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ysin²xlny+y=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 y*sin (x)*log(y) + y = 0
$$y + y \sin^{2}{\left(x \right)} \log{\left(y \right)} = 0$$
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 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
The equation has two roots.
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:
$$y + y \sin^{2}{\left(x \right)} \log{\left(y \right)} = 0$$
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))/
---------------------------------------------------------------------
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)