x*y *\4 - y /                                                                                                           \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                                                               \\/  polar_lift\4 - y / /                                                                                                            \\/  polar_lift\4 - y / /                                                  x           
         |  |--------------------------------------------------------------------------- - --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------  for --------- > 1   
         |  |           _________________________              _________________________              _________________________              _________________________              _________________________              _________________________   /           _________________________              _________________________\                                                                                                                                             2                                                                                                                                                 2                                /           _________________________              _________________________\                                                                                                                                     2                                                                                                                                                 2                                              _________________________ /           _________________________              _________________________\                                                                                                                                 2                                                                                                                                                                  2                                                            _________________________ /           _________________________              _________________________\                                                                                                                                                        2                                                              _________________________ /           _________________________              _________________________\                            _________________________ /           _________________________              _________________________\                         |      2|       
         |  |          /               2                      /               2                      /               2                      /               2                      /               2                      /               2            |          /               2                      /               2         |    ____________________                                        /           _________________________              _________________________\                                                                     /           _________________________              _________________________\                                 |          /               2                      /               2         |    ____________________                                /           _________________________              _________________________\                                                                     /           _________________________              _________________________\                                              /               2          |          /               2                      /               2         |                                                    /           _________________________              _________________________\                                                                                      /           _________________________              _________________________\                                                            /               2          |          /               2                      /               2         |                                                                           /           _________________________              _________________________\                                                              /               2          |          /               2                      /               2         |                           /               2          |          /               2                      /               2         |                         |-4 + y |       
         |  |         /               x                2     /               x                      /               x                2     /               x                      /               x                2     /               x             |         /               x                2     /               x          |   /           /     2\                                         |          /               2                      /               2         |                                                                     |          /               2                      /               2         |                                 |         /               x                2     /               x          |   /           /     2\                                 |          /               2                      /               2         |                                                                     |          /               2                      /               2         |                                             /               x           |         /               x                2     /               x          |           /     2\                                 |          /               2                      /               2         |                                                                                      |          /               2                      /               2         |                                                           /               x           |         /               x                2     /               x          |           /     2\                                                        |          /               2                      /               2         |                                                             /               x           |         /               x                2     /               x          |           /     2\       /               x           |         /               x                2     /               x          |           /     2\                      
         |  |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\4 - y /                                         |         /               x                2     /               x          |                                                                     |         /               x                2     /               x          |                                 |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\4 - y /                                 |         /               x                2     /               x          |                                                                     |         /               x                2     /               x          |                                            /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /                                 |         /               x                2     /               x          |                                                                                      |         /               x                2     /               x          |                                                          /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /                                                        |         /               x                2     /               x          |                                                            /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /      /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /                      
         |  |       /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\    |       /                   /     2\           /                   /     2\ |                                                                |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                     |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                 |       /                   /     2\           /                   /     2\ |                                                        |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                     |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                           /                   /     2\  |       /                   /     2\           /                   /     2\ |                                                    |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                                      |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                         /                   /     2\  |       /                   /     2\           /                   /     2\ |                                                                           |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                           /                   /     2\  |       /                   /     2\           /                   /     2\ |                        /                   /     2\  |       /                   /     2\           /                   /     2\ |                                         
         |  |     \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /    \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                |       /                   /     2\           /                   /     2\ |                                                                     |       /                   /     2\           /                   /     2\ |                                 \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                        |       /                   /     2\           /                   /     2\ |                                                                     |       /                   /     2\           /                   /     2\ |                                         \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                    |       /                   /     2\           /                   /     2\ |                                                                                      |       /                   /     2\           /                   /     2\ |                                                       \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                           |       /                   /     2\           /                   /     2\ |                                                         \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                      \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                         
1        |  |                                                                                                                                                                                                                                                                                                                                                                                       \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                     \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                      \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                     \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                                                          \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                      \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                                                                                               \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                  
- + y +  |  <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 dx
2        |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               ________________________                                                                                                                                                             ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ____________________ /                                                                                 2                     \                                                                                                               ____________________ /                                                                                 2                     \           ____________________ /                                                                                 2                     \          /              2                   /                                                                                 2                     \                                         /              2                   /                                                                                 2                     \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           3   /           /     2\  |                         8*x                                                2*x*y                      |                                                                                                              /           /     2\  |                         8*x                                                2*x*y                      |      2   /           /     2\  |                         8*x                                                2*x*y                      |         /              x           /     2\ |                         8*x                                                2*x*y                      |     /           x           \    2     /              x           /     2\ |                         8*x                                                2*x*y                      |     /           x           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          x *\/  polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        4*x*\/  polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------|   x*y *\/  polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------|   4*   /   1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|   y *   /   1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |        ________________________                            ________________________                   |                                                                                                                                    |        ________________________                            ________________________                   |                                |        ________________________                            ________________________                   |       /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|        /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |       /              2                                    /              2                            |                                                                                                                                    |       /              2                                    /              2                            |                                |       /              2                                    /              2                            |     \/         polar_lift\4 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |      \/         polar_lift\4 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |                                                /     2\     /           x           \                                                                                       2 /     2\     /           x           \                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |      /              x                     /     2\       /              x                     /     2\|                                                                                                                                    |      /              x                     /     2\       /              x                     /     2\|                                |      /              x                     /     2\       /              x                     /     2\|                                             |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\4 - y / /                                              |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\4 - y / /                                            4*x*\4 - y /*asin|-----------------------|                                                                                    x*y *\4 - y /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                                ____________________                                                        ____________________                                                                                                                                                             ____________________                                                    |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                                                                                                                    |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                             |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                                                            |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                                                                           |   ____________________|                                                                                                      |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                               /           /     2\                                                    2   /           /     2\                                                                         /     2\                                                                        2   /           /     2\                                                     |    /                  /     2\                         /                  /     2\                    |                                               2 /     2\                                                                           |    /                  /     2\                         /                  /     2\                    |                                |    /                  /     2\                         /                  /     2\                    |                                             |    /                  /     2\                         /                  /     2\                    |                                                                            |    /                  /     2\                         /                  /     2\                    |                                                                                           |  /           /     2\ |                                                                                                      |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                           4*\/  polar_lift\4 - y /                                                   y *\/  polar_lift\4 - y /                                                                       4*\4 - y /                                                                     3*x *\/  polar_lift\4 - y /                                                     \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                              y *\4 - y /                                                                           \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                             \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                                                            \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                                                                           \\/  polar_lift\4 - y / /                                                                                                      \\/  polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                  - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                        otherwise       
         |  |                                                                                                                                                                                                                                                                                                                              ________________________              ________________________             ________________________              ________________________   /          ________________________              ________________________\                                     ________________________              ________________________                                                                                                         2                                /          ________________________              ________________________\                                                                                                                                  2                                                                                                                                       2                                                                                                                                                              2                                                                                                                                                                                   2                                                             ________________________ /          ________________________              ________________________\                            ________________________ /          ________________________              ________________________\                                                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                             /              2                      /              2                     /              2                      /              2            |         /              2                      /              2         |    ____________________            /              2                      /              2                                        /          ________________________              ________________________\                                 |         /              2                      /              2         |    ____________________                                /          ________________________              ________________________\                                                              /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                                                                                          /          ________________________              ________________________\                                                             /              2          |         /              2                      /              2         |                           /              2          |         /              2                      /              2         |                                                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                            /              x                2     /              x                     /              x                2     /              x             |        /              x                2     /              x          |   /           /     2\            /              x                2     /              x                                         |         /              2                      /              2         |                                 |        /              x                2     /              x          |   /           /     2\                                 |         /              2                      /              2         |                                                              |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                                                                                          |         /              2                      /              2         |                                                            /              x           |        /              x                2     /              x          |           /     2\       /              x           |        /              x                2     /              x          |           /     2\                                                                                                                                                                                                                                                                                                                                        
         |  |                                                                                                                                                                                                                                                                                                                    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\4 - y /    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------                                |        /              x                2     /              x          |                                 |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\4 - y /                                 |        /              x                2     /              x          |                                                              |        /              x                2     /              x          |                                                                                     |        /              x                2     /              x          |                                                                                                          |        /              x                2     /              x          |                                                           /   1 - ------------------ *|- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\4 - y /      /   1 - ------------------ *|- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\4 - y /                                                                                                                                                                                                                                                                                                                                        
         |  |                                                                                                                                                                                                                                                                                                                          /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\                                |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                        |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                              |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                                                     |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                                                                          |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                          /                  /     2\  |      /                  /     2\           /                  /     2\ |                        /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /    \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                               \/         polar_lift\4 - y /         \/         polar_lift\4 - y /                                |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                        |      /                  /     2\           /                  /     2\ |                                                              |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                                                                                          |      /                  /     2\           /                  /     2\ |                                                        \/         polar_lift\4 - y /  \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                      \/         polar_lift\4 - y /  \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                           
         |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                                                                                   \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                              \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                     \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                          \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
        /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
        0
y+∫01{−2ix3(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−6ix2polar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1−2ixy2(4−y2)acosh⁡(xpolar_lift⁡(4−y2))x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)+πxy2(4−y2)x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)−2ixy2(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2+8ix(4−y2)acosh⁡(xpolar_lift⁡(4−y2))x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)−4πx(4−y2)x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)+8ix(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−2iy2(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))acosh⁡(xpolar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2+πy2(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−2iy2(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)−2iy2polar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1+8i(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))acosh⁡(xpolar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−4π(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2+8i(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)+8ipolar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1for x2∣y2−4∣>1x3(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+3x2polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1−xy2(4−y2)asin⁡(xpolar_lift⁡(4−y2))−x2polar_lift⁡(4−y2)+1(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)+xy2(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+4x(4−y2)asin⁡(xpolar_lift⁡(4−y2))−x2polar_lift⁡(4−y2)+1(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)−4x(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+y2(4−y2)−x2polar_lift⁡(4−y2)+1(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))asin⁡(xpolar_lift⁡(4−y2))(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+y2(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)+y2polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1−4(4−y2)−x2polar_lift⁡(4−y2)+1(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))asin⁡(xpolar_lift⁡(4−y2))(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2−4(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)−4polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1otherwise dx+12y + \int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{\pi x y^{2} \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i x \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 \pi x \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{8 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{4 \pi \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{8 i \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 4}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{4 x \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} - \frac{4 \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{4 \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx + \frac{1}{2} 
=
= 
          1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
          /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     
         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  /                                                                                                                                                                                                                                                                                                                                                             _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               _________________________                                                                                                                                                                                                                                                                                                                                                                                                                _________________________                                                                                                                                                                                                                                                                                                     _________________________                                                                                                                                                                                                                                                                                                                                                                                                                                              
         |  |                                                                                                                                                                                                                                                                                                                                                            /               2                   /                                                                                2                      \             ____________________ /                                                                                2                      \                                                                                                                    ____________________ /                                                                                2                      \              /               2                   /                                                                                2                      \                                                                                                                                                  ____________________ /                                                                                2                      \            /               2                   /                                                                                2                      \                                                                                                                                                                                 /               2                   /                                                                                2                      \                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                                                           /               x           /     2\ |                       16*x                                                4*x*y                       |        3   /           /     2\  |                       16*x                                                4*x*y                       |                                                                                                                   /           /     2\  |                       16*x                                                4*x*y                       |       2     /               x           /     2\ |                       16*x                                                4*x*y                       |                                                                                                                                             2   /           /     2\  |                       16*x                                                4*x*y                       |           /               x           /     2\ |                       16*x                                                4*x*y                       |      /           x           \                                                                                                                                           2     /               x           /     2\ |                       16*x                                                4*x*y                       |      /           x           \                                                                                                                                                                                                                                                                                            
         |  |                                                                                                                                                                                                                                                                                                                                                  4*pi*   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|   2*I*x *\/  polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------|                                                                                                           8*I*x*\/  polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------|   pi*y *   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                      2*I*x*y *\/  polar_lift\4 - y / *|-------------------------------------------------- - --------------------------------------------------|   8*I*   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                      2*I*y *   /   -1 + ------------------ *\4 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                            
         |  |                                                                                                                                                                                                                                                                                                                                                         /                   /     2\           |      _________________________                            _________________________                   |                                  |      _________________________                            _________________________                   |                                                                                                                                         |      _________________________                            _________________________                   |           /                   /     2\           |      _________________________                            _________________________                   |                                                                                                                                                                       |      _________________________                            _________________________                   |         /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                               /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                                                                                                                                                                            
         |  |                                                                                                                                                                                                                                                                                                                                                       \/          polar_lift\4 - y /           |     /               2                                    /               2                            |                                  |     /               2                                    /               2                            |                                                                                                                                         |     /               2                                    /               2                            |         \/          polar_lift\4 - y /           |     /               2                                    /               2                            |                                                                                                                                                                       |     /               2                                    /               2                            |       \/          polar_lift\4 - y /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                                                                                                             \/          polar_lift\4 - y /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                  /     2\      /           x           \                                                                                            2 /     2\      /           x           \                                                              
         |  |                                                                                                                                                                                                                                                                                                                                                                                                |    /               x                     /     2\       /               x                     /     2\|                                  |    /               x                     /     2\       /               x                     /     2\|                                                                                                                                         |    /               x                     /     2\       /               x                     /     2\|                                                  |    /               x                     /     2\       /               x                     /     2\|                                                                                                                                                                       |    /               x                     /     2\       /               x                     /     2\|                                                |    /               x                     /     2\       /               x                     /     2\|      \\/  polar_lift\4 - y / /                                                                                                                                                                                      |    /               x                     /     2\       /               x                     /     2\|      \\/  polar_lift\4 - y / /                                            8*I*x*\4 - y /*acosh|-----------------------|                                                                                     2*I*x*y *\4 - y /*acosh|-----------------------|                                                              
         |  |                               ____________________                                                            ____________________                                                          ____________________                                                                                                                                                                               |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                  |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                                                                         |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                  |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                                                                                                       |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                                                                                                                                                     |   /   -1 + ------------------ *polar_lift\4 - y /      /   -1 + ------------------ *polar_lift\4 - y /|                                                                                               |   ____________________|                                                                                                            |   ____________________|                                                              
         |  |                              /           /     2\                                                        2   /           /     2\                                                      2   /           /     2\                                                                           /     2\                                                                                             |  /                   /     2\                         /                   /     2\                    |                                  |  /                   /     2\                         /                   /     2\                    |                                                   2 /     2\                                                                            |  /                   /     2\                         /                   /     2\                    |                                                  |  /                   /     2\                         /                   /     2\                    |                                                                  /     2\                                                                                             |  /                   /     2\                         /                   /     2\                    |                                                |  /                   /     2\                         /                   /     2\                    |                                                                                                2 /     2\                                                                                                           |  /                   /     2\                         /                   /     2\                    |                                                                                               |  /           /     2\ |                                                                                                            |  /           /     2\ |                                                   2          
         |  |                        8*I*\/  polar_lift\4 - y /                                                   6*I*x *\/  polar_lift\4 - y /                                                 2*I*y *\/  polar_lift\4 - y /                                                                       8*I*\4 - y /                                                                                             \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                  \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                              2*I*y *\4 - y /                                                                            \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                  \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                           4*pi*x*\4 - y /                                                                                             \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                                                          pi*x*y *\4 - y /                                                                                                           \\/          polar_lift\4 - y /                       \/          polar_lift\4 - y /                    /                                                                                               \\/  polar_lift\4 - y / /                                                                                                            \\/  polar_lift\4 - y / /                                                  x           
         |  |--------------------------------------------------------------------------- - --------------------------------------------------------------------------- - --------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------  for --------- > 1   
         |  |           _________________________              _________________________              _________________________              _________________________              _________________________              _________________________   /           _________________________              _________________________\                                                                                                                                             2                                                                                                                                                 2                                /           _________________________              _________________________\                                                                                                                                     2                                                                                                                                                 2                                              _________________________ /           _________________________              _________________________\                                                                                                                                 2                                                                                                                                                                  2                                                            _________________________ /           _________________________              _________________________\                                                                                                                                                        2                                                              _________________________ /           _________________________              _________________________\                            _________________________ /           _________________________              _________________________\                         |      2|       
         |  |          /               2                      /               2                      /               2                      /               2                      /               2                      /               2            |          /               2                      /               2         |    ____________________                                        /           _________________________              _________________________\                                                                     /           _________________________              _________________________\                                 |          /               2                      /               2         |    ____________________                                /           _________________________              _________________________\                                                                     /           _________________________              _________________________\                                              /               2          |          /               2                      /               2         |                                                    /           _________________________              _________________________\                                                                                      /           _________________________              _________________________\                                                            /               2          |          /               2                      /               2         |                                                                           /           _________________________              _________________________\                                                              /               2          |          /               2                      /               2         |                           /               2          |          /               2                      /               2         |                         |-4 + y |       
         |  |         /               x                2     /               x                      /               x                2     /               x                      /               x                2     /               x             |         /               x                2     /               x          |   /           /     2\                                         |          /               2                      /               2         |                                                                     |          /               2                      /               2         |                                 |         /               x                2     /               x          |   /           /     2\                                 |          /               2                      /               2         |                                                                     |          /               2                      /               2         |                                             /               x           |         /               x                2     /               x          |           /     2\                                 |          /               2                      /               2         |                                                                                      |          /               2                      /               2         |                                                           /               x           |         /               x                2     /               x          |           /     2\                                                        |          /               2                      /               2         |                                                             /               x           |         /               x                2     /               x          |           /     2\       /               x           |         /               x                2     /               x          |           /     2\                      
         |  |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\4 - y /                                         |         /               x                2     /               x          |                                                                     |         /               x                2     /               x          |                                 |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\4 - y /                                 |         /               x                2     /               x          |                                                                     |         /               x                2     /               x          |                                            /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /                                 |         /               x                2     /               x          |                                                                                      |         /               x                2     /               x          |                                                          /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /                                                        |         /               x                2     /               x          |                                                            /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /      /   -1 + ------------------ *|- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\4 - y /                      
         |  |       /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\           /                   /     2\    |       /                   /     2\           /                   /     2\ |                                                                |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                     |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                 |       /                   /     2\           /                   /     2\ |                                                        |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                     |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                           /                   /     2\  |       /                   /     2\           /                   /     2\ |                                                    |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                                      |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                         /                   /     2\  |       /                   /     2\           /                   /     2\ |                                                                           |- 16*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                           /                   /     2\  |       /                   /     2\           /                   /     2\ |                        /                   /     2\  |       /                   /     2\           /                   /     2\ |                                         
         |  |     \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /         \/          polar_lift\4 - y /    \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                |       /                   /     2\           /                   /     2\ |                                                                     |       /                   /     2\           /                   /     2\ |                                 \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                        |       /                   /     2\           /                   /     2\ |                                                                     |       /                   /     2\           /                   /     2\ |                                         \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                    |       /                   /     2\           /                   /     2\ |                                                                                      |       /                   /     2\           /                   /     2\ |                                                       \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                           |       /                   /     2\           /                   /     2\ |                                                         \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                      \/          polar_lift\4 - y /  \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                         
1        |  |                                                                                                                                                                                                                                                                                                                                                                                       \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                     \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                      \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                     \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                                                          \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                      \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                                                                                               \     \/          polar_lift\4 - y /         \/          polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                  
- + y +  |  <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 dx
2        |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               ________________________                                                                                                                                                             ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ____________________ /                                                                                 2                     \                                                                                                               ____________________ /                                                                                 2                     \           ____________________ /                                                                                 2                     \          /              2                   /                                                                                 2                     \                                         /              2                   /                                                                                 2                     \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           3   /           /     2\  |                         8*x                                                2*x*y                      |                                                                                                              /           /     2\  |                         8*x                                                2*x*y                      |      2   /           /     2\  |                         8*x                                                2*x*y                      |         /              x           /     2\ |                         8*x                                                2*x*y                      |     /           x           \    2     /              x           /     2\ |                         8*x                                                2*x*y                      |     /           x           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          x *\/  polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        4*x*\/  polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------|   x*y *\/  polar_lift\4 - y / *|- ------------------------------------------------- + -------------------------------------------------|   4*   /   1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|   y *   /   1 - ------------------ *\4 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |        ________________________                            ________________________                   |                                                                                                                                    |        ________________________                            ________________________                   |                                |        ________________________                            ________________________                   |       /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|        /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |       /              2                                    /              2                            |                                                                                                                                    |       /              2                                    /              2                            |                                |       /              2                                    /              2                            |     \/         polar_lift\4 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |      \/         polar_lift\4 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |                                                /     2\     /           x           \                                                                                       2 /     2\     /           x           \                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |      /              x                     /     2\       /              x                     /     2\|                                                                                                                                    |      /              x                     /     2\       /              x                     /     2\|                                |      /              x                     /     2\       /              x                     /     2\|                                             |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\4 - y / /                                              |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\4 - y / /                                            4*x*\4 - y /*asin|-----------------------|                                                                                    x*y *\4 - y /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                                ____________________                                                        ____________________                                                                                                                                                             ____________________                                                    |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                                                                                                                    |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                             |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                                                            |     /   1 - ------------------ *polar_lift\4 - y /      /   1 - ------------------ *polar_lift\4 - y /|                                                                                           |   ____________________|                                                                                                      |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                               /           /     2\                                                    2   /           /     2\                                                                         /     2\                                                                        2   /           /     2\                                                     |    /                  /     2\                         /                  /     2\                    |                                               2 /     2\                                                                           |    /                  /     2\                         /                  /     2\                    |                                |    /                  /     2\                         /                  /     2\                    |                                             |    /                  /     2\                         /                  /     2\                    |                                                                            |    /                  /     2\                         /                  /     2\                    |                                                                                           |  /           /     2\ |                                                                                                      |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                                           4*\/  polar_lift\4 - y /                                                   y *\/  polar_lift\4 - y /                                                                       4*\4 - y /                                                                     3*x *\/  polar_lift\4 - y /                                                     \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                              y *\4 - y /                                                                           \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                             \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                                                            \  \/         polar_lift\4 - y /                       \/         polar_lift\4 - y /                    /                                                                                           \\/  polar_lift\4 - y / /                                                                                                      \\/  polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                                                 
         |  |                                                                                                                                                                                                                                                                                                                  - ------------------------------------------------------------------------ + ------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------                                                                                                                                                                                                                                                                                                                        otherwise       
         |  |                                                                                                                                                                                                                                                                                                                              ________________________              ________________________             ________________________              ________________________   /          ________________________              ________________________\                                     ________________________              ________________________                                                                                                         2                                /          ________________________              ________________________\                                                                                                                                  2                                                                                                                                       2                                                                                                                                                              2                                                                                                                                                                                   2                                                             ________________________ /          ________________________              ________________________\                            ________________________ /          ________________________              ________________________\                                                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                             /              2                      /              2                     /              2                      /              2            |         /              2                      /              2         |    ____________________            /              2                      /              2                                        /          ________________________              ________________________\                                 |         /              2                      /              2         |    ____________________                                /          ________________________              ________________________\                                                              /          ________________________              ________________________\                                                                                     /          ________________________              ________________________\                                                                                                          /          ________________________              ________________________\                                                             /              2          |         /              2                      /              2         |                           /              2          |         /              2                      /              2         |                                                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                            /              x                2     /              x                     /              x                2     /              x             |        /              x                2     /              x          |   /           /     2\            /              x                2     /              x                                         |         /              2                      /              2         |                                 |        /              x                2     /              x          |   /           /     2\                                 |         /              2                      /              2         |                                                              |         /              2                      /              2         |                                                                                     |         /              2                      /              2         |                                                                                                          |         /              2                      /              2         |                                                            /              x           |        /              x                2     /              x          |           /     2\       /              x           |        /              x                2     /              x          |           /     2\                                                                                                                                                                                                                                                                                                                                        
         |  |                                                                                                                                                                                                                                                                                                                    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------    |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\4 - y /    - 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------                                |        /              x                2     /              x          |                                 |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\4 - y /                                 |        /              x                2     /              x          |                                                              |        /              x                2     /              x          |                                                                                     |        /              x                2     /              x          |                                                                                                          |        /              x                2     /              x          |                                                           /   1 - ------------------ *|- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\4 - y /      /   1 - ------------------ *|- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\4 - y /                                                                                                                                                                                                                                                                                                                                        
         |  |                                                                                                                                                                                                                                                                                                                          /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\                                |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                        |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                              |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                                                     |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                                                                          |- 8*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                          /                  /     2\  |      /                  /     2\           /                  /     2\ |                        /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                           
         |  |                                                                                                                                                                                                                                                                                                                        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /        \/         polar_lift\4 - y /         \/         polar_lift\4 - y /    \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                               \/         polar_lift\4 - y /         \/         polar_lift\4 - y /                                |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                        |      /                  /     2\           /                  /     2\ |                                                              |      /                  /     2\           /                  /     2\ |                                                                                     |      /                  /     2\           /                  /     2\ |                                                                                                          |      /                  /     2\           /                  /     2\ |                                                        \/         polar_lift\4 - y /  \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                      \/         polar_lift\4 - y /  \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                           
         |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                                                                                   \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                              \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                     \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                          \    \/         polar_lift\4 - y /         \/         polar_lift\4 - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
        /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
        0
y+∫01{−2ix3(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−6ix2polar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1−2ixy2(4−y2)acosh⁡(xpolar_lift⁡(4−y2))x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)+πxy2(4−y2)x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)−2ixy2(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2+8ix(4−y2)acosh⁡(xpolar_lift⁡(4−y2))x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)−4πx(4−y2)x2polar_lift⁡(4−y2)−1(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)+8ix(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−2iy2(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))acosh⁡(xpolar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2+πy2(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−2iy2(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)−2iy2polar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1+8i(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))acosh⁡(xpolar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2−4π(4−y2)x2polar_lift⁡(4−y2)−1(−4xy2x2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2)+16xx2polar_lift⁡(4−y2)−1polar_lift⁡(4−y2))(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)2+8i(4−y2)(4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1)polar_lift⁡(4−y2)+8ipolar_lift⁡(4−y2)4y2x2polar_lift⁡(4−y2)−1−16x2polar_lift⁡(4−y2)−1for x2∣y2−4∣>1x3(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+3x2polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1−xy2(4−y2)asin⁡(xpolar_lift⁡(4−y2))−x2polar_lift⁡(4−y2)+1(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)+xy2(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+4x(4−y2)asin⁡(xpolar_lift⁡(4−y2))−x2polar_lift⁡(4−y2)+1(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)−4x(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))polar_lift⁡(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+y2(4−y2)−x2polar_lift⁡(4−y2)+1(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))asin⁡(xpolar_lift⁡(4−y2))(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2+y2(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)+y2polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1−4(4−y2)−x2polar_lift⁡(4−y2)+1(2xy2−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2)−8x−x2polar_lift⁡(4−y2)+1polar_lift⁡(4−y2))asin⁡(xpolar_lift⁡(4−y2))(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)2−4(4−y2)(2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1)polar_lift⁡(4−y2)−4polar_lift⁡(4−y2)2y2−x2polar_lift⁡(4−y2)+1−8−x2polar_lift⁡(4−y2)+1otherwise dx+12y + \int\limits_{0}^{1} \begin{cases} - \frac{2 i x^{3} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{\pi x y^{2} \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i x \left(4 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 \pi x \left(4 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{8 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} + \frac{8 i \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} - \frac{4 \pi \left(4 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{16 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right)^{2}} + \frac{8 i \left(4 - y^{2}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{8 i \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1} - 16 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 4}\right|} > 1 \\\frac{x^{3} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{x y^{2} \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + \frac{x y^{2} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{4 x \left(4 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{4 x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} - \frac{4 \left(4 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}} - \frac{8 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(4 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right)^{2}} - \frac{4 \left(4 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}} - \frac{4 \sqrt{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1} - 8 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(4 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx + \frac{1}{2} 
1/2 + y + Integral(Piecewise((8*i*sqrt(polar_lift(4 - y^2))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))) - 6*i*x^2*sqrt(polar_lift(4 - y^2))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))) - 2*i*y^2*sqrt(polar_lift(4 - y^2))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))) + 8*i*(4 - y^2)/((-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) - 4*pi*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 - 2*i*x^3*sqrt(polar_lift(4 - y^2))*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 - 2*i*y^2*(4 - y^2)/((-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) + 8*i*x*sqrt(polar_lift(4 - y^2))*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + pi*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 - 4*pi*x*(4 - y^2)/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - 2*i*x*y^2*sqrt(polar_lift(4 - y^2))*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + 8*i*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*acosh(x/sqrt(polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + pi*x*y^2*(4 - y^2)/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - 2*i*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2))*(4 - y^2)*(16*x/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) - 4*x*y^2/(sqrt(-1 + x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*acosh(x/sqrt(polar_lift(4 - y^2)))/(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))^2 + 8*i*x*(4 - y^2)*acosh(x/sqrt(polar_lift(4 - y^2)))/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - 2*i*x*y^2*(4 - y^2)*acosh(x/sqrt(polar_lift(4 - y^2)))/(sqrt(-1 + x^2/polar_lift(4 - y^2))*(-16*sqrt(-1 + x^2/polar_lift(4 - y^2)) + 4*y^2*sqrt(-1 + x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)), x^2/|-4 + y^2| > 1), (-4*sqrt(polar_lift(4 - y^2))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2))) + y^2*sqrt(polar_lift(4 - y^2))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2))) - 4*(4 - y^2)/((-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) + 3*x^2*sqrt(polar_lift(4 - y^2))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2))) + x^3*sqrt(polar_lift(4 - y^2))*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + y^2*(4 - y^2)/((-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*sqrt(polar_lift(4 - y^2))) - 4*x*sqrt(polar_lift(4 - y^2))*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + x*y^2*sqrt(polar_lift(4 - y^2))*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 - 4*sqrt(1 - x^2/polar_lift(4 - y^2))*(4 - y^2)*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*asin(x/sqrt(polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + y^2*sqrt(1 - x^2/polar_lift(4 - y^2))*(4 - y^2)*(-8*x/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)) + 2*x*y^2/(sqrt(1 - x^2/polar_lift(4 - y^2))*polar_lift(4 - y^2)))*asin(x/sqrt(polar_lift(4 - y^2)))/(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))^2 + 4*x*(4 - y^2)*asin(x/sqrt(polar_lift(4 - y^2)))/(sqrt(1 - x^2/polar_lift(4 - y^2))*(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)) - x*y^2*(4 - y^2)*asin(x/sqrt(polar_lift(4 - y^2)))/(sqrt(1 - x^2/polar_lift(4 - y^2))*(-8*sqrt(1 - x^2/polar_lift(4 - y^2)) + 2*y^2*sqrt(1 - x^2/polar_lift(4 - y^2)))*polar_lift(4 - y^2)), True)), (x, 0, 1))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

    © Sr Examen – Калькулятор