Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x/4)*sin(x/4)/(2*(1+cos(x/4)^2))+sin(x/4)^3*cos(x/4)/(2*(1+cos(x/4)^2)^2)
¿Cómo vas a descomponer esta cos(x/4)*sin(x/4)/(2*(1+cos(x/4)^2))+sin(x/4)^3*cos(x/4)/(2*(1+cos(x/4)^2)^2) expresión en fracciones?
Solución
Ha introducido
[src]
/x\ /x\ 3/x\ /x\
cos|-|*sin|-| sin |-|*cos|-|
\4/ \4/ \4/ \4/
--------------- + ----------------
/ 2/x\\ 2
2*|1 + cos |-|| / 2/x\\
\ \4// 2*|1 + cos |-||
\ \4//
$$\frac{\sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)} + \frac{\sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
(cos(x/4)*sin(x/4))/((2*(1 + cos(x/4)^2))) + (sin(x/4)^3*cos(x/4))/((2*(1 + cos(x/4)^2)^2))
Simplificación general
[src]
/x\
2*sin|-|
\2/
-------------
2
/ /x\\
|3 + cos|-||
\ \2//
$$\frac{2 \sin{\left(\frac{x}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
2*sin(x/2)/(3 + cos(x/2))^2
Unión de expresiones racionales
[src]
/ 2/x\ 2/x\\ /x\ /x\
|1 + cos |-| + sin |-||*cos|-|*sin|-|
\ \4/ \4// \4/ \4/
-------------------------------------
2
/ 2/x\\
2*|1 + cos |-||
\ \4//
$$\frac{\left(\sin^{2}{\left(\frac{x}{4} \right)} + \cos^{2}{\left(\frac{x}{4} \right)} + 1\right) \sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
(1 + cos(x/4)^2 + sin(x/4)^2)*cos(x/4)*sin(x/4)/(2*(1 + cos(x/4)^2)^2)
Respuesta numérica
[src]
cos(x/4)*sin(x/4)/(2.0 + 2.0*cos(x/4)^2) + 0.5*sin(x/4)^3*cos(x/4)/(1.0 + cos(x/4)^2)^2
cos(x/4)*sin(x/4)/(2.0 + 2.0*cos(x/4)^2) + 0.5*sin(x/4)^3*cos(x/4)/(1.0 + cos(x/4)^2)^2
Combinatoria
[src]
/ 2/x\ 2/x\\ /x\ /x\
|1 + cos |-| + sin |-||*cos|-|*sin|-|
\ \4/ \4// \4/ \4/
-------------------------------------
2
/ 2/x\\
2*|1 + cos |-||
\ \4//
$$\frac{\left(\sin^{2}{\left(\frac{x}{4} \right)} + \cos^{2}{\left(\frac{x}{4} \right)} + 1\right) \sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
(1 + cos(x/4)^2 + sin(x/4)^2)*cos(x/4)*sin(x/4)/(2*(1 + cos(x/4)^2)^2)
Denominador común
[src]
3/x\ /x\ 3/x\ /x\ /x\ /x\
cos |-|*sin|-| + sin |-|*cos|-| + cos|-|*sin|-|
\4/ \4/ \4/ \4/ \4/ \4/
-----------------------------------------------
4/x\ 2/x\
2 + 2*cos |-| + 4*cos |-|
\4/ \4/
$$\frac{\sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)} + \sin{\left(\frac{x}{4} \right)} \cos^{3}{\left(\frac{x}{4} \right)} + \sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \cos^{4}{\left(\frac{x}{4} \right)} + 4 \cos^{2}{\left(\frac{x}{4} \right)} + 2}$$
(cos(x/4)^3*sin(x/4) + sin(x/4)^3*cos(x/4) + cos(x/4)*sin(x/4))/(2 + 2*cos(x/4)^4 + 4*cos(x/4)^2)
Denominador racional
[src]
2
3/x\ / 2/x\\ /x\ / 2/x\\ /x\ /x\
sin |-|*|2 + 2*cos |-||*cos|-| + 2*|1 + cos |-|| *cos|-|*sin|-|
\4/ \ \4// \4/ \ \4// \4/ \4/
---------------------------------------------------------------
2
/ 2/x\\ / 2/x\\
2*|1 + cos |-|| *|2 + 2*cos |-||
\ \4// \ \4//
$$\frac{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)} + \left(2 \cos^{2}{\left(\frac{x}{4} \right)} + 2\right) \sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(2 \cos^{2}{\left(\frac{x}{4} \right)} + 2\right)}$$
(sin(x/4)^3*(2 + 2*cos(x/4)^2)*cos(x/4) + 2*(1 + cos(x/4)^2)^2*cos(x/4)*sin(x/4))/(2*(1 + cos(x/4)^2)^2*(2 + 2*cos(x/4)^2))
Compilar la expresión
[src]
/x\ /x\ 3/x\ /x\
cos|-|*sin|-| sin |-|*cos|-|
\4/ \4/ \4/ \4/
------------- + ----------------
2/x\ 2
2 + 2*cos |-| / 2/x\\
\4/ 2*|1 + cos |-||
\ \4//
$$\frac{\sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \cos^{2}{\left(\frac{x}{4} \right)} + 2} + \frac{\sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
cos(x/4)*sin(x/4)/(2 + 2*cos(x/4)^2) + sin(x/4)^3*cos(x/4)/(2*(1 + cos(x/4)^2)^2)
Potencias
[src]
/x\ /x\ 3/x\ /x\
cos|-|*sin|-| sin |-|*cos|-|
\4/ \4/ \4/ \4/
------------- + ----------------
2/x\ 2
2 + 2*cos |-| / 2/x\\
\4/ 2*|1 + cos |-||
\ \4//
$$\frac{\sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \cos^{2}{\left(\frac{x}{4} \right)} + 2} + \frac{\sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
/ -I*x I*x\ 3 / -I*x I*x\
| ----- ---| / -I*x I*x\ / -I*x I*x\ | ----- ---|
| 4 4 | | ----- ---| | ----- ---| | 4 4 |
|e e | | 4 4 | | 4 4 | |e e |
I*|------ + ----|*\- e + e / I*\- e + e / *|------ + ----|
\ 2 2 / \ 2 2 /
- ----------------------------------- + ------------------------------------
/ 2\ 2
| / -I*x I*x\ | / 2\
| | ----- ---| | | / -I*x I*x\ |
| | 4 4 | | | | ----- ---| |
| |e e | | | | 4 4 | |
2*|2 + 2*|------ + ----| | | |e e | |
\ \ 2 2 / / 16*|1 + |------ + ----| |
\ \ 2 2 / /
$$- \frac{i \left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right) \left(e^{\frac{i x}{4}} - e^{- \frac{i x}{4}}\right)}{2 \left(2 \left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right)^{2} + 2\right)} + \frac{i \left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right) \left(e^{\frac{i x}{4}} - e^{- \frac{i x}{4}}\right)^{3}}{16 \left(\left(\frac{e^{\frac{i x}{4}}}{2} + \frac{e^{- \frac{i x}{4}}}{2}\right)^{2} + 1\right)^{2}}$$
-i*(exp(-i*x/4)/2 + exp(i*x/4)/2)*(-exp(-i*x/4) + exp(i*x/4))/(2*(2 + 2*(exp(-i*x/4)/2 + exp(i*x/4)/2)^2)) + i*(-exp(-i*x/4) + exp(i*x/4))^3*(exp(-i*x/4)/2 + exp(i*x/4)/2)/(16*(1 + (exp(-i*x/4)/2 + exp(i*x/4)/2)^2)^2)
Parte trigonométrica
[src]
4/x\ / 2/x\\ /x\ 8/x\ 3/x\ / 2/x\\
2*cos |-|*|1 - tan |-||*tan|-| 16*cos |-|*tan |-|*|1 - tan |-||
\8/ \ \8// \8/ \8/ \8/ \ \8//
------------------------------ + --------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{4}{\left(\frac{x}{8} \right)} \tan{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{8}{\left(\frac{x}{8} \right)} \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
/x\
6*sin|-| + sin(x)
\2/
------------------
3
/ /pi x\\
|3 + sin|-- + -||
\ \2 2//
$$\frac{6 \sin{\left(\frac{x}{2} \right)} + \sin{\left(x \right)}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3\right)^{3}}$$
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 8/| | \2 8/|
2*|1 - ------------| 16*|1 - ------------|
| 2/x\ | | 2/x\ |
| csc |-| | | csc |-| |
\ \8/ / \ \8/ /
------------------------------------- + ---------------------------------------
/ 1 \ /x\ 3/pi x\ 2
|3 + -----------|*csc|-|*csc |-- - -| / 1 \ 3/x\ 5/pi x\
| /pi x\| \8/ \2 8/ |3 + -----------| *csc |-|*csc |-- - -|
| csc|-- - -|| | /pi x\| \8/ \2 8/
\ \2 2// | csc|-- - -||
\ \2 2//
$$\frac{2 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right) \csc{\left(\frac{x}{8} \right)} \csc^{3}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} + \frac{16 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{2} \csc^{3}{\left(\frac{x}{8} \right)} \csc^{5}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}$$
4/x \ 8/x \ / 2/x\\ /x\ 3/x\ 8/x \ 16/x \ / 2/x\\
32*cot |--|*sin |--|*|-1 + cot |-||*cot|-| 4096*cot |-|*cot |--|*sin |--|*|-1 + cot |-||
\16/ \16/ \ \8// \8/ \8/ \16/ \16/ \ \8//
------------------------------------------ + ----------------------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{32 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{8}{\left(\frac{x}{16} \right)} \cot^{4}{\left(\frac{x}{16} \right)} \cot{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{4096 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{16}{\left(\frac{x}{16} \right)} \cot^{8}{\left(\frac{x}{16} \right)} \cot^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
/x\ /x\
2*cot|-| 12*cot|-|
\2/ \4/
----------- + -----------
2/x\ 2/x\
1 + cot |-| 1 + cot |-|
\2/ \4/
-------------------------
3
/ 2/x\\
| -1 + cot |-||
| \4/|
|3 + ------------|
| 2/x\ |
| 1 + cot |-| |
\ \4/ /
$$\frac{\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{12 \cot{\left(\frac{x}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{3}}$$
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \8/ | | \8/ |
2*|1 - --------------| 16*|1 - --------------|
| 2/ pi x\| | 2/ pi x\|
| sec |- -- + -|| | sec |- -- + -||
\ \ 2 8// \ \ 2 8//
---------------------------------- + ------------------------------------
/ 1 \ 3/x\ / pi x\ 2
|3 + ------|*sec |-|*sec|- -- + -| / 1 \ 5/x\ 3/ pi x\
| /x\| \8/ \ 2 8/ |3 + ------| *sec |-|*sec |- -- + -|
| sec|-|| | /x\| \8/ \ 2 8/
\ \2// | sec|-||
\ \2//
$$\frac{2 \left(- \frac{\sec^{2}{\left(\frac{x}{8} \right)}}{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + 1\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right) \sec^{3}{\left(\frac{x}{8} \right)} \sec{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + \frac{16 \left(- \frac{\sec^{2}{\left(\frac{x}{8} \right)}}{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + 1\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{5}{\left(\frac{x}{8} \right)} \sec^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}$$
/ 2/x\\ /x\ 3/x\ / 2/x\\
2*|1 - tan |-||*tan|-| 4*tan |-|*|1 - tan |-||
\ \8// \8/ \8/ \ \8//
------------------------------------- + ------------------------------------
/ 2\ 2
| / 2/x\\ | / 2\
2 | 2*|1 - tan |-|| | | / 2/x\\ |
/ 2/x\\ | \ \8// | 4 | |1 - tan |-|| |
|1 + tan |-|| *|2 + ----------------| / 2/x\\ | \ \8// |
\ \8// | 2 | |1 + tan |-|| *|1 + --------------|
| / 2/x\\ | \ \8// | 2|
| |1 + tan |-|| | | / 2/x\\ |
\ \ \8// / | |1 + tan |-|| |
\ \ \8// /
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \tan{\left(\frac{x}{8} \right)}}{\left(\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + 2\right) \left(\tan^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{8} \right)} + 1\right)^{4}}$$
4 8
/ 2/x \\ 8/x \ / 2/x\\ /x\ / 2/x \\ 16/x \ 3/x\ / 2/x\\
2*|1 - tan |--|| *cos |--|*|1 - tan |-||*tan|-| 16*|1 - tan |--|| *cos |--|*tan |-|*|1 - tan |-||
\ \16// \16/ \ \8// \8/ \ \16// \16/ \8/ \ \8//
----------------------------------------------- + --------------------------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{16 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{8} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{16}{\left(\frac{x}{16} \right)} \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \cos^{8}{\left(\frac{x}{16} \right)} \tan{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3}$$
4/x \ / 1 \ 8/x \ / 1 \
32*tan |--|*|-1 + -------| 4096*tan |--|*|-1 + -------|
\16/ | 2/x\| \16/ | 2/x\|
| tan |-|| | tan |-||
\ \8// \ \8//
---------------------------------------- + ------------------------------------------
/ 2/x\\ 2
4 | 1 - tan |-|| / 2/x\\
/ 2/x \\ | \4/| /x\ 8 | 1 - tan |-||
|1 + tan |--|| *|3 + -----------|*tan|-| / 2/x \\ | \4/| 3/x\
\ \16// | 2/x\| \8/ |1 + tan |--|| *|3 + -----------| *tan |-|
| 1 + tan |-|| \ \16// | 2/x\| \8/
\ \4// | 1 + tan |-||
\ \4//
$$\frac{32 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \tan^{4}{\left(\frac{x}{16} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4} \tan{\left(\frac{x}{8} \right)}} + \frac{4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \tan^{8}{\left(\frac{x}{16} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8} \tan^{3}{\left(\frac{x}{8} \right)}}$$
/x\ /pi x\ 3/x\ /pi x\
sin|-|*sin|-- + -| sin |-|*sin|-- + -|
\4/ \2 4/ \4/ \2 4/
------------------ + ---------------------
2/pi x\ 2
2 + 2*sin |-- + -| / 2/pi x\\
\2 4/ 2*|1 + sin |-- + -||
\ \2 4//
$$\frac{\sin{\left(\frac{x}{4} \right)} \sin{\left(\frac{x}{4} + \frac{\pi}{2} \right)}}{2 \sin^{2}{\left(\frac{x}{4} + \frac{\pi}{2} \right)} + 2} + \frac{\sin^{3}{\left(\frac{x}{4} \right)} \sin{\left(\frac{x}{4} + \frac{\pi}{2} \right)}}{2 \left(\sin^{2}{\left(\frac{x}{4} + \frac{\pi}{2} \right)} + 1\right)^{2}}$$
4/x \ / 2/x\\ /x\ 3/x\ 8/x \ / 2/x\\
32*cot |--|*|-1 + cot |-||*cot|-| 4096*cot |-|*cot |--|*|-1 + cot |-||
\16/ \ \8// \8/ \8/ \16/ \ \8//
---------------------------------- + ------------------------------------
/ 2/x\\ 2
4 | -1 + cot |-|| / 2/x\\
/ 2/x \\ | \4/| 8 | -1 + cot |-||
|1 + cot |--|| *|3 + ------------| / 2/x \\ | \4/|
\ \16// | 2/x\ | |1 + cot |--|| *|3 + ------------|
| 1 + cot |-| | \ \16// | 2/x\ |
\ \4/ / | 1 + cot |-| |
\ \4/ /
$$\frac{32 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \cot^{4}{\left(\frac{x}{16} \right)} \cot{\left(\frac{x}{8} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4}} + \frac{4096 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \cot^{8}{\left(\frac{x}{16} \right)} \cot^{3}{\left(\frac{x}{8} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8}}$$
/ 2/x\ \ / 2/x\ \
| cos |-| | | cos |-| |
3/ pi x\ | \8/ | /x\ 3/x\ 5/ pi x\ | \8/ |
2*cos |- -- + -|*|-1 + --------------|*cos|-| 16*cos |-|*cos |- -- + -|*|-1 + --------------|
\ 2 8/ | 2/ pi x\| \8/ \8/ \ 2 8/ | 2/ pi x\|
| cos |- -- + -|| | cos |- -- + -||
\ \ 2 8// \ \ 2 8//
--------------------------------------------- + -----------------------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{2 \left(\frac{\cos^{2}{\left(\frac{x}{8} \right)}}{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(\frac{\cos^{2}{\left(\frac{x}{8} \right)}}{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} - 1\right) \cos^{3}{\left(\frac{x}{8} \right)} \cos^{5}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
1 1
---------------------------------- + --------------------------------------
/ 2 \ /x\ / pi x\ 2
|2 + -------|*sec|-|*sec|- -- + -| / 1 \ /x\ 3/ pi x\
| 2/x\| \4/ \ 2 4/ 2*|1 + -------| *sec|-|*sec |- -- + -|
| sec |-|| | 2/x\| \4/ \ 2 4/
\ \4// | sec |-||
\ \4//
$$\frac{1}{\left(2 + \frac{2}{\sec^{2}{\left(\frac{x}{4} \right)}}\right) \sec{\left(\frac{x}{4} \right)} \sec{\left(\frac{x}{4} - \frac{\pi}{2} \right)}} + \frac{1}{2 \left(1 + \frac{1}{\sec^{2}{\left(\frac{x}{4} \right)}}\right)^{2} \sec{\left(\frac{x}{4} \right)} \sec^{3}{\left(\frac{x}{4} - \frac{\pi}{2} \right)}}$$
3/x\ / 2/x\\ /x\ 5/x\ 3/x\ / 2/x\\
2*cos |-|*|1 - tan |-||*sin|-| 16*cos |-|*sin |-|*|1 - tan |-||
\8/ \ \8// \8/ \8/ \8/ \ \8//
------------------------------ + --------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \sin{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \sin^{3}{\left(\frac{x}{8} \right)} \cos^{5}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
/x\ /x\ 3/x\ /x\
cos|-|*sin|-| sin |-|*cos|-|
\4/ \4/ \4/ \4/
------------- + ----------------
2/x\ 2
2 + 2*cos |-| / 2/x\\
\4/ 2*|1 + cos |-||
\ \4//
$$\frac{\sin{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \cos^{2}{\left(\frac{x}{4} \right)} + 2} + \frac{\sin^{3}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
3/x\ / 1 \ /x\ 3/x\ 5/x\ / 1 \
2*sin |-|*|-1 + -------|*cos|-| 16*cos |-|*sin |-|*|-1 + -------|
\8/ | 2/x\| \8/ \8/ \8/ | 2/x\|
| tan |-|| | tan |-||
\ \8// \ \8//
------------------------------- + ---------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{2 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \sin^{3}{\left(\frac{x}{8} \right)} \cos{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \sin^{5}{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
1 6
----------- + -----------
/ pi\ /x pi\
sec|x - --| sec|- - --|
\ 2 / \2 2 /
-------------------------
3
/ 1 \
|3 + ------|
| /x\|
| sec|-||
\ \2//
$$\frac{\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{6}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{3}}$$
/ 2/x\ \ / 2/x\ \
| sin |-| | | sin |-| |
2/x\ | \4/ | /x\ 3/x\ 2/x\ | \4/ |
sin |-|*|-1 + ---------|*sin|-| 2*sin |-|*sin |-|*|-1 + ---------|
\8/ | 4/x\| \4/ \4/ \8/ | 4/x\|
| 4*sin |-|| | 4*sin |-||
\ \8// \ \8//
------------------------------- + ----------------------------------
/pi x\ 2
3 + sin|-- + -| / /pi x\\
\2 2/ |3 + sin|-- + -||
\ \2 2//
$$\frac{\left(-1 + \frac{\sin^{2}{\left(\frac{x}{4} \right)}}{4 \sin^{4}{\left(\frac{x}{8} \right)}}\right) \sin^{2}{\left(\frac{x}{8} \right)} \sin{\left(\frac{x}{4} \right)}}{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3} + \frac{2 \left(-1 + \frac{\sin^{2}{\left(\frac{x}{4} \right)}}{4 \sin^{4}{\left(\frac{x}{8} \right)}}\right) \sin^{2}{\left(\frac{x}{8} \right)} \sin^{3}{\left(\frac{x}{4} \right)}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3\right)^{2}}$$
/ 2/x\ \ / 2/x\ \
| csc |-| | | csc |-| |
| \8/ | | \8/ |
2*|-1 + ------------| 16*|-1 + ------------|
| 2/pi x\| | 2/pi x\|
| csc |-- - -|| | csc |-- - -||
\ \2 8// \ \2 8//
------------------------------------- + ---------------------------------------
/ 1 \ 3/x\ /pi x\ 2
|3 + -----------|*csc |-|*csc|-- - -| / 1 \ 5/x\ 3/pi x\
| /pi x\| \8/ \2 8/ |3 + -----------| *csc |-|*csc |-- - -|
| csc|-- - -|| | /pi x\| \8/ \2 8/
\ \2 2// | csc|-- - -||
\ \2 2//
$$\frac{2 \left(\frac{\csc^{2}{\left(\frac{x}{8} \right)}}{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} - 1\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(\frac{x}{8} \right)} \csc{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} + \frac{16 \left(\frac{\csc^{2}{\left(\frac{x}{8} \right)}}{\csc^{2}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}} - 1\right)}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{2} \csc^{5}{\left(\frac{x}{8} \right)} \csc^{3}{\left(- \frac{x}{8} + \frac{\pi}{2} \right)}}$$
/x\ / pi x\ 3/ pi x\ /x\
cos|-|*cos|- -- + -| cos |- -- + -|*cos|-|
\4/ \ 2 4/ \ 2 4/ \4/
-------------------- + ---------------------
2/x\ 2
2 + 2*cos |-| / 2/x\\
\4/ 2*|1 + cos |-||
\ \4//
$$\frac{\cos{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{4} - \frac{\pi}{2} \right)}}{2 \cos^{2}{\left(\frac{x}{4} \right)} + 2} + \frac{\cos{\left(\frac{x}{4} \right)} \cos^{3}{\left(\frac{x}{4} - \frac{\pi}{2} \right)}}{2 \left(\cos^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
/x pi\ / pi\
6*cos|- - --| + cos|x - --|
\2 2 / \ 2 /
---------------------------
3
/ /x\\
|3 + cos|-||
\ \2//
$$\frac{6 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{3}}$$
/ 2/x\\ /x\ 3/x\ / 2/x\\
2*|-1 + cot |-||*cot|-| 4*cot |-|*|-1 + cot |-||
\ \8// \8/ \8/ \ \8//
-------------------------------------- + -------------------------------------
/ 2\ 2
| / 2/x\\ | / 2\
2 | 2*|-1 + cot |-|| | | / 2/x\\ |
/ 2/x\\ | \ \8// | 4 | |-1 + cot |-|| |
|1 + cot |-|| *|2 + -----------------| / 2/x\\ | \ \8// |
\ \8// | 2 | |1 + cot |-|| *|1 + ---------------|
| / 2/x\\ | \ \8// | 2|
| |1 + cot |-|| | | / 2/x\\ |
\ \ \8// / | |1 + cot |-|| |
\ \ \8// /
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \cot{\left(\frac{x}{8} \right)}}{\left(\frac{2 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + 2\right) \left(\cot^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + \frac{4 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \cot^{3}{\left(\frac{x}{8} \right)}}{\left(\frac{\left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{8} \right)} + 1\right)^{4}}$$
/x\
2*sin|-|
\2/
-------------
2
/ /x\\
|3 + cos|-||
\ \2//
$$\frac{2 \sin{\left(\frac{x}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
2/x\ 4/pi x\ | \8/| 6/x\ 8/pi x\ | \8/|
4*sin |-|*sin |-- + -|*|1 - ---------| 128*sin |-|*sin |-- + -|*|1 - ---------|
\8/ \2 8/ | 2/x\ | \8/ \2 8/ | 2/x\ |
| sin |-| | | sin |-| |
\ \4/ / \ \4/ /
-------------------------------------- + ----------------------------------------
/ /pi x\\ /x\ 2
|3 + sin|-- + -||*sin|-| / /pi x\\ 3/x\
\ \2 2// \4/ |3 + sin|-- + -|| *sin |-|
\ \2 2// \4/
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{x}{8} \right)}}{\sin^{2}{\left(\frac{x}{4} \right)}} + 1\right) \sin^{2}{\left(\frac{x}{8} \right)} \sin^{4}{\left(\frac{x}{8} + \frac{\pi}{2} \right)}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3\right) \sin{\left(\frac{x}{4} \right)}} + \frac{128 \left(- \frac{4 \sin^{4}{\left(\frac{x}{8} \right)}}{\sin^{2}{\left(\frac{x}{4} \right)}} + 1\right) \sin^{6}{\left(\frac{x}{8} \right)} \sin^{8}{\left(\frac{x}{8} + \frac{\pi}{2} \right)}}{\left(\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + 3\right)^{2} \sin^{3}{\left(\frac{x}{4} \right)}}$$
/ 2/ pi x\\ / 2/ pi x\\
| cos |- -- + -|| | cos |- -- + -||
3/x\ | \ 2 8/| / pi x\ 5/x\ 3/ pi x\ | \ 2 8/|
2*cos |-|*|1 - --------------|*cos|- -- + -| 16*cos |-|*cos |- -- + -|*|1 - --------------|
\8/ | 2/x\ | \ 2 8/ \8/ \ 2 8/ | 2/x\ |
| cos |-| | | cos |-| |
\ \8/ / \ \8/ /
-------------------------------------------- + ----------------------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{2 \left(1 - \frac{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{3}{\left(\frac{x}{8} \right)} \cos{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \frac{\cos^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{5}{\left(\frac{x}{8} \right)} \cos^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
4/x\ / 2/x\\ /x\ 3/x\ 8/x\ / 2/x\\
2*sin |-|*|-1 + cot |-||*cot|-| 16*cot |-|*sin |-|*|-1 + cot |-||
\8/ \ \8// \8/ \8/ \8/ \ \8//
------------------------------- + ---------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{4}{\left(\frac{x}{8} \right)} \cot{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right) \sin^{8}{\left(\frac{x}{8} \right)} \cot^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
1 1
--------------------------- + -------------------------------
/ 2 \ /x\ /x\ 2
|2 + -------|*csc|-|*sec|-| / 1 \ 3/x\ /x\
| 2/x\| \4/ \4/ 2*|1 + -------| *csc |-|*sec|-|
| sec |-|| | 2/x\| \4/ \4/
\ \4// | sec |-||
\ \4//
$$\frac{1}{\left(2 + \frac{2}{\sec^{2}{\left(\frac{x}{4} \right)}}\right) \csc{\left(\frac{x}{4} \right)} \sec{\left(\frac{x}{4} \right)}} + \frac{1}{2 \left(1 + \frac{1}{\sec^{2}{\left(\frac{x}{4} \right)}}\right)^{2} \csc^{3}{\left(\frac{x}{4} \right)} \sec{\left(\frac{x}{4} \right)}}$$
4 8
/ 2/x \\ 8/x \ / 2/x\\ /x\ / 2/x \\ 16/x \ 3/x\ / 2/x\\
2*|-1 + cot |--|| *sin |--|*|1 - tan |-||*tan|-| 16*|-1 + cot |--|| *sin |--|*tan |-|*|1 - tan |-||
\ \16// \16/ \ \8// \8/ \ \16// \16/ \8/ \ \8//
------------------------------------------------ + ---------------------------------------------------
/x\ 2
3 + cos|-| / /x\\
\2/ |3 + cos|-||
\ \2//
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{16} \right)} \tan{\left(\frac{x}{8} \right)}}{\cos{\left(\frac{x}{2} \right)} + 3} + \frac{16 \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{x}{16} \right)} \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$
8/x \ 4/x \ / 1 \ 16/x \ 8/x \ / 1 \
32*cos |--|*tan |--|*|-1 + -------| 4096*cos |--|*tan |--|*|-1 + -------|
\16/ \16/ | 2/x\| \16/ \16/ | 2/x\|
| tan |-|| | tan |-||
\ \8// \ \8//
----------------------------------- + --------------------------------------
/ /x\\ /x\ 2
|3 + cos|-||*tan|-| / /x\\ 3/x\
\ \2// \8/ |3 + cos|-|| *tan |-|
\ \2// \8/
$$\frac{32 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{8}{\left(\frac{x}{16} \right)} \tan^{4}{\left(\frac{x}{16} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right) \tan{\left(\frac{x}{8} \right)}} + \frac{4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{8} \right)}}\right) \cos^{16}{\left(\frac{x}{16} \right)} \tan^{8}{\left(\frac{x}{16} \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{2} \tan^{3}{\left(\frac{x}{8} \right)}}$$
/x\ /x\
2*tan|-| 12*tan|-|
\2/ \4/
----------- + -----------
2/x\ 2/x\
1 + tan |-| 1 + tan |-|
\2/ \4/
-------------------------
3
/ 2/x\\
| 1 - tan |-||
| \4/|
|3 + -----------|
| 2/x\|
| 1 + tan |-||
\ \4//
$$\frac{\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{12 \tan{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{3}}$$
4 8
/ 2/x \\ / 1 \ / 2/x \\ / 1 \
2*|-1 + cot |--|| *|1 - -------| 16*|-1 + cot |--|| *|1 - -------|
\ \16// | 2/x\| \ \16// | 2/x\|
| cot |-|| | cot |-||
\ \8// \ \8//
----------------------------------------- + -------------------------------------------
/ 2/x\\ 2
4 | -1 + cot |-|| / 2/x\\
/ 2/x \\ | \4/| /x\ 8 | -1 + cot |-||
|1 + cot |--|| *|3 + ------------|*cot|-| / 2/x \\ | \4/| 3/x\
\ \16// | 2/x\ | \8/ |1 + cot |--|| *|3 + ------------| *cot |-|
| 1 + cot |-| | \ \16// | 2/x\ | \8/
\ \4/ / | 1 + cot |-| |
\ \4/ /
$$\frac{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{8} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{4}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4} \cot{\left(\frac{x}{8} \right)}} + \frac{16 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{8} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{16} \right)} - 1\right)^{8}}{\left(\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\cot^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8} \cot^{3}{\left(\frac{x}{8} \right)}}$$
1 1
------------------------------------- + -----------------------------------------
/ 2 \ /x\ /pi x\ 2
|2 + ------------|*csc|-|*csc|-- - -| / 1 \ 3/x\ /pi x\
| 2/pi x\| \4/ \2 4/ 2*|1 + ------------| *csc |-|*csc|-- - -|
| csc |-- - -|| | 2/pi x\| \4/ \2 4/
\ \2 4// | csc |-- - -||
\ \2 4//
$$\frac{1}{\left(2 + \frac{2}{\csc^{2}{\left(- \frac{x}{4} + \frac{\pi}{2} \right)}}\right) \csc{\left(\frac{x}{4} \right)} \csc{\left(- \frac{x}{4} + \frac{\pi}{2} \right)}} + \frac{1}{2 \left(1 + \frac{1}{\csc^{2}{\left(- \frac{x}{4} + \frac{\pi}{2} \right)}}\right)^{2} \csc^{3}{\left(\frac{x}{4} \right)} \csc{\left(- \frac{x}{4} + \frac{\pi}{2} \right)}}$$
/x\
6*sin|-| + sin(x)
\2/
-----------------
3
/ /x\\
|3 + cos|-||
\ \2//
$$\frac{6 \sin{\left(\frac{x}{2} \right)} + \sin{\left(x \right)}}{\left(\cos{\left(\frac{x}{2} \right)} + 3\right)^{3}}$$
/ 2/ pi x\\ / 2/ pi x\\
| sec |- -- + -|| | sec |- -- + -||
| \ 2 8/| | \ 2 8/|
2*|-1 + --------------| 16*|-1 + --------------|
| 2/x\ | | 2/x\ |
| sec |-| | | sec |-| |
\ \8/ / \ \8/ /
---------------------------------- + ------------------------------------
/ 1 \ /x\ 3/ pi x\ 2
|3 + ------|*sec|-|*sec |- -- + -| / 1 \ 3/x\ 5/ pi x\
| /x\| \8/ \ 2 8/ |3 + ------| *sec |-|*sec |- -- + -|
| sec|-|| | /x\| \8/ \ 2 8/
\ \2// | sec|-||
\ \2//
$$\frac{2 \left(-1 + \frac{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{8} \right)} \sec^{3}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}} + \frac{16 \left(-1 + \frac{\sec^{2}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{8} \right)}}\right)}{\left(3 + \frac{1}{\sec{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{3}{\left(\frac{x}{8} \right)} \sec^{5}{\left(\frac{x}{8} - \frac{\pi}{2} \right)}}$$
1 6
------ + ------
csc(x) /x\
csc|-|
\2/
------------------
3
/ 1 \
|3 + -----------|
| /pi x\|
| csc|-- - -||
\ \2 2//
$$\frac{\frac{1}{\csc{\left(x \right)}} + \frac{6}{\csc{\left(\frac{x}{2} \right)}}}{\left(3 + \frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right)^{3}}$$
4 8
/ 2/x \\ / 2/x\\ /x\ / 2/x \\ 3/x\ / 2/x\\
2*|1 - tan |--|| *|1 - tan |-||*tan|-| 16*|1 - tan |--|| *tan |-|*|1 - tan |-||
\ \16// \ \8// \8/ \ \16// \8/ \ \8//
-------------------------------------- + ----------------------------------------
/ 2/x\\ 2
4 | 1 - tan |-|| / 2/x\\
/ 2/x \\ | \4/| 8 | 1 - tan |-||
|1 + tan |--|| *|3 + -----------| / 2/x \\ | \4/|
\ \16// | 2/x\| |1 + tan |--|| *|3 + -----------|
| 1 + tan |-|| \ \16// | 2/x\|
\ \4// | 1 + tan |-||
\ \4//
$$\frac{16 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{8} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \tan^{3}{\left(\frac{x}{8} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right)^{2} \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{8}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{16} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right) \tan{\left(\frac{x}{8} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} + 3\right) \left(\tan^{2}{\left(\frac{x}{16} \right)} + 1\right)^{4}}$$
2*(1 - tan(x/16)^2)^4*(1 - tan(x/8)^2)*tan(x/8)/((1 + tan(x/16)^2)^4*(3 + (1 - tan(x/4)^2)/(1 + tan(x/4)^2))) + 16*(1 - tan(x/16)^2)^8*tan(x/8)^3*(1 - tan(x/8)^2)/((1 + tan(x/16)^2)^8*(3 + (1 - tan(x/4)^2)/(1 + tan(x/4)^2))^2)