Интеграл sqrt(1-x^2-y^2) (dx)

Преподаватель очень удивится увидев твоё верное решение 😼

Виды выражений

обычное уравнение sqrt(1-x^2-y^2) = 0

неявная функция sqrt(1-x^2-y^2)

производная неявной sqrt(1-x^2-y^2)

канонический вид sqrt(1-x^2-y^2)

система уравнений sqrt(1-x^2-y^2)

Неопределенный интеграл sqrt(1-x^2-y^2)

построить график sqrt(1-x^2-y^2)

предел функции sqrt(1-x^2-y^2)

производная sqrt(1-x^2-y^2)

упростить выражение sqrt(1-x^2-y^2)

обычный калькулятор sqrt(1-x^2-y^2)

Решение

Вы ввели [src]

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0

$$\int\limits_{0}^{1} \sqrt{- x^{2} - y^{2} + 1}\, dx$$

Подробное решение

SqrtQuadraticRule(a=1 - y**2, b=0, c=-1, context=sqrt(-x**2 - y**2 + 1), symbol=x)

Теперь упростить:
$\frac{x \sqrt{- x^{2} - y^{2} + 1}}{2} + \frac{i \left(y^{2} - 1\right) \log{\left(- 2 x + 2 i \sqrt{- x^{2} - y^{2} + 1} \right)}}{2}$
Добавляем постоянную интегрирования:
$\frac{x \sqrt{- x^{2} - y^{2} + 1}}{2} + \frac{i \left(y^{2} - 1\right) \log{\left(- 2 x + 2 i \sqrt{- x^{2} - y^{2} + 1} \right)}}{2}+ \mathrm{constant}$

Ответ:

$\frac{x \sqrt{- x^{2} - y^{2} + 1}}{2} + \frac{i \left(y^{2} - 1\right) \log{\left(- 2 x + 2 i \sqrt{- x^{2} - y^{2} + 1} \right)}}{2}+ \mathrm{constant}$

Ответ [src]

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 |  |                                                                                                                                                                                                                                                                                                                                                                                                 /              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         |                                                          /              x           |        /              x                2     /              x          |           /     2\                                                      |         /              2                      /              2         |                                                            /              x           |        /              x                2     /              x          |           /     2\                                                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                                                                         - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\1 - y /    - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------                                |        /              x                2     /              x          |                                 |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\1 - y /                                |        /              x                2     /              x          |                                                             |        /              x                2     /              x          |                                                                                    |        /              x                2     /              x          |                                                         /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\1 - y /                                                      |        /              x                2     /              x          |                                                           /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\1 - y /                                                                                                                                                                                                                                                                                                                                                                                                              
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 |  |                                                                                                                                                                                                                                                                                                                                                                                             \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /    \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                               \/         polar_lift\1 - y /         \/         polar_lift\1 - y /                                |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                       |      /                  /     2\           /                  /     2\ |                                                             |      /                  /     2\           /                  /     2\ |                                                                                    |      /                  /     2\           /                  /     2\ |                                                      \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                         |      /                  /     2\           /                  /     2\ |                                                        \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                 
 |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                             \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                    \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                        \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
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/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
0

$$\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(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{6 i x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} - \frac{2 i x y^{2} \cdot \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \cdot \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{\pi x y^{2} \cdot \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \cdot \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 i x y^{2} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i x \left(1 - y^{2}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \cdot \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{\pi x \left(1 - y^{2}\right)}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \cdot \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{2 i x \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \cdot \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{\pi y^{2} \cdot \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{2 i y^{2} \cdot \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}}{\sqrt{\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - 1} \sqrt{\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + 1} \cdot \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{2 i y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{acosh}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} - \frac{\pi \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \left(- \frac{4 x y^{2}}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{4 x}{\sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right)}{\left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right)^{2}} + \frac{2 i \left(1 - y^{2}\right) \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}}{\sqrt{\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - 1} \sqrt{\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + 1} \cdot \left(4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{2 i \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{4 y^{2} \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1} - 4 \sqrt{\frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - 1}} & \text{for}\: \frac{x^{2}}{\left|{y^{2} - 1}\right|} > 1 \\\frac{x^{3} \cdot \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{3 x^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{x y^{2} \cdot \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \cdot \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + \frac{x y^{2} \cdot \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{x \left(1 - y^{2}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \cdot \left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{x \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \cdot \left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \cdot \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} + \frac{y^{2} \cdot \left(1 - y^{2}\right)}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} + \frac{y^{2} \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} - \frac{\left(1 - y^{2}\right) \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \cdot \left(\frac{2 x y^{2}}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}} - \frac{2 x}{\sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} \operatorname{polar\_lift}{\left(1 - y^{2} \right)}}\right) \operatorname{asin}{\left(\frac{x}{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} \right)}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right)^{2}} - \frac{1 - y^{2}}{\left(2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}\right) \sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}} - \frac{\sqrt{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}}}{2 y^{2} \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1} - 2 \sqrt{- \frac{x^{2}}{\operatorname{polar\_lift}{\left(1 - y^{2} \right)}} + 1}} & \text{otherwise} \end{cases}\, dx$$

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 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
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 |  |                                                                                                                                                                                                                                              /               x           /     2\ |                       4*x                                                 4*x*y                       |        3   /           /     2\  |                       4*x                                                 4*x*y                       |           /           /     2\  |                       4*x                                                 4*x*y                       |       2     /               x           /     2\ |                       4*x                                                 4*x*y                       |                                                                                                                                            2   /           /     2\  |                       4*x                                                 4*x*y                       |           /               x           /     2\ |                       4*x                                                 4*x*y                       |      /           x           \                                                                                                                                          2     /               x           /     2\ |                       4*x                                                 4*x*y                       |      /           x           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
 |  |                                                                                                                                                                                                                                       pi*   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|   2*I*x *\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|   2*I*x*\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|   pi*y *   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|                                                                                                                                     2*I*x*y *\/  polar_lift\1 - y / *|-------------------------------------------------- - --------------------------------------------------|   2*I*   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                     2*I*y *   /   -1 + ------------------ *\1 - y /*|-------------------------------------------------- - --------------------------------------------------|*acosh|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          
 |  |                                                                                                                                                                                                                                            /                   /     2\           |      _________________________                            _________________________                   |                                  |      _________________________                            _________________________                   |                                 |      _________________________                            _________________________                   |           /                   /     2\           |      _________________________                            _________________________                   |                                                                                                                                                                      |      _________________________                            _________________________                   |         /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                              /                   /     2\           |      _________________________                            _________________________                   |      |   ____________________|                                                                                                                                                                                                                _________________________                                                                                                                                                                                                                                                                                          _________________________                                                                                              
 |  |                                                                                                                                                                                                                                          \/          polar_lift\1 - y /           |     /               2                                    /               2                            |                                  |     /               2                                    /               2                            |                                 |     /               2                                    /               2                            |         \/          polar_lift\1 - y /           |     /               2                                    /               2                            |                                                                                                                                                                      |     /               2                                    /               2                            |       \/          polar_lift\1 - y /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                                                                                                            \/          polar_lift\1 - y /           |     /               2                                    /               2                            |      |  /           /     2\ |                                                  /     2\      /           x           \                                                                                                                      /               2                                                                                                                                    2 /     2\      /           x           \                                                                                                                     /               2                                                                                                       
 |  |                                                                                                                                                                                                                                                                                   |    /               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\1 - y / /                                                                                                                                                                                     |    /               x                     /     2\       /               x                     /     2\|      \\/  polar_lift\1 - y / /                                            2*I*x*\1 - y /*acosh|-----------------------|                                                                                                                     /               x           /     2\                                                                                                           2*I*x*y *\1 - y /*acosh|-----------------------|                                                                                                              2     /               x           /     2\                                                                                     
 |  |                              ____________________                                                           ____________________                                                         ____________________                                                                     |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                  |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                 |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                  |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                                                                                                                                      |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                                                                                                                                                                                    |   /   -1 + ------------------ *polar_lift\1 - y /      /   -1 + ------------------ *polar_lift\1 - y /|                                                                                               |   ____________________|                                                                                                             2*I*   /   -1 + ------------------ *\1 - y /                                                                                                                                  |   ____________________|                                                                                                         2*I*y *   /   -1 + ------------------ *\1 - 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*I*\/  polar_lift\1 - y /                                                  6*I*x *\/  polar_lift\1 - y /                                                2*I*y *\/  polar_lift\1 - y /                                                                      \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                  \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                 \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                  \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                            pi*x*\1 - y /                                                                                             \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                                                          pi*x*y *\1 - y /                                                                                                          \\/          polar_lift\1 - y /                       \/          polar_lift\1 - y /                    /                                                                                               \\/  polar_lift\1 - y / /                                                                                                                 \/          polar_lift\1 - y /                                                                                                                                            \\/  polar_lift\1 - y / /                                                                                                                \/          polar_lift\1 - 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         |    ____________________      |-1 + y |       
 |  |        /               x                2     /               x                     /               x                2     /               x                     /               x                2     /               x                                                 |         /               2                      /               2         |                                                                     |         /               2                      /               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             |        /               x                2     /               x          |   /           /     2\        /               x           |        /               x                2     /               x          |           /     2\        /                x                  /                 x             |        /               x                2     /               x          |   /           /     2\                       
 |  |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------    - 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------                                        |        /               x                2     /               x          |                                                                     |        /               x                2     /               x          |                                                               |        /               x                2     /               x          |                                                                      |        /               x                2     /               x          |                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                                 |        /               x                2     /               x          |                                                                                       |        /               x                2     /               x          |                                                           /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /                                                        |        /               x                2     /               x          |                                                             /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /       /  1 + ----------------------- *    /  -1 + ----------------------- *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\1 - y /       /   -1 + ------------------ *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*polar_lift\1 - y /       /  1 + ----------------------- *    /  -1 + ----------------------- *|- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |*\/  polar_lift\1 - y /                       
 |  |      /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\          /                   /     2\           /                   /     2\                                        |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                     |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                               |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                      |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                    |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                                                       |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                          /                   /     2\  |      /                   /     2\           /                   /     2\ |                                                                           |- 4*   /   -1 + ------------------  + 4*y *   /   -1 + ------------------ |                                                            /                   /     2\  |      /                   /     2\           /                   /     2\ |                         /          ____________________     /           ____________________  |      /                   /     2\           /                   /     2\ |                             /                   /     2\  |      /                   /     2\           /                   /     2\ |                         /          ____________________     /           ____________________  |      /                   /     2\           /                   /     2\ |                                              
 |  |    \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /        \/          polar_lift\1 - y /         \/          polar_lift\1 - y /                                        |      /                   /     2\           /                   /     2\ |                                                                     |      /                   /     2\           /                   /     2\ |                                                               |      /                   /     2\           /                   /     2\ |                                                                      |      /                   /     2\           /                   /     2\ |                                          \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                    |      /                   /     2\           /                   /     2\ |                                                                                       |      /                   /     2\           /                   /     2\ |                                                        \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                           |      /                   /     2\           /                   /     2\ |                                                          \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                        /          /           /     2\     /           /           /     2\   \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                           \/          polar_lift\1 - y /  \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                        /          /           /     2\     /           /           /     2\   \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                              
 |  |                                                                                                                                                                                                                                                                           \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                     \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                               \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                      \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                          \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                       \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                                                                               \    \/          polar_lift\1 - y /         \/          polar_lift\1 - y / /                                                                                                                                                                                            \/         \/  polar_lift\1 - y /   \/          \/  polar_lift\1 - y /                                                                                                                                                                                                                                            \/         \/  polar_lift\1 - y /   \/          \/  polar_lift\1 - y /                                                                                                                             
 |  <                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        dx
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ________________________                                                                                                                                                                                                                                                                                            ________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ____________________ /                                                                                 2                     \                                                                                                             ____________________ /                                                                                 2                     \           ____________________ /                                                                                 2                     \        /              2                   /                                                                                 2                     \                                                                                                                                                                        /              2                   /                                                                                 2                     \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                3   /           /     2\  |                         2*x                                                2*x*y                      |                                                                                                            /           /     2\  |                         2*x                                                2*x*y                      |      2   /           /     2\  |                         2*x                                                2*x*y                      |       /              x           /     2\ |                         2*x                                                2*x*y                      |     /           x           \                                                                                                                                   2     /              x           /     2\ |                         2*x                                                2*x*y                      |     /           x           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               x *\/  polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------|                                                                                                        x*\/  polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------|   x*y *\/  polar_lift\1 - y / *|- ------------------------------------------------- + -------------------------------------------------|      /   1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                  y *   /   1 - ------------------ *\1 - y /*|- ------------------------------------------------- + -------------------------------------------------|*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |        ________________________                            ________________________                   |                                                                                                                                  |        ________________________                            ________________________                   |                                |        ________________________                            ________________________                   |     /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                       /                  /     2\           |        ________________________                            ________________________                   |     |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |       /              2                                    /              2                            |                                                                                                                                  |       /              2                                    /              2                            |                                |       /              2                                    /              2                            |   \/         polar_lift\1 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |                                               /     2\     /           x           \                                                \/         polar_lift\1 - y /           |       /              2                                    /              2                            |     |  /           /     2\ |                                              2 /     2\     /           x           \                                                                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |      /              x                     /     2\       /              x                     /     2\|                                                                                                                                  |      /              x                     /     2\       /              x                     /     2\|                                |      /              x                     /     2\       /              x                     /     2\|                                           |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\1 - y / /                                             x*\1 - y /*asin|-----------------------|                                                                                        |      /              x                     /     2\       /              x                     /     2\|     \\/  polar_lift\1 - y / /                                           x*y *\1 - y /*asin|-----------------------|                                                                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                    ____________________                                                         ____________________                                                                                                                                                             ____________________                                                    |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                                                                                                                  |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                           |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                                                                          |   ____________________|                                                                                        |     /   1 - ------------------ *polar_lift\1 - y /      /   1 - ------------------ *polar_lift\1 - y /|                                                                                           |   ____________________|                                                                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                   /           /     2\                                                     2   /           /     2\                                                                              2                                                                          2   /           /     2\                                                     |    /                  /     2\                         /                  /     2\                    |                                               2 /     2\                                                                         |    /                  /     2\                         /                  /     2\                    |                                |    /                  /     2\                         /                  /     2\                    |                                           |    /                  /     2\                         /                  /     2\                    |                                                                                          |  /           /     2\ |                                                                                        |    /                  /     2\                         /                  /     2\                    |                                                                                           |  /           /     2\ |                                                                                                                                                                                                                                                                                                                                                                                                                                                       
 |  |                                                                                                                                                                                                                                                                                                                                                                                                                 \/  polar_lift\1 - y /                                                    y *\/  polar_lift\1 - y /                                                                         1 - y                                                                        3*x *\/  polar_lift\1 - y /                                                     \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                              y *\1 - y /                                                                         \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                           \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                                                                          \\/  polar_lift\1 - y / /                                                                                        \  \/         polar_lift\1 - y /                       \/         polar_lift\1 - y /                    /                                                                                           \\/  polar_lift\1 - 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         |                                                          /              x           |        /              x                2     /              x          |           /     2\                                                      |         /              2                      /              2         |                                                            /              x           |        /              x                2     /              x          |           /     2\                                                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                                                                         - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------    - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------    |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\1 - y /    - 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------                                |        /              x                2     /              x          |                                 |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*\/  polar_lift\1 - y /                                |        /              x                2     /              x          |                                                             |        /              x                2     /              x          |                                                                                    |        /              x                2     /              x          |                                                         /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\1 - y /                                                      |        /              x                2     /              x          |                                                           /   1 - ------------------ *|- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |*polar_lift\1 - y /                                                                                                                                                                                                                                                                                                                                                                                                              
 |  |                                                                                                                                                                                                                                                                                                                                                                                               /                  /     2\           /                  /     2\          /                  /     2\           /                  /     2\    |      /                  /     2\           /                  /     2\ |                                 /                  /     2\           /                  /     2\                                |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                 |      /                  /     2\           /                  /     2\ |                                                       |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                             |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                                                    |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                        /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                         |- 2*   /   1 - ------------------  + 2*y *   /   1 - ------------------ |                                                          /                  /     2\  |      /                  /     2\           /                  /     2\ |                                                                                                                                                                                                                                                                                                                                                                                                                                 
 |  |                                                                                                                                                                                                                                                                                                                                                                                             \/         polar_lift\1 - y /         \/         polar_lift\1 - y /        \/         polar_lift\1 - y /         \/         polar_lift\1 - y /    \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                               \/         polar_lift\1 - y /         \/         polar_lift\1 - y /                                |      /                  /     2\           /                  /     2\ |                                 \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                       |      /                  /     2\           /                  /     2\ |                                                             |      /                  /     2\           /                  /     2\ |                                                                                    |      /                  /     2\           /                  /     2\ |                                                      \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                         |      /                  /     2\           /                  /     2\ |                                                        \/         polar_lift\1 - y /  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                 
 |  \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                  \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                             \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                    \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                        \    \/         polar_lift\1 - y /         \/         polar_lift\1 - y / /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
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  /                                                                                          
 |                                _____________                                              
 |    _____________              /      2    2      /     2\    /              _____________\
 |   /      2    2           x*\/  1 - x  - y       |1   y |    |             /      2    2 |
 | \/  1 - x  - y   dx = C + ------------------ - I*|- - --|*log\-2*x + 2*I*\/  1 - x  - y  /
 |                                   2              \2   2 /                                 
/

$$\int \sqrt{- x^{2} - y^{2} + 1}\, dx = C + \frac{x \sqrt{- x^{2} - y^{2} + 1}}{2} - i \left(\frac{1}{2} - \frac{y^{2}}{2}\right) \log{\left(- 2 x + 2 i \sqrt{- x^{2} - y^{2} + 1} \right)}$$

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