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1/6x3^3-2x la ecuación
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Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$- 2 x + \frac{x_{3}^{3}}{6} = 0$$
de
$$a x_{3}^{3} + b x_{3}^{2} + c x_{3} + d = 0$$
como ecuación cúbica reducida
$$x_{3}^{3} + \frac{b x_{3}^{2}}{a} + \frac{c x_{3}}{a} + \frac{d}{a} = 0$$
$$- 12 x + x_{3}^{3} = 0$$
$$p x_{3}^{2} + q x_{3} + v + x_{3}^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - 12 x$$
Fórmulas de Cardano-Vieta
$$x_{3 1} + x_{3 2} + x_{3 3} = - p$$
$$x_{3 1} x_{3 2} + x_{3 1} x_{3 3} + x_{3 2} x_{3 3} = q$$
$$x_{3 1} x_{3 2} x_{3 3} = v$$
$$x_{3 1} + x_{3 2} + x_{3 3} = 0$$
$$x_{3 1} x_{3 2} + x_{3 1} x_{3 3} + x_{3 2} x_{3 3} = 0$$
$$x_{3 1} x_{3 2} x_{3 3} = - 12 x$$
$$- 2 x + \frac{x_{3}^{3}}{6} = 0$$
de
$$a x_{3}^{3} + b x_{3}^{2} + c x_{3} + d = 0$$
como ecuación cúbica reducida
$$x_{3}^{3} + \frac{b x_{3}^{2}}{a} + \frac{c x_{3}}{a} + \frac{d}{a} = 0$$
$$- 12 x + x_{3}^{3} = 0$$
$$p x_{3}^{2} + q x_{3} + v + x_{3}^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - 12 x$$
Fórmulas de Cardano-Vieta
$$x_{3 1} + x_{3 2} + x_{3 3} = - p$$
$$x_{3 1} x_{3 2} + x_{3 1} x_{3 3} + x_{3 2} x_{3 3} = q$$
$$x_{3 1} x_{3 2} x_{3 3} = v$$
$$x_{3 1} + x_{3 2} + x_{3 3} = 0$$
$$x_{3 1} x_{3 2} + x_{3 1} x_{3 3} + x_{3 2} x_{3 3} = 0$$
$$x_{3 1} x_{3 2} x_{3 3} = - 12 x$$
Gráfica
Respuesta rápida
[src]
_________________ _________________
3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 3 ____ 6 / 2 2 /atan2(im(x), re(x))\
x31 = \/ 12 *\/ im (x) + re (x) *cos|-------------------| + I*\/ 12 *\/ im (x) + re (x) *sin|-------------------|
\ 3 / \ 3 /
$$x_{31} = \sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}$$
/ _________________ _________________ \ _________________ _________________
| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\
| \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| \/ 12 *\/ im (x) + re (x) *cos|-------------------| 2 *3 *\/ im (x) + re (x) *sin|-------------------|
| \ 3 / \ 3 /| \ 3 / \ 3 /
x32 = I*|- ---------------------------------------------------- - -------------------------------------------------------| - ---------------------------------------------------- + -------------------------------------------------------
\ 2 2 / 2 2
$$x_{32} = i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}$$
/ _________________ _________________ \ _________________ _________________
| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\
| \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| \/ 12 *\/ im (x) + re (x) *cos|-------------------| 2 *3 *\/ im (x) + re (x) *sin|-------------------|
| \ 3 / \ 3 /| \ 3 / \ 3 /
x33 = I*|- ---------------------------------------------------- + -------------------------------------------------------| - ---------------------------------------------------- - -------------------------------------------------------
\ 2 2 / 2 2
$$x_{33} = i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}$$
x33 = i*(-12^(1/3)*(re(x)^2 + im(x)^2)^(1/6)*sin(atan2(im(x, re(x))/3)/2 + 2^(2/3)*3^(5/6)*(re(x)^2 + im(x)^2)^(1/6)*cos(atan2(im(x), re(x))/3)/2) - 2^(2/3)*3^(5/6)*(re(x)^2 + im(x)^2)^(1/6)*sin(atan2(im(x), re(x))/3)/2 - 12^(1/3)*(re(x)^2 + im(x)^2)^(1/6)*cos(atan2(im(x), re(x))/3)/2)
Suma y producto de raíces
[src]
suma
/ _________________ _________________ \ _________________ _________________ / _________________ _________________ \ _________________ _________________
| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\ | 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\
_________________ _________________ | \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| \/ 12 *\/ im (x) + re (x) *cos|-------------------| 2 *3 *\/ im (x) + re (x) *sin|-------------------| | \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| \/ 12 *\/ im (x) + re (x) *cos|-------------------| 2 *3 *\/ im (x) + re (x) *sin|-------------------|
3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ | \ 3 / \ 3 /| \ 3 / \ 3 / | \ 3 / \ 3 /| \ 3 / \ 3 /
\/ 12 *\/ im (x) + re (x) *cos|-------------------| + I*\/ 12 *\/ im (x) + re (x) *sin|-------------------| + I*|- ---------------------------------------------------- - -------------------------------------------------------| - ---------------------------------------------------- + ------------------------------------------------------- + I*|- ---------------------------------------------------- + -------------------------------------------------------| - ---------------------------------------------------- - -------------------------------------------------------
\ 3 / \ 3 / \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(\left(\sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}\right) + \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right)\right) + \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right)$$
=
/ _________________ _________________ \ / _________________ _________________ \ | 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| | 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| | \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| | \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| _________________ | \ 3 / \ 3 /| | \ 3 / \ 3 /| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ I*|- ---------------------------------------------------- + -------------------------------------------------------| + I*|- ---------------------------------------------------- - -------------------------------------------------------| + I*\/ 12 *\/ im (x) + re (x) *sin|-------------------| \ 2 2 / \ 2 2 / \ 3 /
$$i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}$$
producto
/ / _________________ _________________ \ _________________ _________________ \ / / _________________ _________________ \ _________________ _________________ \
| | 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| | | 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\| 3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 2/3 5/6 6 / 2 2 /atan2(im(x), re(x))\|
/ _________________ _________________ \ | | \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| \/ 12 *\/ im (x) + re (x) *cos|-------------------| 2 *3 *\/ im (x) + re (x) *sin|-------------------|| | | \/ 12 *\/ im (x) + re (x) *sin|-------------------| 2 *3 *\/ im (x) + re (x) *cos|-------------------|| \/ 12 *\/ im (x) + re (x) *cos|-------------------| 2 *3 *\/ im (x) + re (x) *sin|-------------------||
|3 ____ 6 / 2 2 /atan2(im(x), re(x))\ 3 ____ 6 / 2 2 /atan2(im(x), re(x))\| | | \ 3 / \ 3 /| \ 3 / \ 3 /| | | \ 3 / \ 3 /| \ 3 / \ 3 /|
|\/ 12 *\/ im (x) + re (x) *cos|-------------------| + I*\/ 12 *\/ im (x) + re (x) *sin|-------------------||*|I*|- ---------------------------------------------------- - -------------------------------------------------------| - ---------------------------------------------------- + -------------------------------------------------------|*|I*|- ---------------------------------------------------- + -------------------------------------------------------| - ---------------------------------------------------- - -------------------------------------------------------|
\ \ 3 / \ 3 // \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(\sqrt[3]{12} i \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}\right) \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) \left(i \left(- \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} + \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right) - \frac{2^{\frac{2}{3}} \cdot 3^{\frac{5}{6}} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[3]{12} \sqrt[6]{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{2}\right)$$
=
/ / im(x) /atan2(im(x), re(x))\\ \
| 3*I*|-------------------- + sin|-------------------|| |
| /atan2(im(x), re(x))\ | _________________ \ 3 /| |
_________________ | 3*cos|-------------------| | / 2 2 | |
/ 2 2 | 3/atan2(im(x), re(x))\ \ 3 / 3/atan2(im(x), re(x))\ \\/ im (x) + re (x) / 3*re(x) |
12*\/ im (x) + re (x) *|cos |-------------------| - -------------------------- - I*sin |-------------------| + ----------------------------------------------------- + ----------------------|
| \ 3 / 4 \ 3 / 4 _________________|
| / 2 2 |
\ 4*\/ im (x) + re (x) /
$$12 \sqrt{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} \left(\frac{3 i \left(\sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \frac{\operatorname{im}{\left(x\right)}}{\sqrt{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}\right)}{4} - i \sin^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} + \cos^{3}{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)} - \frac{3 \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(x\right)},\operatorname{re}{\left(x\right)} \right)}}{3} \right)}}{4} + \frac{3 \operatorname{re}{\left(x\right)}}{4 \sqrt{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}}}\right)$$
12*sqrt(im(x)^2 + re(x)^2)*(cos(atan2(im(x), re(x))/3)^3 - 3*cos(atan2(im(x), re(x))/3)/4 - i*sin(atan2(im(x), re(x))/3)^3 + 3*i*(im(x)/sqrt(im(x)^2 + re(x)^2) + sin(atan2(im(x), re(x))/3))/4 + 3*re(x)/(4*sqrt(im(x)^2 + re(x)^2)))