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ysin²xlny+y=0 equation

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Numerical solution:

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The solution

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     2                  
y*sin (x)*log(y) + y = 0
$$y + y \sin^{2}{\left(x \right)} \log{\left(y \right)} = 0$$
Simplify
Detail solution
Given the equation
$$y + y \sin^{2}{\left(x \right)} \log{\left(y \right)} = 0$$
transform
$$y \log{\left(y \right)} \sin^{2}{\left(x \right)} + y - 1 = 0$$
$$y + y \sin^{2}{\left(x \right)} \log{\left(y \right)} - 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = y \log{\left(y \right)}$$
$$b = 0$$
$$c = y - 1$$
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (y*log(y)) * (-1 + y) = -4*y*(-1 + y)*log(y)

The equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

or
$$w_{1} = \frac{\sqrt{- y \left(y - 1\right) \log{\left(y \right)}}}{y \log{\left(y \right)}}$$
$$w_{2} = - \frac{\sqrt{- y \left(y - 1\right) \log{\left(y \right)}}}{y \log{\left(y \right)}}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
substitute w:
The graph
Rapid solution [src] 
                                                                                                                                                                 2            2                                                                                               2           2                                                                                                 2            2                                                                                               2           2                                                                                                                                                              
                                                                                                                                                              cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                       cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                                                                                       
                                                                                                                     --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------      --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------                                                                                                               
                                                                                                                        4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                                                                                                                      
        /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \  cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))      cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))    /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \
y1 = cos|---------------------------------------------------------------------------------------------------------|*e                                                                                                                                                                                                                      + I*e                                                                                                                                                                                                                     *sin|---------------------------------------------------------------------------------------------------------|
        |   4            4              4           4               2            2           2            2       |                                                                                                                                                                                                                                                                                                                                                                                                                                                      |   4            4              4           4               2            2           2            2       |
        \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/                                                                                                                                                                                                                                                                                                                                                                                                                                                      \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/
$$y_{1} = i e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \sin{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)} + e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \cos{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)}$$ 
y1 = i*exp(-sin(re(x))^2*cosh(im(x))^2/(sin(re(x))^4*cosh(im(x))^4 + 2*sin(re(x))^2*cos(re(x))^2*sinh(im(x))^2*cosh(im(x))^2 + cos(re(x))^4*sinh(im(x))^4) + cos(re(x))^2*sinh(im(x))^2/(sin(re(x))^4*cosh(im(x))^4 + 2*sin(re(x))^2*cos(re(x))^2*sinh(im(x))^2*cosh(im(x))^2 + cos(re(x))^4*sinh(im(x))^4))*sin(2*sin(re(x))*cos(re(x))*sinh(im(x))*cosh(im(x))/(sin(re(x))^4*cosh(im(x))^4 + 2*sin(re(x))^2*cos(re(x))^2*sinh(im(x))^2*cosh(im(x))^2 + cos(re(x))^4*sinh(im(x))^4)) + exp(-sin(re(x))^2*cosh(im(x))^2/(sin(re(x))^4*cosh(im(x))^4 + 2*sin(re(x))^2*cos(re(x))^2*sinh(im(x))^2*cosh(im(x))^2 + cos(re(x))^4*sinh(im(x))^4) + cos(re(x))^2*sinh(im(x))^2/(sin(re(x))^4*cosh(im(x))^4 + 2*sin(re(x))^2*cos(re(x))^2*sinh(im(x))^2*cosh(im(x))^2 + cos(re(x))^4*sinh(im(x))^4))*cos(2*sin(re(x))*cos(re(x))*sinh(im(x))*cosh(im(x))/(sin(re(x))^4*cosh(im(x))^4 + 2*sin(re(x))^2*cos(re(x))^2*sinh(im(x))^2*cosh(im(x))^2 + cos(re(x))^4*sinh(im(x))^4))
Sum and product of roots [src] 
sum
                                                                                                                                                            2            2                                                                                               2           2                                                                                                 2            2                                                                                               2           2                                                                                                                                                              
                                                                                                                                                         cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                       cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                                                                                       
                                                                                                                --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------      --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------                                                                                                               
                                                                                                                   4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                                                                                                                      
   /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \  cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))      cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))    /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \
cos|---------------------------------------------------------------------------------------------------------|*e                                                                                                                                                                                                                      + I*e                                                                                                                                                                                                                     *sin|---------------------------------------------------------------------------------------------------------|
   |   4            4              4           4               2            2           2            2       |                                                                                                                                                                                                                                                                                                                                                                                                                                                      |   4            4              4           4               2            2           2            2       |
   \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/                                                                                                                                                                                                                                                                                                                                                                                                                                                      \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/
$$i e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \sin{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)} + e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \cos{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)}$$ 
=
                                                                                                                                                            2            2                                                                                               2           2                                                                                                 2            2                                                                                               2           2                                                                                                                                                              
                                                                                                                                                         cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                       cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                                                                                       
                                                                                                                --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------      --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------                                                                                                               
                                                                                                                   4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                                                                                                                      
   /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \  cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))      cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))    /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \
cos|---------------------------------------------------------------------------------------------------------|*e                                                                                                                                                                                                                      + I*e                                                                                                                                                                                                                     *sin|---------------------------------------------------------------------------------------------------------|
   |   4            4              4           4               2            2           2            2       |                                                                                                                                                                                                                                                                                                                                                                                                                                                      |   4            4              4           4               2            2           2            2       |
   \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/                                                                                                                                                                                                                                                                                                                                                                                                                                                      \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/
$$i e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \sin{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)} + e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \cos{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)}$$ 
product
                                                                                                                                                            2            2                                                                                               2           2                                                                                                 2            2                                                                                               2           2                                                                                                                                                              
                                                                                                                                                         cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                       cos (re(x))*sinh (im(x))                                                                                    cosh (im(x))*sin (re(x))                                                                                                                                                       
                                                                                                                --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------      --------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------                                                                                                               
                                                                                                                   4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                4            4              4           4               2            2           2            2             4            4              4           4               2            2           2            2                                                                                                                      
   /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \  cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))      cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))   cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))    /                             2*cos(re(x))*cosh(im(x))*sin(re(x))*sinh(im(x))                             \
cos|---------------------------------------------------------------------------------------------------------|*e                                                                                                                                                                                                                      + I*e                                                                                                                                                                                                                     *sin|---------------------------------------------------------------------------------------------------------|
   |   4            4              4           4               2            2           2            2       |                                                                                                                                                                                                                                                                                                                                                                                                                                                      |   4            4              4           4               2            2           2            2       |
   \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/                                                                                                                                                                                                                                                                                                                                                                                                                                                      \cos (re(x))*sinh (im(x)) + cosh (im(x))*sin (re(x)) + 2*cos (re(x))*cosh (im(x))*sin (re(x))*sinh (im(x))/
$$i e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \sin{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)} + e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} + \frac{\cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}}} \cos{\left(\frac{2 \sin{\left(\operatorname{re}{\left(x\right)} \right)} \cos{\left(\operatorname{re}{\left(x\right)} \right)} \sinh{\left(\operatorname{im}{\left(x\right)} \right)} \cosh{\left(\operatorname{im}{\left(x\right)} \right)}}{\sin^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{4}{\left(\operatorname{im}{\left(x\right)} \right)} + 2 \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \cos^{4}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{4}{\left(\operatorname{im}{\left(x\right)} \right)}} \right)}$$ 
=
                                                                                           /    2           2             2            2       \        
                                                                                          -\cosh (im(x))*sin (re(x)) - cos (re(x))*sinh (im(x))/        
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                                                                                                                                                       2
                                                                                   /    2              2           2             2            2       \ 
/     /  2*sin(2*re(x))*sinh(2*im(x))  \      /  2*sin(2*re(x))*sinh(2*im(x))  \\  \sinh (im(x)) + cosh (im(x))*sin (re(x)) - sin (re(x))*sinh (im(x))/ 
|I*sin|--------------------------------| + cos|--------------------------------||*e                                                                     
|     |                               2|      |                               2||                                                                       
\     \(-cosh(2*im(x)) + cos(2*re(x))) /      \(-cosh(2*im(x)) + cos(2*re(x))) //
$$\left(i \sin{\left(\frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} - \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)^{2}} \right)} + \cos{\left(\frac{2 \sin{\left(2 \operatorname{re}{\left(x\right)} \right)} \sinh{\left(2 \operatorname{im}{\left(x\right)} \right)}}{\left(\cos{\left(2 \operatorname{re}{\left(x\right)} \right)} - \cosh{\left(2 \operatorname{im}{\left(x\right)} \right)}\right)^{2}} \right)}\right) e^{- \frac{\sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} - \cos^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}}{\left(- \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \sin^{2}{\left(\operatorname{re}{\left(x\right)} \right)} \cosh^{2}{\left(\operatorname{im}{\left(x\right)} \right)} + \sinh^{2}{\left(\operatorname{im}{\left(x\right)} \right)}\right)^{2}}}$$ 
(i*sin(2*sin(2*re(x))*sinh(2*im(x))/(-cosh(2*im(x)) + cos(2*re(x)))^2) + cos(2*sin(2*re(x))*sinh(2*im(x))/(-cosh(2*im(x)) + cos(2*re(x)))^2))*exp(-(cosh(im(x))^2*sin(re(x))^2 - cos(re(x))^2*sinh(im(x))^2)/(sinh(im(x))^2 + cosh(im(x))^2*sin(re(x))^2 - sin(re(x))^2*sinh(im(x))^2)^2)
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