Господин Экзамен
sin(x)^6+cos(x)^6+3*sin(x)^2*cos(x)^2 если x=-3
Решение
Вы ввели
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6 6 2 2 sin (x) + cos (x) + 3*sin (x)*cos (x)
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}$$
sin(x)^6 + cos(x)^6 + 3*sin(x)^2*cos(x)^2
Общее упрощение
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3 6 6 3*cos(4*x) - + cos (x) + sin (x) - ---------- 8 8
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} - \frac{3 \cos{\left(4 x \right)}}{8} + \frac{3}{8}$$
3/8 + cos(x)^6 + sin(x)^6 - 3*cos(4*x)/8
Подстановка условия
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sin(x)^6 + cos(x)^6 + 3*sin(x)^2*cos(x)^2 при x = -3
подставляем
6 6 2 2 sin (x) + cos (x) + 3*sin (x)*cos (x)
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} + 3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}$$
3 6 6 3*cos(4*x) - + cos (x) + sin (x) - ---------- 8 8
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} - \frac{3 \cos{\left(4 x \right)}}{8} + \frac{3}{8}$$
переменные
x = -3
$$x = -3$$
3 6 6 3*cos(4*(-3)) - + cos ((-3)) + sin ((-3)) - ------------- 8 8
$$\sin^{6}{\left((-3) \right)} + \cos^{6}{\left((-3) \right)} - \frac{3 \cos{\left(4 (-3) \right)}}{8} + \frac{3}{8}$$
3 6 6 3*cos(4*-3) - + cos (-3) + sin (-3) - ----------- 8 8
$$- \frac{3 \cos{\left(4 \left(-3\right) \right)}}{8} + \sin^{6}{\left(-3 \right)} + \frac{3}{8} + \cos^{6}{\left(-3 \right)}$$
3 6 6 3*cos(12) - + cos (3) + sin (3) - --------- 8 8
$$- \frac{3 \cos{\left(12 \right)}}{8} + \sin^{6}{\left(3 \right)} + \frac{3}{8} + \cos^{6}{\left(3 \right)}$$
3/8 + cos(3)^6 + sin(3)^6 - 3*cos(12)/8
Численный ответ
[src]
cos(x)^6 + sin(x)^6 + 3.0*cos(x)^2*sin(x)^2
cos(x)^6 + sin(x)^6 + 3.0*cos(x)^2*sin(x)^2
Степени
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2
/ I*x -I*x\ 2
6 6 |e e | / -I*x I*x\
/ I*x -I*x\ / -I*x I*x\ 3*|---- + -----| *\- e + e /
|e e | \- e + e / \ 2 2 /
|---- + -----| - ----------------- - -----------------------------------
\ 2 2 / 64 4
$$\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{6} - \frac{\left(e^{i x} - e^{- i x}\right)^{6}}{64} - \frac{3 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} \left(e^{i x} - e^{- i x}\right)^{2}}{4}$$
(exp(i*x)/2 + exp(-i*x)/2)^6 - (-exp(-i*x) + exp(i*x))^6/64 - 3*(exp(i*x)/2 + exp(-i*x)/2)^2*(-exp(-i*x) + exp(i*x))^2/4
Тригонометрическая часть
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1
$$1$$
3 6 6 3 - + cos (x) + sin (x) - ---------- 8 8*sec(4*x)
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} + \frac{3}{8} - \frac{3}{8 \sec{\left(4 x \right)}}$$
3 6 6 3*cos(4*x) - + cos (x) + sin (x) - ---------- 8 8
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} - \frac{3 \cos{\left(4 x \right)}}{8} + \frac{3}{8}$$
6 6/x\ 6/x\
1 - sin (x) + 64*cos |-|*sin |-|
\2/ \2/
$$64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} - \sin^{6}{\left(x \right)} + 1$$
6 6 3*(1 - cos(4*x))
cos (x) + sin (x) + ----------------
8
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} + \frac{3 \cdot \left(- \cos{\left(4 x \right)} + 1\right)}{8}$$
3 6 6/ pi\ 3*cos(4*x) - + cos (x) + cos |x - --| - ---------- 8 \ 2 / 8
$$\cos^{6}{\left(x \right)} + \cos^{6}{\left(x - \frac{\pi}{2} \right)} - \frac{3 \cos{\left(4 x \right)}}{8} + \frac{3}{8}$$
/pi \
3*sin|-- + 4*x|
3 6 6 \2 /
- + cos (x) + sin (x) - ---------------
8 8
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} - \frac{3 \sin{\left(4 x + \frac{\pi}{2} \right)}}{8} + \frac{3}{8}$$
3 6 6 3
- + cos (x) + sin (x) - ---------------
8 /pi \
8*csc|-- - 4*x|
\2 /
$$\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)} + \frac{3}{8} - \frac{3}{8 \csc{\left(- 4 x + \frac{\pi}{2} \right)}}$$
1 1 3 ------- + ------- + --------------- 6 6 2 2 csc (x) sec (x) csc (x)*sec (x)
$$\frac{3}{\csc^{2}{\left(x \right)} \sec^{2}{\left(x \right)}} + \frac{1}{\sec^{6}{\left(x \right)}} + \frac{1}{\csc^{6}{\left(x \right)}}$$
/pi \
3*sin|-- + 4*x|
3 6 6/ pi\ \2 /
- + sin (x) + sin |x + --| - ---------------
8 \ 2 / 8
$$\sin^{6}{\left(x \right)} + \sin^{6}{\left(x + \frac{\pi}{2} \right)} - \frac{3 \sin{\left(4 x + \frac{\pi}{2} \right)}}{8} + \frac{3}{8}$$
6 6/ pi\ 2 2/ pi\
cos (x) + cos |x - --| + 3*cos (x)*cos |x - --|
\ 2 / \ 2 /
$$\cos^{6}{\left(x \right)} + \cos^{6}{\left(x - \frac{\pi}{2} \right)} + 3 \cos^{2}{\left(x \right)} \cos^{2}{\left(x - \frac{\pi}{2} \right)}$$
6 6/ pi\ 2 2/ pi\
sin (x) + sin |x + --| + 3*sin (x)*sin |x + --|
\ 2 / \ 2 /
$$\sin^{6}{\left(x \right)} + \sin^{6}{\left(x + \frac{\pi}{2} \right)} + 3 \sin^{2}{\left(x \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)}$$
3 1 1 3
- + ------- + ------------ - ----------
8 6 6/ pi\ 8*sec(4*x)
sec (x) sec |x - --|
\ 2 /
$$\frac{3}{8} - \frac{3}{8 \sec{\left(4 x \right)}} + \frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{6}{\left(x \right)}}$$
3 1 1 3
- + ------- + ------------ - ---------------
8 6 6/pi \ /pi \
csc (x) csc |-- - x| 8*csc|-- - 4*x|
\2 / \2 /
$$\frac{3}{8} - \frac{3}{8 \csc{\left(- 4 x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(x \right)}}$$
1 1 3
------- + ------------ + --------------------
6 6/ pi\ 2 2/ pi\
sec (x) sec |x - --| sec (x)*sec |x - --|
\ 2 / \ 2 /
$$\frac{3}{\sec^{2}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{6}{\left(x \right)}}$$
1 1 3
------- + ------------ + --------------------
6 6/pi \ 2 2/pi \
sec (x) sec |-- - x| sec (x)*sec |-- - x|
\2 / \2 /
$$\frac{3}{\sec^{2}{\left(x \right)} \sec^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{6}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{6}{\left(x \right)}}$$
1 1 3
------- + ------------ + --------------------
6 6/pi \ 2 2/pi \
csc (x) csc |-- - x| csc (x)*csc |-- - x|
\2 / \2 /
$$\frac{3}{\csc^{2}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(x \right)}}$$
5 3*cos(4*x) /1 cos(2*x)\ /1 cos(2*x)\ - + ---------- + 3*|- + --------|*|- - --------| 8 8 \2 2 / \2 2 /
$$3 \cdot \left(- \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) + \frac{3 \cos{\left(4 x \right)}}{8} + \frac{5}{8}$$
1 1 3
------------ + ------------ + -------------------------
6 6/pi \ 2 2/pi \
csc (pi - x) csc |-- - x| csc (pi - x)*csc |-- - x|
\2 / \2 /
$$\frac{3}{\csc^{2}{\left(- x + \pi \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{6}{\left(- x + \pi \right)}}$$
6 6 6/x\ 2 2/x\
cos (x) + (1 + cos(x)) *tan |-| + 6*cos (x)*sin |-|*(1 + cos(x))
\2/ \2/
$$\left(\cos{\left(x \right)} + 1\right)^{6} \tan^{6}{\left(\frac{x}{2} \right)} + \cos^{6}{\left(x \right)} + 6 \left(\cos{\left(x \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(x \right)}$$
6 12/x\
64*cos (x)*cos |-|
3*(1 - cos(4*x)) 6/x\ 6/x\ \2/
---------------- + 64*cos |-|*sin |-| + -------------------
8 \2/ \2/ 6
(1 + cos(x))
$$64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} + \frac{64 \cos^{12}{\left(\frac{x}{2} \right)} \cos^{6}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right)^{6}} + \frac{3 \cdot \left(- \cos{\left(4 x \right)} + 1\right)}{8}$$
6 2 / 2 \ 3 / 2/x\\ 12/x\ 12/x\ 6/x\ 3*cos (2*x)*\1 - tan (2*x)/ - + |1 - tan |-|| *cos |-| + 64*cos |-|*tan |-| - --------------------------- 8 \ \2// \2/ \2/ \2/ 8
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} + 64 \cos^{12}{\left(\frac{x}{2} \right)} \tan^{6}{\left(\frac{x}{2} \right)} - \frac{3 \cdot \left(- \tan^{2}{\left(2 x \right)} + 1\right) \cos^{2}{\left(2 x \right)}}{8} + \frac{3}{8}$$
6 2 / 2/x\\ 12/x\ 6/x\ 6/x\ / 2/x\\ 6/x\ 2/x\ |1 - tan |-|| *cos |-| + 64*cos |-|*sin |-| + 12*|1 - tan |-|| *cos |-|*sin |-| \ \2// \2/ \2/ \2/ \ \2// \2/ \2/
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} + 64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} + 12 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \sin^{2}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)}$$
6 2 / 2/x\\ 12/x\ 12/x\ 6/x\ / 2/x\\ 8/x\ 2/x\ |1 - tan |-|| *cos |-| + 64*cos |-|*tan |-| + 12*|1 - tan |-|| *cos |-|*tan |-| \ \2// \2/ \2/ \2/ \ \2// \2/ \2/
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} + 64 \cos^{12}{\left(\frac{x}{2} \right)} \tan^{6}{\left(\frac{x}{2} \right)} + 12 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}$$
6
/ 2/x\\ 6/x\
|1 - tan |-|| 64*tan |-| / 2 \
3 \ \2// \2/ 3*\1 - tan (2*x)/
- + -------------- + -------------- - -----------------
8 6 6 / 2 \
/ 2/x\\ / 2/x\\ 8*\1 + tan (2*x)/
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} - \frac{3 \cdot \left(- \tan^{2}{\left(2 x \right)} + 1\right)}{8 \left(\tan^{2}{\left(2 x \right)} + 1\right)} + \frac{3}{8} + \frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}$$
6 2 2 / 2/x\\ 12/x\ 6/x\ 6/x\ / /x\\ / /x\\ 6/x\ 2/x\ |1 - tan |-|| *cos |-| + 64*cos |-|*sin |-| + 12*|1 + tan|-|| *|-1 + tan|-|| *cos |-|*sin |-| \ \2// \2/ \2/ \2/ \ \2// \ \2// \2/ \2/
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} + 12 \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \sin^{2}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} + 64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)}$$
2
/ 2 2 \ / 2 2 \ / 2 2 \ 2 2
5 3*\cos (x) - sin (x)/ |1 cos (x) sin (x)| |1 sin (x) cos (x)| 3*cos (x)*sin (x)
- + ---------------------- + 3*|- + ------- - -------|*|- + ------- - -------| - -----------------
8 8 \2 2 2 / \2 2 2 / 2
$$- \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{2} + \frac{3 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{2}}{8} + 3 \left(- \frac{\sin^{2}{\left(x \right)}}{2} + \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) \left(\frac{\sin^{2}{\left(x \right)}}{2} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{2}\right) + \frac{5}{8}$$
6 2
/ 2/x\\ 6/x\ / 2/x\\ 2/x\
|1 - tan |-|| 64*tan |-| 12*|1 - tan |-|| *tan |-|
\ \2// \2/ \ \2// \2/
-------------- + -------------- + -------------------------
6 6 4
/ 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \2// \ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} + \frac{12 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}$$
2 2
6/x pi\ / /x\\ / /x\\ 4/x\
64*tan |- + --| 3*|1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*x))
\2 4 / 12/x\ 6/x\ \ \2// \ \2// \2/
------------------- + 64*cos |-|*tan |-| + -----------------------------------------------------
6 \2/ \2/ 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$64 \cos^{12}{\left(\frac{x}{2} \right)} \tan^{6}{\left(\frac{x}{2} \right)} + \frac{3 \cdot \left(- \cos{\left(2 x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{2} + \frac{64 \tan^{6}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}}$$
2 2
6/x pi\ / /x\\ / /x\\ 4/x\
64*tan |- + --| 3*|1 + tan|-|| *|-1 + tan|-|| *cos |-|*(1 - cos(2*x))
\2 4 / 6/x\ 12/x\ \ \2// \ \2// \2/
------------------- + 64*cot |-|*sin |-| + -----------------------------------------------------
6 \2/ \2/ 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$64 \sin^{12}{\left(\frac{x}{2} \right)} \cot^{6}{\left(\frac{x}{2} \right)} + \frac{3 \cdot \left(- \cos{\left(2 x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{2} + \frac{64 \tan^{6}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}}$$
2 2
/ 2/x\\ / 2/x pi\\ 4/x\
6 3*|-1 + cot |-|| *|-1 + tan |- + --|| *sin |-|
/ 2/x\\ 12/x\ 6/x\ 6/x\ \ \2// \ \2 4 // \2/
|-1 + cot |-|| *sin |-| + 64*cos |-|*sin |-| + ----------------------------------------------
\ \2// \2/ \2/ \2/ 2
/ 2/x pi\\
|1 + tan |- + --||
\ \2 4 //
$$\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} + 64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} + \frac{3 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
6 2
/ 1 \ / 1 \
|1 - -------| 12*|1 - -------|
| 2/x\| | 2/x\|
| cot |-|| | cot |-||
\ \2// 64 \ \2//
-------------- + ---------------------- + ----------------------
6 6 4
/ 1 \ / 1 \ 6/x\ / 1 \ 2/x\
|1 + -------| |1 + -------| *cot |-| |1 + -------| *cot |-|
| 2/x\| | 2/x\| \2/ | 2/x\| \2/
| cot |-|| | cot |-|| | cot |-||
\ \2// \ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{6}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{6}} + \frac{12 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cot^{2}{\left(\frac{x}{2} \right)}} + \frac{64}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \cot^{6}{\left(\frac{x}{2} \right)}}$$
6 2 2
/ 2/x pi\\ 6 / 2/x pi\\ / 2/x\\ 2 4/x\
6 |1 - cot |- + --|| *(1 + sin(x)) 3*|1 - cot |- + --|| *|1 - tan |-|| *(1 + sin(x)) *cos |-|
/ 2/x\\ 12/x\ \ \2 4 // \ \2 4 // \ \2// \2/
|1 - tan |-|| *cos |-| + --------------------------------- + ----------------------------------------------------------
\ \2// \2/ 64 4
$$\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} + \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6} \left(\sin{\left(x \right)} + 1\right)^{6}}{64} + \frac{3 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(x \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}{4}$$
6 2 / 2/x pi\\ / 2/x pi\\ | cos |- - --|| | cos |- - --|| | \2 2 /| 12/x\ 6/x\ 6/x pi\ | \2 2 /| 6/x\ 2/x pi\ |1 - ------------| *cos |-| + 64*cos |-|*cos |- - --| + 12*|1 - ------------| *cos |-|*cos |- - --| | 2/x\ | \2/ \2/ \2 2 / | 2/x\ | \2/ \2 2 / | cos |-| | | cos |-| | \ \2/ / \ \2/ /
$$\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} + 64 \cos^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + 12 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{6}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}$$
6/x\ 6/x pi\ 2/x\ 2/x pi\
64*tan |-| 64*tan |- + --| 48*tan |-|*tan |- + --|
\2/ \2 4 / \2/ \2 4 /
-------------- + ------------------- + ----------------------------------
6 6 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + tan |-|| |1 + tan |- + --|| |1 + tan |-|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{64 \tan^{6}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}} + \frac{48 \tan^{2}{\left(\frac{x}{2} \right)} \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}$$
6/x\ 6/x pi\ 2/x\ 2/x pi\
64*cot |-| 64*tan |- + --| 48*cot |-|*tan |- + --|
\2/ \2 4 / \2/ \2 4 /
-------------- + ------------------- + ----------------------------------
6 6 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} + \frac{48 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{64 \tan^{6}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}}$$
6 2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
|1 - ------------| 12*|1 - ------------|
| 2/x pi\| | 2/x pi\|
| sec |- - --|| | sec |- - --||
\ \2 2 // 64 \ \2 2 //
------------------- + -------------------- + ----------------------
12/x\ 6/x\ 6/x pi\ 6/x\ 2/x pi\
sec |-| sec |-|*sec |- - --| sec |-|*sec |- - --|
\2/ \2/ \2 2 / \2/ \2 2 /
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6}}{\sec^{12}{\left(\frac{x}{2} \right)}} + \frac{12 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\sec^{6}{\left(\frac{x}{2} \right)} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{64}{\sec^{6}{\left(\frac{x}{2} \right)} \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
6 12 12 2 8 / 2/x\\ / 2/x\\ 24/x\ / 2/x\\ 24/x\ 6/x\ / 2/x\\ / 2/x\\ 16/x\ 2/x\ |1 - tan |-|| *|1 - tan |-|| *cos |-| + 64*|1 - tan |-|| *cos |-|*tan |-| + 12*|1 - tan |-|| *|1 - tan |-|| *cos |-|*tan |-| \ \2// \ \4// \4/ \ \4// \4/ \2/ \ \2// \ \4// \4/ \2/
$$\left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12} \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{24}{\left(\frac{x}{4} \right)} + 64 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12} \cos^{24}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{2} \right)} + 12 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{16}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{2} \right)}$$
2
/ 4/x\\
| 4*sin |-||
6 | \2/| 4/x\ 8/pi x\
/ 4/x\\ 12/x\ 12/pi x\ 48*|1 - ---------| *sin |-|*sin |-- + -|
| 4*sin |-|| 4096*sin |-|*sin |-- + -| | 2 | \2/ \2 2/
| \2/| 12/pi x\ \2/ \2 2/ \ sin (x) /
|1 - ---------| *sin |-- + -| + --------------------------- + ----------------------------------------
| 2 | \2 2/ 6 2
\ sin (x) / sin (x) sin (x)
$$\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{6} \sin^{12}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} + \frac{4096 \sin^{12}{\left(\frac{x}{2} \right)} \sin^{12}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin^{6}{\left(x \right)}} + \frac{48 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)} \sin^{8}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin^{2}{\left(x \right)}}$$
6 2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
|1 - ------------| 12*|1 - ------------|
| 2/x\ | | 2/x\ |
| csc |-| | | csc |-| |
\ \2/ / 64 \ \2/ /
------------------- + -------------------- + ----------------------
12/pi x\ 6/x\ 6/pi x\ 2/x\ 6/pi x\
csc |-- - -| csc |-|*csc |-- - -| csc |-|*csc |-- - -|
\2 2/ \2/ \2 2/ \2/ \2 2/
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{6}}{\csc^{12}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{12 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\csc^{2}{\left(\frac{x}{2} \right)} \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{64}{\csc^{6}{\left(\frac{x}{2} \right)} \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
// 1 for 2*x mod pi = 0\
|| |
3*|< 2 / 2 \ | // 1 for x mod 2*pi = 0\
||sin (2*x)*\-1 + cot (2*x)/ otherwise | // 0 for x mod pi = 0\ || |
3 \\ / || | || 6 |
- - --------------------------------------------------- + |< 6 | + |
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\frac{3 \left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\cot^{2}{\left(2 x \right)} - 1\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right)}{8}\right) + \frac{3}{8}$$
6 6 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|-1 + cot |-|| |-1 + tan |- + --|| 3*|-1 + cot |-|| *|-1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
--------------- + -------------------- + --------------------------------------
6 6 2 2
/ 2/x\\ / 2/x pi\\ / 2/x\\ / 2/x pi\\
|1 + cot |-|| |1 + tan |- + --|| |1 + cot |-|| *|1 + tan |- + --||
\ \2// \ \2 4 // \ \2// \ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}} + \frac{3 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}$$
6 6 2 2
/ 2/x pi\\ / 2/x\\ / 2/x pi\\ / 2/x\\
|1 - cot |- + --|| |1 - tan |-|| 3*|1 - cot |- + --|| *|1 - tan |-||
\ \2 4 // \ \2// \ \2 4 // \ \2//
------------------- + -------------- + ------------------------------------
6 6 2 2
/ 2/x pi\\ / 2/x\\ / 2/x pi\\ / 2/x\\
|1 + cot |- + --|| |1 + tan |-|| |1 + cot |- + --|| *|1 + tan |-||
\ \2 4 // \ \2// \ \2 4 // \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} + \frac{3 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}}$$
// 0 for x mod pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || 6 | // 1 for x mod 2*pi = 0\
|| | || | ||(-1 + cos(x)) | || |
3*|< 2 |*|< 2 | + |<-------------- otherwise | + |< 6 |
||sin (x) otherwise | ||cos (x) otherwise | || 6/x\ | ||cos (x) otherwise |
\\ / \\ / || tan |-| | \\ /
|| \2/ |
\\ /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\left(\cos{\left(x \right)} - 1\right)^{6}}{\tan^{6}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || 6 |
3*|< 2 |*|< 2 | + |< 6 | + |
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || 6 |
3*|< 2 |*|< 2 | + |< 6 | + |
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
6 12 12 2 8
/ 2/x\\ / 2/x\\ / 2/x\\ 6/x\ / 2/x\\ / 2/x\\ 2/x\
|1 - tan |-|| *|1 - tan |-|| 64*|1 - tan |-|| *tan |-| 12*|1 - tan |-|| *|1 - tan |-|| *tan |-|
\ \2// \ \4// \ \4// \2/ \ \2// \ \4// \2/
------------------------------ + -------------------------- + ----------------------------------------
12 12 8
/ 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \4// \ \4// \ \4//
$$\frac{\left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12} \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} + \frac{64 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12} \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} + \frac{12 \left(- \tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 6 12/x\ |
|| | || | || | ||64*cos (x)*sin |-| |
3*|< 2 |*|< 2 | + |< 6 | + |< \2/ |
||sin (x) otherwise | ||cos (x) otherwise | ||sin (x) otherwise | ||------------------- otherwise |
\\ / \\ / \\ / || 6 |
|| (-1 + cos(x)) |
\\ /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{64 \sin^{12}{\left(\frac{x}{2} \right)} \cos^{6}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 6 |
|| | || | || | ||/ 2 \ |
3*|< 2 |*|< 2 | + |< 6 | + |<| sin (x) | 12/x\ |
||sin (x) otherwise | ||cos (x) otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 6 |
|| | || | || | ||/ 2 \ |
3*|< 2 |*|< 2/ pi\ | + |< 6 | + |<| sin (x) | 12/x\ |
||sin (x) otherwise | ||sin |x + --| otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ \ 2 / / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || 6 |
|| | || | || | || / 1 \ 24/x\ 12/x\ |
3*|< 2 |*|< 2 | + |< 6 | + |<4096*|-1 + -------| *cos |-|*tan |-| otherwise |
||sin (x) otherwise | ||cos (x) otherwise | ||sin (x) otherwise | || | 2/x\| \4/ \4/ |
\\ / \\ / \\ / || | tan |-|| |
|| \ \2// |
\\ /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \cos^{24}{\left(\frac{x}{4} \right)} \tan^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for 2*x mod pi = 0\
|| |
|| 2 | // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
3*|<-1 + cot (2*x) | || | || |
||-------------- otherwise | || 6/x\ | || 6 |
|| 2 | || 64*cot |-| | ||/ 2/x\\ |
3 \\1 + cot (2*x) / || \2/ | |||-1 + cot |-|| |
- - --------------------------------------- + |<-------------- otherwise | + |<\ \2// |
8 8 || 6 | ||--------------- otherwise |
||/ 2/x\\ | || 6 |
|||1 + cot |-|| | || / 2/x\\ |
||\ \2// | || |1 + cot |-|| |
\\ / \\ \ \2// /
$$\left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) - \left(\frac{3 \left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 x \right)} - 1}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}\right)}{8}\right) + \frac{3}{8}$$
6 2
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 4/x\
|1 - ---------| 12/x\ 48*|1 - ---------| *sin |-|
| 2 | 4096*sin |-| | 2 | \2/
\ sin (x) / \2/ \ sin (x) /
---------------- + ------------------------ + ---------------------------
6 6 4
/ 4/x\\ / 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 6 | \2/| 2
|1 + ---------| |1 + ---------| *sin (x) |1 + ---------| *sin (x)
| 2 | | 2 | | 2 |
\ sin (x) / \ sin (x) / \ sin (x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{6}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{6}} + \frac{48 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{4} \sin^{2}{\left(x \right)}} + \frac{4096 \sin^{12}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{6} \sin^{6}{\left(x \right)}}$$
// 1 for x mod 2*pi = 0\
|| |
|| 6 |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||/ 2/x\ \ |
|| | || | || | ||| cos |-| | |
3*|< 2/ pi\ |*|< 2 | + |< 6/ pi\ | + |<| \2/ | 12/x pi\ |
||cos |x - --| otherwise | ||cos (x) otherwise | ||cos |x - --| otherwise | |||-1 + ------------| *cos |- - --| otherwise |
\\ \ 2 / / \\ / \\ \ 2 / / ||| 2/x pi\| \2 2 / |
||| cos |- - --|| |
||\ \2 2 // |
\\ /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{2}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos^{6}{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{6} \cos^{12}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 6 |
||/ 2/x\ \ |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||| csc |-| | |
|| | || | || | ||| \2/ | |
|| 1 | || 1 | || 1 | |||-1 + ------------| |
3*|<------- otherwise |*|<------------ otherwise | + |<------- otherwise | + |<| 2/pi x\| |
|| 2 | || 2/pi \ | || 6 | ||| csc |-- - -|| |
||csc (x) | ||csc |-- - x| | ||csc (x) | ||\ \2 2// |
\\ / \\ \2 / / \\ / ||-------------------- otherwise |
|| 12/x\ |
|| csc |-| |
|| \2/ |
\\ /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc^{6}{\left(x \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{6}}{\csc^{12}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // 1 for x mod 2*pi = 0\ // / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0| || | || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / | || 6 | || \ 2 / |
3*|< 2 |*|< | + |
$$\left(3 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{x}{2} \right)} + 4 \cos^{2}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0| // 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / | || | || \ 2 / |
3*|< 2 |*|< | + |< 6 | + |< |
||sin (x) otherwise | || 2 2/x pi\ | ||sin (x) otherwise | || 6 6/x pi\ |
\\ / ||(1 + sin(x)) *cot |- + --| otherwise | \\ / ||(1 + sin(x)) *cot |- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{2} \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right)^{6} \cot^{6}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 6 |
||/ 2/x pi\\ |
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ ||| sec |- - --|| |
|| | || | || | ||| \2 2 /| |
|| 1 | || 1 | || 1 | |||-1 + ------------| |
3*|<------------ otherwise |*|<------- otherwise | + |<------------ otherwise | + |<| 2/x\ | |
|| 2/ pi\ | || 2 | || 6/ pi\ | ||| sec |-| | |
||sec |x - --| | ||sec (x) | ||sec |x - --| | ||\ \2/ / |
\\ \ 2 / / \\ / \\ \ 2 / / ||-------------------- otherwise |
|| 12/x pi\ |
|| sec |- - --| |
|| \2 2 / |
\\ /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{6}}{\sec^{12}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
6 2
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/x pi\
|1 - ------------| 12*|1 - ------------| *cos |- - --|
| 2/x\ | 6/x pi\ | 2/x\ | \2 2 /
| cos |-| | 64*cos |- - --| | cos |-| |
\ \2/ / \2 2 / \ \2/ /
------------------- + --------------------------- + -----------------------------------
6 6 4
/ 2/x pi\\ / 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 6/x\ | \2 2 /| 2/x\
|1 + ------------| |1 + ------------| *cos |-| |1 + ------------| *cos |-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/
| cos |-| | | cos |-| | | cos |-| |
\ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{6}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{6}} + \frac{12 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{2}{\left(\frac{x}{2} \right)}} + \frac{64 \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \cos^{6}{\left(\frac{x}{2} \right)}}$$
6 2
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/x\
|1 - ------------| 12*|1 - ------------| *sec |-|
| 2/x pi\| 6/x\ | 2/x pi\| \2/
| sec |- - --|| 64*sec |-| | sec |- - --||
\ \2 2 // \2/ \ \2 2 //
------------------- + -------------------------------- + --------------------------------
6 6 4
/ 2/x\ \ / 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| | | sec |-| |
| \2/ | | \2/ | 6/x pi\ | \2/ | 2/x pi\
|1 + ------------| |1 + ------------| *sec |- - --| |1 + ------------| *sec |- - --|
| 2/x pi\| | 2/x pi\| \2 2 / | 2/x pi\| \2 2 /
| sec |- - --|| | sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6}} + \frac{12 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{4} \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{64 \sec^{6}{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6} \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
6 2
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/pi x\
|1 - ------------| 12*|1 - ------------| *csc |-- - -|
| 2/x\ | 6/pi x\ | 2/x\ | \2 2/
| csc |-| | 64*csc |-- - -| | csc |-| |
\ \2/ / \2 2/ \ \2/ /
------------------- + --------------------------- + -----------------------------------
6 6 4
/ 2/pi x\\ / 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 6/x\ | \2 2/| 2/x\
|1 + ------------| |1 + ------------| *csc |-| |1 + ------------| *csc |-|
| 2/x\ | | 2/x\ | \2/ | 2/x\ | \2/
| csc |-| | | csc |-| | | csc |-| |
\ \2/ / \ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{6}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{6}} + \frac{12 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \csc^{2}{\left(\frac{x}{2} \right)}} + \frac{64 \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \csc^{6}{\left(\frac{x}{2} \right)}}$$
// x \ // x \ // x \
|| 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0|
6 || 2 | || 2 | 2 || 2 |
/ 2/x\\ || | 6/x\ || | / 2/x\\ 2/x\ || |
|1 - tan |-|| *|< 12 | + 64*tan |-|*|< 12 | + 12*|1 - tan |-|| *tan |-|*|< 8 |
\ \2// ||/ 2/x\\ 24/x\ | \2/ ||/ 2/x\\ 24/x\ | \ \2// \2/ ||/ 2/x\\ 16/x\ |
|||-1 + cot |-|| *sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise |
||\ \4// \4/ | ||\ \4// \4/ | ||\ \4// \4/ |
\\ / \\ / \\ /
$$\left(\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6} \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{12} \sin^{24}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(64 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{12} \sin^{24}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{6}{\left(\frac{x}{2} \right)}\right) + \left(12 \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(\frac{x}{2} \right)}\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | || | || | || | || 2/x\ | || 2 | || 6/x\ | || 6 | || 4*cot |-| | ||/ 2/x\\ | || 64*cot |-| | ||/ 2/x\\ | || \2/ | |||-1 + cot |-|| | || \2/ | |||-1 + cot |-|| | 3*|<-------------- otherwise |*|<\ \2// | + |<-------------- otherwise | + |<\ \2// | || 2 | ||--------------- otherwise | || 6 | ||--------------- otherwise | ||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 6 | |||1 + cot |-|| | || / 2/x\\ | |||1 + cot |-|| | || / 2/x\\ | ||\ \2// | || |1 + cot |-|| | ||\ \2// | || |1 + cot |-|| | \\ / \\ \ \2// / \\ / \\ \ \2// /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || 6 |
|| 2/x\ | || 2 | || 6/x\ | || / 1 \ 12/x\ |
|| 4*tan |-| | ||/ 2/x\\ | || 64*tan |-| | ||4096*|-1 + -------| *tan |-| |
|| \2/ | |||1 - tan |-|| | || \2/ | || | 2/x\| \4/ |
3*|<-------------- otherwise |*|<\ \2// | + |<-------------- otherwise | + |< | tan |-|| |
|| 2 | ||-------------- otherwise | || 6 | || \ \2// |
||/ 2/x\\ | || 2 | ||/ 2/x\\ | ||----------------------------- otherwise |
|||1 + tan |-|| | ||/ 2/x\\ | |||1 + tan |-|| | || 12 |
||\ \2// | |||1 + tan |-|| | ||\ \2// | || / 2/x\\ |
\\ / \\\ \2// / \\ / || |1 + tan |-|| |
\\ \ \4// /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4096 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \tan^{12}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 6 |
|| | ||/ 1 \ | || | ||/ 1 \ |
|| 4 | |||-1 + -------| | || 64 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/x\| | ||---------------------- otherwise | ||| 2/x\| |
|| 2 | ||| tan |-|| | || 6 | ||| tan |-|| |
3*|
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \tan^{6}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{6}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ // / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0| // 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / | || | || \ 2 / |
|| 2/x\ | || | || 6/x\ | || |
|| 4*cot |-| | || 2/x pi\ | || 64*cot |-| | || 6/x pi\ |
|| \2/ | || 4*cot |- + --| | || \2/ | || 64*cot |- + --| |
3*|<-------------- otherwise |*|< \2 4 / | + |<-------------- otherwise | + |< \2 4 / |
|| 2 | ||------------------- otherwise | || 6 | ||------------------- otherwise |
||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 6 |
|||1 + cot |-|| | ||/ 2/x pi\\ | |||1 + cot |-|| | ||/ 2/x pi\\ |
||\ \2// | |||1 + cot |- + --|| | ||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // | \\ / ||\ \2 4 // |
\\ / \\ /
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || // x \ |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||/ 0 for x mod pi = 0 | || 6 || 0 for - mod pi = 0| |
3*|<| |*|<| | + |<| | + |
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin^{6}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\4096 \sin^{24}{\left(\frac{x}{4} \right)} \cot^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// x \ // x \
|| 1 for - mod 2*pi = 0| || 1 for - mod 2*pi = 0|
|| 2 | || 2 |
|| | || |
|| 12 | 2 || 8 |
||/ 2/x\\ | / 1 \ ||/ 2/x\\ |
64*|<|-1 + cot |-|| | 12*|1 - -------| *|<|-1 + cot |-|| |
// x \ ||\ \4// | | 2/x\| ||\ \4// |
|| 1 for - mod 2*pi = 0| ||---------------- otherwise | | cot |-|| ||--------------- otherwise |
|| 2 | || 12 | \ \2// || 8 |
|| | ||/ 2/x\\ | || / 2/x\\ |
6 || 12 | |||1 + cot |-|| | || |1 + cot |-|| |
/ 1 \ ||/ 2/x\\ | \\\ \4// / \\ \ \4// /
|1 - -------| *|<|-1 + cot |-|| | + ------------------------------------------ + --------------------------------------------------------
| 2/x\| ||\ \4// | 6/x\ 2/x\
| cot |-|| ||---------------- otherwise | cot |-| cot |-|
\ \2// || 12 | \2/ \2/
||/ 2/x\\ |
|||1 + cot |-|| |
\\\ \4// /
$$\left(\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{12}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{12 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) + \left(\frac{64 \left(\begin{cases} 1 & \text{for}\: \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{12}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right)}{\cot^{6}{\left(\frac{x}{2} \right)}}\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0| // 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / | || | || \ 2 / |
|| 2 | || | || 6 | || |
||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 6 |
|||-1 + cot |-|| | ||/ 2/x pi\\ | |||-1 + cot |-|| | ||/ 2/x pi\\ |
3*|<\ \2// |*|<|-1 + tan |- + --|| | + |<\ \2// | + |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // | ||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise | || 6 | ||-------------------- otherwise |
|| / 2/x\\ | || 2 | || / 2/x\\ | || 6 |
|| |1 + cot |-|| | ||/ 2/x pi\\ | || |1 + cot |-|| | ||/ 2/x pi\\ |
\\ \ \2// / |||1 + tan |- + --|| | \\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // / \\\ \2 4 // /
$$\left(3 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 0 for x mod pi = 0\ // 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\ || | // 1 for x mod 2*pi = 0\
|| 2 | || | || 6 | || |
|| sin (x) | || 2 | || sin (x) | || 6 |
||------------------------ otherwise | ||/ 2 4/x\\ | ||------------------------- otherwise | ||/ 2 4/x\\ |
|| 2 | |||sin (x) - 4*sin |-|| | || 6 | |||sin (x) - 4*sin |-|| |
3*|
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{6}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{6}}{\left(4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 6 |
|| | ||/ 2 \ | || | ||/ 2 \ |
|| 2 | ||| sin (x) | | || 6 | ||| sin (x) | |
|| sin (x) | |||-1 + ---------| | || sin (x) | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/x\| | ||------------------------- otherwise | ||| 4/x\| |
|| 2 | ||| 4*sin |-|| | || 6 | ||| 4*sin |-|| |
3*|
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{2}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin^{6}{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ // 0 for x mod pi = 0\ || |
|| | || | || | || // x \ |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 | ||/ 0 for x mod pi = 0 | || || 0 for - mod pi = 0| |
||| | ||| | ||| | || || 2 | |
||| 2/x\ | ||| 2 | ||| 6/x\ | || || | |
||| 4*cot |-| | |||/ 2/x\\ | ||| 64*cot |-| | || 6 || 12/x\ | |
3*|<| \2/ |*|<||-1 + cot |-|| | + |<| \2/ | + |
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \left(\begin{cases} 0 & \text{for}\: \frac{x}{2} \bmod \pi = 0 \\\frac{4096 \cot^{12}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 6 |
|| | ||/ 2/x\ \ | || | ||/ 2/x\ \ |
|| 2/x\ | ||| cos |-| | | || 6/x\ | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | | || 64*cos |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| | || \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/x pi\| | ||-------------------------------- otherwise | ||| 2/x pi\| |
|| 2 | ||| cos |- - --|| | || 6 | ||| cos |- - --|| |
3*|
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \cos^{6}{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6} \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{6}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 6 |
|| | ||/ 2/x pi\\ | || | ||/ 2/x pi\\ |
|| 2/x pi\ | ||| sec |- - --|| | || 6/x pi\ | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| | || 64*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| | || \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/x\ | | ||--------------------------- otherwise | ||| 2/x\ | |
|| 2 | ||| sec |-| | | || 6 | ||| sec |-| | |
3*|
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{6} \sec^{6}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{6}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{6}} & \text{otherwise} \end{cases}\right)$$
// 1 for x mod 2*pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
// 0 for x mod pi = 0\ || 2 | // 0 for x mod pi = 0\ || 6 |
|| | ||/ 2/x\ \ | || | ||/ 2/x\ \ |
|| 2/x\ | ||| csc |-| | | || 6/x\ | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | | || 64*csc |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| | || \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi x\| | ||-------------------------------- otherwise | ||| 2/pi x\| |
|| 2 | ||| csc |-- - -|| | || 6 | ||| csc |-- - -|| |
3*|
$$\left(3 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{64 \csc^{6}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{6} \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{6}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)$$
3*Piecewise((0, Mod(x = pi, 0)), (4*csc(x/2)^2/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2*csc(pi/2 - x/2)^2), True))*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^2, True)) + Piecewise((0, Mod(x = pi, 0)), (64*csc(x/2)^6/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^6*csc(pi/2 - x/2)^6), True)) + Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^6/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2)^6, True))