Производная cos(x)/(Abs(cos(x)))

Вы ввели

 cos(x) 
--------
|cos(x)|

$$\frac{\cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|}$$

Первая производная

             //d                                  d                               \                          /  d                                  d                               \                       \       
             ||--(im(x))*cos(re(x))*sinh(im(x)) - --(re(x))*cosh(im(x))*sin(re(x))|*cos(re(x))*cosh(im(x)) - |- --(im(x))*cosh(im(x))*sin(re(x)) - --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))|*cos(x)
   sin(x)    \\dx                                 dx                              /                          \  dx                                 dx                              /                       /       
- -------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  |cos(x)|                                                                                                         3                                                                                               
                                                                                                           |cos(x)|                                                                                                

$$- \frac{\cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{3}} \left(\left(- \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(- \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right) - \frac{\sin{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|}$$

Вторая производная

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          |||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*sin(re(x))*sinh(im(x)) - ||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cosh(im(x))*sin(re(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cos(re(x))*sinh(im(x)) + |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cosh(im(x))*sin(re(x)) - |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cos(re(x))*sinh(im(x))|*cos(x)     //d                                  d                               \                          /d                                  d                               \                       \            //d                                  d                               \                          /d                                  d                               \                       \        
          ||\dx       /                             2                                   2                                 \dx       /                             dx        dx                              |                          |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              |                          \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                              |          2*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))|*sin(x)   3*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))| *cos(x)
          \\                                      dx                                  dx                                                                                                                    /                          \                                      dx                                                                        dx                                                                              /                                                                                                                                                                                                                                                                                                                                                                                                                                                           /            \\dx                                 dx                              /                          \dx                                 dx                              /                       /            \\dx                                 dx                              /                          \dx                                 dx                              /                       /        
-cos(x) - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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                                                                                                                                                                                                                                                                                                                                                                                                                                                      |cos(x)|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |cos(x)|                                                                                                                                                                                                  |cos(x)|                                                                                                
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |cos(x)|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           

$$\frac{1}{\left|{\cos{\left (x \right )}}\right|} \left(\frac{3 \cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{4}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right)^{2} - \frac{2 \sin{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{2}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right) - \frac{\cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{2}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} - \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )}\right) - \cos{\left (x \right )}\right)$$

Третья производная

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  ||/d        \                            d                                   d                                  /d        \                             /d        \  d                                    d          d                                    /d        \  d                                    d          d                                    d          d                                    d          d                               |                          | d                                  /d        \                           /d        \                            d                                    d          d                                    /d        \  d                                    /d        \  d                                    d          d                                    d          d                                    d          d                               |                          /d        \  /d                                  d                  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                     |/d        \                            d                                   d                                  /d        \                             d         d                               | d                                    |/d        \                            d                                   d                                  /d        \                             d         d                               | d                                    |/d        \                            d                                  /d        \                            d                                    d         d                               | d                                    /d                                  d                               \ d         d                                    /d                                  d                               \ d         d                               |                                                      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           /d                                  d                               \ d                                  /d                                  d                               \ d                               |                                                                                                                                                                                                          3            //d                                  d                               \                          /d                                  d                               \                       \ ||/d        \                            d                                   d                                  /d        \                             d         d                               |                          |/d        \                            d                                  /d        \                            d                                    d    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  |||--(im(x))| *cosh(im(x))*sin(re(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *cos(re(x))*sinh(im(x)) - 3*|--(re(x))| *--(im(x))*cosh(im(x))*sin(re(x)) - 3*--(re(x))*---(re(x))*sin(re(x))*sinh(im(x)) + 3*|--(im(x))| *--(re(x))*cos(re(x))*sinh(im(x)) + 3*--(im(x))*---(im(x))*sin(re(x))*sinh(im(x)) + 3*--(im(x))*---(re(x))*cos(re(x))*cosh(im(x)) + 3*--(re(x))*---(im(x))*cos(re(x))*cosh(im(x))|*sin(re(x))*sinh(im(x)) - |---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*sinh(im(x)) - |--(re(x))| *cosh(im(x))*sin(re(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) - 3*--(im(x))*---(im(x))*cos(re(x))*cosh(im(x)) + 3*|--(im(x))| *--(re(x))*cosh(im(x))*sin(re(x)) + 3*|--(re(x))| *--(im(x))*cos(re(x))*sinh(im(x)) + 3*--(im(x))*---(re(x))*sin(re(x))*sinh(im(x)) + 3*--(re(x))*---(im(x))*sin(re(x))*sinh(im(x)) + 3*--(re(x))*---(re(x))*cos(re(x))*cosh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))| *|--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x)) + |--(re(x))| *|--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*---(im(x))*cosh(im(x))*sin(re(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*---(re(x))*cos(re(x))*sinh(im(x)) + |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *|--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(re(x))| *|--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x)) - |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*---(im(x))*cos(re(x))*sinh(im(x)) - 2*||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*--(im(x))*cos(re(x))*sinh(im(x)) + 2*||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*--(im(x))*cosh(im(x))*sin(re(x)) + 2*||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*--(re(x))*cos(re(x))*sinh(im(x)) + 2*||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*--(re(x))*cosh(im(x))*sin(re(x)) + 2*|--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x)) + 2*|--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*cos(x)     //d                                  d                               \                          /d                                  d                               \                       \             //d                                  d                               \                          /d                                  d                               \                       \          3*|||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*sin(re(x))*sinh(im(x)) - ||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cosh(im(x))*sin(re(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cos(re(x))*sinh(im(x)) + |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cosh(im(x))*sin(re(x)) - |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cos(re(x))*sinh(im(x))|*sin(x)      //d                                  d                               \                          /d                                  d                               \                       \           9*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))|*|||--(im(x))| *sin(re(x))*sinh(im(x)) + ---(im(x))*cosh(im(x))*sin(re(x)) + ---(re(x))*cos(re(x))*sinh(im(x)) - |--(re(x))| *sin(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*cos(re(x))*cosh(im(x))|*sin(re(x))*sinh(im(x)) - ||--(re(x))| *cos(re(x))*cosh(im(x)) + ---(re(x))*cosh(im(x))*sin(re(x)) - |--(im(x))| *cos(re(x))*cosh(im(x)) - ---(im(x))*cos(re(x))*sinh(im(x)) + 2*--(im(x))*--(re(x))*sin(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cosh(im(x))*sin(re(x)) + |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cos(re(x))*sinh(im(x)) + |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(re(x))*cosh(im(x))*sin(re(x)) - |--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*--(im(x))*cos(re(x))*sinh(im(x))|*cos(x)         
  ||\dx       /                             3                                   3                                 \dx       /                             \dx       /  dx                                   dx          2                                   \dx       /  dx                                   dx          2                                   dx          2                                   dx          2                              |                          |  3                                 \dx       /                           \dx       /                             3                                   dx          2                                   \dx       /  dx                                   \dx       /  dx                                   dx          2                                   dx          2                                   dx          2                              |                          \dx       /  \dx                                 dx                              /                          \dx       /  \dx                                 dx                              /                          \dx                                 dx                              /   2                                 \dx                                 dx                              /   2                                 \dx                                 dx                              /   2                                 \dx       /  \dx                                 dx                              /                          \dx       /  \dx                                 dx                              /                          \dx                                 dx                              /   2                                   |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              | dx                                   |\dx       /                             2                                   2                                 \dx       /                             dx        dx                              | dx                                   |\dx       /                             2                                   2                                 \dx       /                             dx        dx                              | dx                                   |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              | dx                                   \dx                                 dx                              / dx        dx                                   \dx                                 dx                              / dx        dx                              |          9*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))| *sin(x)   3*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))|*cos(x)     ||\dx       /                             2                                   2                                 \dx       /                             dx        dx                              |                          |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              |                          \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                              |          15*||--(re(x))*cosh(im(x))*sin(re(x)) - --(im(x))*cos(re(x))*sinh(im(x))|*cos(re(x))*cosh(im(x)) - |--(im(x))*cosh(im(x))*sin(re(x)) + --(re(x))*cos(re(x))*sinh(im(x))|*sin(re(x))*sinh(im(x))| *cos(x)     \\dx                                 dx                              /                          \dx                                 dx                              /                       / ||\dx       /                             2                                   2                                 \dx       /                             dx        dx                              |                          |\dx       /                             2                                 \dx       /                             2                                   dx        dx                              |                          \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                                 \dx                                 dx                              / dx                              |                
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- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + sin(x)
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                                                                                                                                                                                                                                                                                                                                    |cos(x)|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |cos(x)|                                     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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |cos(x)|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           

$$\frac{1}{\left|{\cos{\left (x \right )}}\right|} \left(\frac{15 \cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{6}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right)^{3} - \frac{9 \sin{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{4}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right)^{2} - \frac{9 \cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{4}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right) \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} - \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )}\right) - \frac{3 \cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{2}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )}\right) + \frac{3 \sin{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{2}} \left(\left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} - \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )}\right) - \frac{\cos{\left (x \right )}}{\left|{\cos{\left (x \right )}}\right|^{2}} \left(2 \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} - \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \left(\sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + 2 \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + 2 \left(- \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} + \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} + 2 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} + 2 \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} - 2 \left(2 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Re{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{2}}{d x^{2}} \Im{x} + \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} - \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{2}\right) \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} + \left(- 3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} + 3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x} - 3 \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} \frac{d}{d x} \Im{x} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{3} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Im{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{3} + 3 \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}^{2} + \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Re{x} + 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Im{x} + 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x}\right) \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} - \left(3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Im{x} + 3 \sin{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Re{x} - \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{3} + 3 \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d}{d x} \Im{x}^{2} + \sin{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Re{x} + 3 \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x}^{2} \frac{d}{d x} \Im{x} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x}^{3} - \cos{\left (\Re{x} \right )} \sinh{\left (\Im{x} \right )} \frac{d^{3}}{d x^{3}} \Im{x} + 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Re{x} \frac{d^{2}}{d x^{2}} \Re{x} - 3 \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )} \frac{d}{d x} \Im{x} \frac{d^{2}}{d x^{2}} \Im{x}\right) \cos{\left (\Re{x} \right )} \cosh{\left (\Im{x} \right )}\right) + \sin{\left (x \right )}\right)$$