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2*(sin(3*x)*cos(5*x)-sin(5*x)*cos(3*x)) если x=1
Решение
Вы ввели
[src]
2*(sin(3*x)*cos(5*x) - sin(5*x)*cos(3*x))
$$2 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)$$
2*(sin(3*x)*cos(5*x) - sin(5*x)*cos(3*x))
Подстановка условия
[src]
2*(sin(3*x)*cos(5*x) - sin(5*x)*cos(3*x)) при x = 1
подставляем
2*(sin(3*x)*cos(5*x) - sin(5*x)*cos(3*x))
$$2 \left(\sin{\left(3 x \right)} \cos{\left(5 x \right)} - \sin{\left(5 x \right)} \cos{\left(3 x \right)}\right)$$
-2*sin(2*x)
$$- 2 \sin{\left(2 x \right)}$$
переменные
x = 1
$$x = 1$$
-2*sin(2*(1))
$$- 2 \sin{\left(2 (1) \right)}$$
-2*sin(2*1)
$$- 2 \sin{\left(2 \cdot 1 \right)}$$
-2*sin(2)
$$- 2 \sin{\left(2 \right)}$$
-2*sin(2)
Рациональный знаменатель
[src]
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
$$2 \sin{\left(3 x \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)} \cos{\left(3 x \right)}$$
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
Комбинаторика
[src]
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
$$2 \sin{\left(3 x \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)} \cos{\left(3 x \right)}$$
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
Общий знаменатель
[src]
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
$$2 \sin{\left(3 x \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)} \cos{\left(3 x \right)}$$
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
Объединение рациональных выражений
[src]
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
$$2 \sin{\left(3 x \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)} \cos{\left(3 x \right)}$$
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
Численный ответ
[src]
2.0*cos(5*x)*sin(3*x) - 2.0*cos(3*x)*sin(5*x)
2.0*cos(5*x)*sin(3*x) - 2.0*cos(3*x)*sin(5*x)
Собрать выражение
[src]
-2*sin(2*x)
$$- 2 \sin{\left(2 x \right)}$$
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
$$2 \sin{\left(3 x \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)} \cos{\left(3 x \right)}$$
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
Степени
[src]
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
$$2 \sin{\left(3 x \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)} \cos{\left(3 x \right)}$$
/ -3*I*x 3*I*x\ / -5*I*x 5*I*x\ |e e | / -5*I*x 5*I*x\ |e e | / -3*I*x 3*I*x\ I*|------- + ------|*\- e + e / - I*|------- + ------|*\- e + e / \ 2 2 / \ 2 2 /
$$i \left(\frac{e^{3 i x}}{2} + \frac{e^{- 3 i x}}{2}\right) \left(e^{5 i x} - e^{- 5 i x}\right) - i \left(e^{3 i x} - e^{- 3 i x}\right) \left(\frac{e^{5 i x}}{2} + \frac{e^{- 5 i x}}{2}\right)$$
i*(exp(-3*i*x)/2 + exp(3*i*x)/2)*(-exp(-5*i*x) + exp(5*i*x)) - i*(exp(-5*i*x)/2 + exp(5*i*x)/2)*(-exp(-3*i*x) + exp(3*i*x))
Тригонометрическая часть
[src]
-2*sin(2*x)
$$- 2 \sin{\left(2 x \right)}$$
-2 -------- csc(2*x)
$$- \frac{2}{\csc{\left(2 x \right)}}$$
/ pi\
-2*cos|2*x - --|
\ 2 /
$$- 2 \cos{\left(2 x - \frac{\pi}{2} \right)}$$
-2 ------------- / pi\ sec|2*x - --| \ 2 /
$$- \frac{2}{\sec{\left(2 x - \frac{\pi}{2} \right)}}$$
-4*tan(x)
-----------
2
1 + tan (x)
$$- \frac{4 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
-2*cos(3*x)*sin(5*x) + 2*cos(5*x)*sin(3*x)
$$2 \sin{\left(3 x \right)} \cos{\left(5 x \right)} - 2 \sin{\left(5 x \right)} \cos{\left(3 x \right)}$$
2 2 - ----------------- + ----------------- csc(5*x)*sec(3*x) csc(3*x)*sec(5*x)
$$- \frac{2}{\csc{\left(5 x \right)} \sec{\left(3 x \right)}} + \frac{2}{\csc{\left(3 x \right)} \sec{\left(5 x \right)}}$$
// 0 for 2*x mod pi = 0\ -2*|< | \\sin(2*x) otherwise /
$$- 2 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)$$
/pi \ /pi \
- 2*sin(5*x)*sin|-- + 3*x| + 2*sin(3*x)*sin|-- + 5*x|
\2 / \2 /
$$2 \sin{\left(3 x \right)} \sin{\left(5 x + \frac{\pi}{2} \right)} - 2 \sin{\left(5 x \right)} \sin{\left(3 x + \frac{\pi}{2} \right)}$$
/ pi\ / pi\
- 2*cos(3*x)*cos|5*x - --| + 2*cos(5*x)*cos|3*x - --|
\ 2 / \ 2 /
$$- 2 \cos{\left(3 x \right)} \cos{\left(5 x - \frac{\pi}{2} \right)} + 2 \cos{\left(5 x \right)} \cos{\left(3 x - \frac{\pi}{2} \right)}$$
2 2
- ---------------------- + ----------------------
/ pi\ / pi\
sec(3*x)*sec|5*x - --| sec(5*x)*sec|3*x - --|
\ 2 / \ 2 /
$$\frac{2}{\sec{\left(5 x \right)} \sec{\left(3 x - \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(3 x \right)} \sec{\left(5 x - \frac{\pi}{2} \right)}}$$
2 2
- ---------------------- + ----------------------
/pi \ /pi \
sec(3*x)*sec|-- - 5*x| sec(5*x)*sec|-- - 3*x|
\2 / \2 /
$$\frac{2}{\sec{\left(5 x \right)} \sec{\left(- 3 x + \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(3 x \right)} \sec{\left(- 5 x + \frac{\pi}{2} \right)}}$$
2 2
- ---------------------- + ----------------------
/pi \ /pi \
csc(5*x)*csc|-- - 3*x| csc(3*x)*csc|-- - 5*x|
\2 / \2 /
$$- \frac{2}{\csc{\left(5 x \right)} \csc{\left(- 3 x + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(3 x \right)} \csc{\left(- 5 x + \frac{\pi}{2} \right)}}$$
// 0 for 2*x mod pi = 0\ || | || 2*cot(x) | -2*|<----------- otherwise | || 2 | ||1 + cot (x) | \\ /
$$- 2 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2 2
- --------------------------- + ---------------------------
/pi \ /pi \
csc(pi - 5*x)*csc|-- - 3*x| csc(pi - 3*x)*csc|-- - 5*x|
\2 / \2 /
$$\frac{2}{\csc{\left(- 3 x + \pi \right)} \csc{\left(- 5 x + \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(- 5 x + \pi \right)} \csc{\left(- 3 x + \frac{\pi}{2} \right)}}$$
/5*x\ /3*x\
- 2*(1 + cos(5*x))*cos(3*x)*tan|---| + 2*(1 + cos(3*x))*cos(5*x)*tan|---|
\ 2 / \ 2 /
$$2 \left(\cos{\left(3 x \right)} + 1\right) \cos{\left(5 x \right)} \tan{\left(\frac{3 x}{2} \right)} - 2 \left(\cos{\left(5 x \right)} + 1\right) \cos{\left(3 x \right)} \tan{\left(\frac{5 x}{2} \right)}$$
2/5*x\ / 2/pi 3*x\\ / 2/5*x\\ 2/3*x\ / 2/pi 5*x\\ / 2/3*x\\
cos |---|*|1 - cot |-- + ---||*|1 - tan |---||*(1 + sin(3*x)) - cos |---|*|1 - cot |-- + ---||*|1 - tan |---||*(1 + sin(5*x))
\ 2 / \ \4 2 // \ \ 2 // \ 2 / \ \4 2 // \ \ 2 //
$$- \left(- \tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(5 x \right)} + 1\right) \cos^{2}{\left(\frac{3 x}{2} \right)} + \left(- \tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(3 x \right)} + 1\right) \cos^{2}{\left(\frac{5 x}{2} \right)}$$
/ 2/3*x\\ /5*x\ / 2/5*x\\ /3*x\
4*|1 - tan |---||*tan|---| 4*|1 - tan |---||*tan|---|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------------- + -------------------------------
/ 2/3*x\\ / 2/5*x\\ / 2/3*x\\ / 2/5*x\\
|1 + tan |---||*|1 + tan |---|| |1 + tan |---||*|1 + tan |---||
\ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 //
$$- \frac{4 \cdot \left(- \tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \tan{\left(\frac{5 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} + \frac{4 \cdot \left(- \tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right) \tan{\left(\frac{3 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)}$$
/5*x\ /pi 3*x\ /3*x\ /pi 5*x\
8*cot|---|*tan|-- + ---| 8*cot|---|*tan|-- + ---|
\ 2 / \4 2 / \ 2 / \4 2 /
- ------------------------------------ + ------------------------------------
/ 2/5*x\\ / 2/pi 3*x\\ / 2/3*x\\ / 2/pi 5*x\\
|1 + cot |---||*|1 + tan |-- + ---|| |1 + cot |---||*|1 + tan |-- + ---||
\ \ 2 // \ \4 2 // \ \ 2 // \ \4 2 //
$$\frac{8 \tan{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{3 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{3 x}{2} \right)} + 1\right)} - \frac{8 \tan{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{5 x}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)}$$
/5*x\ /pi 3*x\ /3*x\ /pi 5*x\
8*tan|---|*tan|-- + ---| 8*tan|---|*tan|-- + ---|
\ 2 / \4 2 / \ 2 / \4 2 /
- ------------------------------------ + ------------------------------------
/ 2/5*x\\ / 2/pi 3*x\\ / 2/3*x\\ / 2/pi 5*x\\
|1 + tan |---||*|1 + tan |-- + ---|| |1 + tan |---||*|1 + tan |-- + ---||
\ \ 2 // \ \4 2 // \ \ 2 // \ \4 2 //
$$- \frac{8 \tan{\left(\frac{5 x}{2} \right)} \tan{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{8 \tan{\left(\frac{3 x}{2} \right)} \tan{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 1 \ / 1 \
4*|1 - ---------| 4*|1 - ---------|
| 2/3*x\| | 2/5*x\|
| cot |---|| | cot |---||
\ \ 2 // \ \ 2 //
- ---------------------------------------- + ----------------------------------------
/ 1 \ / 1 \ /5*x\ / 1 \ / 1 \ /3*x\
|1 + ---------|*|1 + ---------|*cot|---| |1 + ---------|*|1 + ---------|*cot|---|
| 2/3*x\| | 2/5*x\| \ 2 / | 2/3*x\| | 2/5*x\| \ 2 /
| cot |---|| | cot |---|| | cot |---|| | cot |---||
\ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 //
$$- \frac{4 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{3 x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{5 x}{2} \right)}}\right) \cot{\left(\frac{5 x}{2} \right)}} + \frac{4 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{5 x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{5 x}{2} \right)}}\right) \cot{\left(\frac{3 x}{2} \right)}}$$
// 0 for 5*x mod pi = 0\ // 1 for 3*x mod 2*pi = 0\ // 0 for 3*x mod pi = 0\ // 1 for 5*x mod 2*pi = 0\
- 2*|< |*|< | + 2*|< |*|< |
\\sin(5*x) otherwise / \\cos(3*x) otherwise / \\sin(3*x) otherwise / \\cos(5*x) otherwise /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\cos{\left(5 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\sin{\left(5 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/3*x\\ / 2/pi 5*x\\ / 2/5*x\\ / 2/pi 3*x\\
2*|-1 + cot |---||*|-1 + tan |-- + ---|| 2*|-1 + cot |---||*|-1 + tan |-- + ---||
\ \ 2 // \ \4 2 // \ \ 2 // \ \4 2 //
- ---------------------------------------- + ----------------------------------------
/ 2/3*x\\ / 2/pi 5*x\\ / 2/5*x\\ / 2/pi 3*x\\
|1 + cot |---||*|1 + tan |-- + ---|| |1 + cot |---||*|1 + tan |-- + ---||
\ \ 2 // \ \4 2 // \ \ 2 // \ \4 2 //
$$\frac{2 \left(\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} - \frac{2 \left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{3 x}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{3 x}{2} \right)} + 1\right)}$$
// 0 for 5*x mod pi = 0\ // 0 for 3*x mod pi = 0\
|| | // 1 for 3*x mod 2*pi = 0\ || | // 1 for 5*x mod 2*pi = 0\
- 2*|< / pi\ |*|< | + 2*|< / pi\ |*|< |
||cos|5*x - --| otherwise | \\cos(3*x) otherwise / ||cos|3*x - --| otherwise | \\cos(5*x) otherwise /
\\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\cos{\left(3 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\cos{\left(5 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\cos{\left(5 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*x mod 2*pi = 0\ // 1 for 5*x mod 2*pi = 0\
// 0 for 5*x mod pi = 0\ || | // 0 for 3*x mod pi = 0\ || |
- 2*|< |*|< /pi \ | + 2*|< |*|< /pi \ |
\\sin(5*x) otherwise / ||sin|-- + 3*x| otherwise | \\sin(3*x) otherwise / ||sin|-- + 5*x| otherwise |
\\ \2 / / \\ \2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\sin{\left(5 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\sin{\left(5 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\sin{\left(3 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi 5*x\\ / 2/3*x\\ / 2/pi 3*x\\ / 2/5*x\\
2*|1 - cot |-- + ---||*|1 - tan |---|| 2*|1 - cot |-- + ---||*|1 - tan |---||
\ \4 2 // \ \ 2 // \ \4 2 // \ \ 2 //
- -------------------------------------- + --------------------------------------
/ 2/pi 5*x\\ / 2/3*x\\ / 2/pi 3*x\\ / 2/5*x\\
|1 + cot |-- + ---||*|1 + tan |---|| |1 + cot |-- + ---||*|1 + tan |---||
\ \4 2 // \ \ 2 // \ \4 2 // \ \ 2 //
$$- \frac{2 \cdot \left(- \tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{2 \cdot \left(- \tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for 3*x mod 2*pi = 0\ || 1 for |5*x + ----| mod 2*pi = 0| // 1 for 5*x mod 2*pi = 0\ || 1 for |3*x + ----| mod 2*pi = 0|
- 2*|< |*|< \ 2 / | + 2*|< |*|< \ 2 / |
\\cos(3*x) otherwise / || | \\cos(5*x) otherwise / || |
\\sin(5*x) otherwise / \\sin(3*x) otherwise /
$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(5 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(5 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\cos{\left(5 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(3 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 5*x mod pi = 0\ // 0 for 3*x mod pi = 0\
|| | // 1 for 3*x mod 2*pi = 0\ || | // 1 for 5*x mod 2*pi = 0\
|| 1 | || | || 1 | || |
- 2*|<------------- otherwise |*|< 1 | + 2*|<------------- otherwise |*|< 1 |
|| / pi\ | ||-------- otherwise | || / pi\ | ||-------- otherwise |
||sec|5*x - --| | \\sec(3*x) / ||sec|3*x - --| | \\sec(5*x) /
\\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(5 x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{1}{\sec{\left(5 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*x mod 2*pi = 0\ // 1 for 5*x mod 2*pi = 0\
// 0 for 5*x mod pi = 0\ || | // 0 for 3*x mod pi = 0\ || |
|| | || 1 | || | || 1 |
- 2*|< 1 |*|<------------- otherwise | + 2*|< 1 |*|<------------- otherwise |
||-------- otherwise | || /pi \ | ||-------- otherwise | || /pi \ |
\\csc(5*x) / ||csc|-- - 3*x| | \\csc(3*x) / ||csc|-- - 5*x| |
\\ \2 / / \\ \2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{1}{\csc{\left(3 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 5 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{1}{\csc{\left(5 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 5*x mod pi = 0\ // 0 for 3*x mod pi = 0\
|| | || |
||1 - cos(5*x) | // 1 for 3*x mod 2*pi = 0\ ||1 - cos(3*x) | // 1 for 5*x mod 2*pi = 0\
- 2*|<------------ otherwise |*|< | + 2*|<------------ otherwise |*|< |
|| /5*x\ | \\cos(3*x) otherwise / || /3*x\ | \\cos(5*x) otherwise /
|| tan|---| | || tan|---| |
\\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{- \cos{\left(3 x \right)} + 1}{\tan{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\cos{\left(5 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{- \cos{\left(5 x \right)} + 1}{\tan{\left(\frac{5 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \ // /pi \ \
|| 0 for |-- + 3*x| mod pi = 0| || 0 for |-- + 5*x| mod pi = 0|
// 0 for 5*x mod pi = 0\ || \2 / | // 0 for 3*x mod pi = 0\ || \2 / |
- 2*|< |*|< | + 2*|< |*|< |
\\sin(5*x) otherwise / || /pi 3*x\ | \\sin(3*x) otherwise / || /pi 5*x\ |
||(1 + sin(3*x))*cot|-- + ---| otherwise | ||(1 + sin(5*x))*cot|-- + ---| otherwise |
\\ \4 2 / / \\ \4 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(5 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(5 x \right)} + 1\right) \cot{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\sin{\left(5 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(3 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(3 x \right)} + 1\right) \cot{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4/3*x\\ / 4/5*x\\
| 4*sin |---|| | 4*sin |---||
2/5*x\ | \ 2 /| 2/3*x\ | \ 2 /|
8*sin |---|*|1 - -----------| 8*sin |---|*|1 - -----------|
\ 2 / | 2 | \ 2 / | 2 |
\ sin (3*x) / \ sin (5*x) /
- -------------------------------------------- + --------------------------------------------
/ 4/3*x\\ / 4/5*x\\ / 4/3*x\\ / 4/5*x\\
| 4*sin |---|| | 4*sin |---|| | 4*sin |---|| | 4*sin |---||
| \ 2 /| | \ 2 /| | \ 2 /| | \ 2 /|
|1 + -----------|*|1 + -----------|*sin(5*x) |1 + -----------|*|1 + -----------|*sin(3*x)
| 2 | | 2 | | 2 | | 2 |
\ sin (3*x) / \ sin (5*x) / \ sin (3*x) / \ sin (5*x) /
$$- \frac{8 \left(- \frac{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}{\sin^{2}{\left(3 x \right)}} + 1\right) \sin^{2}{\left(\frac{5 x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}{\sin^{2}{\left(3 x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{5 x}{2} \right)}}{\sin^{2}{\left(5 x \right)}} + 1\right) \sin{\left(5 x \right)}} + \frac{8 \left(- \frac{4 \sin^{4}{\left(\frac{5 x}{2} \right)}}{\sin^{2}{\left(5 x \right)}} + 1\right) \sin^{2}{\left(\frac{3 x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}{\sin^{2}{\left(3 x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{5 x}{2} \right)}}{\sin^{2}{\left(5 x \right)}} + 1\right) \sin{\left(3 x \right)}}$$
// 0 for 5*x mod pi = 0\ // 1 for 3*x mod 2*pi = 0\ // 0 for 3*x mod pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || | || | || |
|| /5*x\ | || 2/3*x\ | || /3*x\ | || 2/5*x\ |
|| 2*cot|---| | ||-1 + cot |---| | || 2*cot|---| | ||-1 + cot |---| |
- 2*|< \ 2 / |*|< \ 2 / | + 2*|< \ 2 / |*|< \ 2 / |
||------------- otherwise | ||-------------- otherwise | ||------------- otherwise | ||-------------- otherwise |
|| 2/5*x\ | || 2/3*x\ | || 2/3*x\ | || 2/5*x\ |
||1 + cot |---| | ||1 + cot |---| | ||1 + cot |---| | ||1 + cot |---| |
\\ \ 2 / / \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 5*x mod pi = 0\ // 1 for 3*x mod 2*pi = 0\ // 0 for 3*x mod pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || | || | || |
|| /5*x\ | || 2/3*x\ | || /3*x\ | || 2/5*x\ |
|| 2*tan|---| | ||1 - tan |---| | || 2*tan|---| | ||1 - tan |---| |
- 2*|< \ 2 / |*|< \ 2 / | + 2*|< \ 2 / |*|< \ 2 / |
||------------- otherwise | ||------------- otherwise | ||------------- otherwise | ||------------- otherwise |
|| 2/5*x\ | || 2/3*x\ | || 2/3*x\ | || 2/5*x\ |
||1 + tan |---| | ||1 + tan |---| | ||1 + tan |---| | ||1 + tan |---| |
\\ \ 2 / / \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{5 x}{2} \right)} + 1}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{3 x}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 5*x mod pi = 0\ // 1 for 3*x mod 2*pi = 0\ // 0 for 3*x mod pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || | || | || |
- 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\cos{\left(5 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\sin{\left(5 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\cos{\left(3 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*x mod 2*pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || |
// 0 for 5*x mod pi = 0\ || 1 | // 0 for 3*x mod pi = 0\ || 1 |
|| | ||-1 + --------- | || | ||-1 + --------- |
|| 2 | || 2/3*x\ | || 2 | || 2/5*x\ |
||------------------------ otherwise | || tan |---| | ||------------------------ otherwise | || tan |---| |
- 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 x}{2} \right)}}\right) \tan{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{5 x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{5 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{5 x}{2} \right)}}\right) \tan{\left(\frac{5 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \ // /pi \ \
// 0 for 5*x mod pi = 0\ || 0 for |-- + 3*x| mod pi = 0| // 0 for 3*x mod pi = 0\ || 0 for |-- + 5*x| mod pi = 0|
|| | || \2 / | || | || \2 / |
|| /5*x\ | || | || /3*x\ | || |
|| 2*cot|---| | || /pi 3*x\ | || 2*cot|---| | || /pi 5*x\ |
- 2*|< \ 2 / |*|< 2*cot|-- + ---| | + 2*|< \ 2 / |*|< 2*cot|-- + ---| |
||------------- otherwise | || \4 2 / | ||------------- otherwise | || \4 2 / |
|| 2/5*x\ | ||------------------ otherwise | || 2/3*x\ | ||------------------ otherwise |
||1 + cot |---| | || 2/pi 3*x\ | ||1 + cot |---| | || 2/pi 5*x\ |
\\ \ 2 / / ||1 + cot |-- + ---| | \\ \ 2 / / ||1 + cot |-- + ---| |
\\ \4 2 / / \\ \4 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(5 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(3 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/3*x\ \ / 2/5*x\ \
| sec |---| | | sec |---| |
| \ 2 / | /5*x\ | \ 2 / | /3*x\
4*|1 - ----------------|*sec|---| 4*|1 - ----------------|*sec|---|
| 2/ pi 3*x\| \ 2 / | 2/ pi 5*x\| \ 2 /
| sec |- -- + ---|| | sec |- -- + ---||
\ \ 2 2 // \ \ 2 2 //
- ------------------------------------------------------------- + -------------------------------------------------------------
/ 2/3*x\ \ / 2/5*x\ \ / 2/3*x\ \ / 2/5*x\ \
| sec |---| | | sec |---| | | sec |---| | | sec |---| |
| \ 2 / | | \ 2 / | / pi 5*x\ | \ 2 / | | \ 2 / | / pi 3*x\
|1 + ----------------|*|1 + ----------------|*sec|- -- + ---| |1 + ----------------|*|1 + ----------------|*sec|- -- + ---|
| 2/ pi 3*x\| | 2/ pi 5*x\| \ 2 2 / | 2/ pi 3*x\| | 2/ pi 5*x\| \ 2 2 /
| sec |- -- + ---|| | sec |- -- + ---|| | sec |- -- + ---|| | sec |- -- + ---||
\ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 //
$$- \frac{4 \left(- \frac{\sec^{2}{\left(\frac{3 x}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{5 x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{3 x}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{5 x}{2} \right)}}{\sec^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} + \frac{4 \left(- \frac{\sec^{2}{\left(\frac{5 x}{2} \right)}}{\sec^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{3 x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{3 x}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{5 x}{2} \right)}}{\sec^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}$$
/ 2/ pi 3*x\\ / 2/ pi 5*x\\
| cos |- -- + ---|| | cos |- -- + ---||
| \ 2 2 /| / pi 5*x\ | \ 2 2 /| / pi 3*x\
4*|1 - ----------------|*cos|- -- + ---| 4*|1 - ----------------|*cos|- -- + ---|
| 2/3*x\ | \ 2 2 / | 2/5*x\ | \ 2 2 /
| cos |---| | | cos |---| |
\ \ 2 / / \ \ 2 / /
- ------------------------------------------------------ + ------------------------------------------------------
/ 2/ pi 3*x\\ / 2/ pi 5*x\\ / 2/ pi 3*x\\ / 2/ pi 5*x\\
| cos |- -- + ---|| | cos |- -- + ---|| | cos |- -- + ---|| | cos |- -- + ---||
| \ 2 2 /| | \ 2 2 /| /5*x\ | \ 2 2 /| | \ 2 2 /| /3*x\
|1 + ----------------|*|1 + ----------------|*cos|---| |1 + ----------------|*|1 + ----------------|*cos|---|
| 2/3*x\ | | 2/5*x\ | \ 2 / | 2/3*x\ | | 2/5*x\ | \ 2 /
| cos |---| | | cos |---| | | cos |---| | | cos |---| |
\ \ 2 / / \ \ 2 / / \ \ 2 / / \ \ 2 / /
$$- \frac{4 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} \right)}}\right) \cos{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{5 x}{2} \right)}}\right) \cos{\left(\frac{5 x}{2} \right)}} + \frac{4 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{5 x}{2} \right)}}\right) \cos{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{5 x}{2} \right)}}\right) \cos{\left(\frac{3 x}{2} \right)}}$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for 3*x mod 2*pi = 0\ || 1 for |5*x + ----| mod 2*pi = 0| // 1 for 5*x mod 2*pi = 0\ || 1 for |3*x + ----| mod 2*pi = 0|
|| | || \ 2 / | || | || \ 2 / |
|| 2/3*x\ | || | || 2/5*x\ | || |
||-1 + cot |---| | || 2/pi 5*x\ | ||-1 + cot |---| | || 2/pi 3*x\ |
- 2*|< \ 2 / |*|<-1 + tan |-- + ---| | + 2*|< \ 2 / |*|<-1 + tan |-- + ---| |
||-------------- otherwise | || \4 2 / | ||-------------- otherwise | || \4 2 / |
|| 2/3*x\ | ||------------------- otherwise | || 2/5*x\ | ||------------------- otherwise |
||1 + cot |---| | || 2/pi 5*x\ | ||1 + cot |---| | || 2/pi 3*x\ |
\\ \ 2 / / || 1 + tan |-- + ---| | \\ \ 2 / / || 1 + tan |-- + ---| |
\\ \4 2 / / \\ \4 2 / /
$$\left(- 2 \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(5 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{5 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(3 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi 3*x\\ / 2/pi 5*x\\
| csc |-- - ---|| | csc |-- - ---||
| \2 2 /| /pi 5*x\ | \2 2 /| /pi 3*x\
4*|1 - --------------|*csc|-- - ---| 4*|1 - --------------|*csc|-- - ---|
| 2/3*x\ | \2 2 / | 2/5*x\ | \2 2 /
| csc |---| | | csc |---| |
\ \ 2 / / \ \ 2 / /
- -------------------------------------------------- + --------------------------------------------------
/ 2/pi 3*x\\ / 2/pi 5*x\\ / 2/pi 3*x\\ / 2/pi 5*x\\
| csc |-- - ---|| | csc |-- - ---|| | csc |-- - ---|| | csc |-- - ---||
| \2 2 /| | \2 2 /| /5*x\ | \2 2 /| | \2 2 /| /3*x\
|1 + --------------|*|1 + --------------|*csc|---| |1 + --------------|*|1 + --------------|*csc|---|
| 2/3*x\ | | 2/5*x\ | \ 2 / | 2/3*x\ | | 2/5*x\ | \ 2 /
| csc |---| | | csc |---| | | csc |---| | | csc |---| |
\ \ 2 / / \ \ 2 / / \ \ 2 / / \ \ 2 / /
$$- \frac{4 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 x}{2} \right)}}\right) \csc{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 x}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{5 x}{2} \right)}}\right) \csc{\left(\frac{5 x}{2} \right)}} + \frac{4 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{5 x}{2} \right)}}\right) \csc{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 x}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{5 x}{2} \right)}}\right) \csc{\left(\frac{3 x}{2} \right)}}$$
// 0 for 5*x mod pi = 0\ // 1 for 3*x mod 2*pi = 0\ // 0 for 3*x mod pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || | || | || |
|| -2*sin(10*x) + 4*sin(5*x) | || -2 - 2*cos(6*x) + 4*cos(3*x) | || -2*sin(6*x) + 4*sin(3*x) | || -2 - 2*cos(10*x) + 4*cos(5*x) |
- 2*|<--------------------------------- otherwise |*|<-------------------------------- otherwise | + 2*|<-------------------------------- otherwise |*|<--------------------------------- otherwise |
|| 2 | || 2 | || 2 | || 2 |
||1 - cos(10*x) + 2*(1 - cos(5*x)) | ||1 - cos(6*x) + 2*(1 - cos(3*x)) | ||1 - cos(6*x) + 2*(1 - cos(3*x)) | ||1 - cos(10*x) + 2*(1 - cos(5*x)) |
\\ / \\ / \\ / \\ /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{4 \sin{\left(3 x \right)} - 2 \sin{\left(6 x \right)}}{2 \left(- \cos{\left(3 x \right)} + 1\right)^{2} - \cos{\left(6 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{4 \cos{\left(5 x \right)} - 2 \cos{\left(10 x \right)} - 2}{2 \left(- \cos{\left(5 x \right)} + 1\right)^{2} - \cos{\left(10 x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{4 \sin{\left(5 x \right)} - 2 \sin{\left(10 x \right)}}{2 \left(- \cos{\left(5 x \right)} + 1\right)^{2} - \cos{\left(10 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{4 \cos{\left(3 x \right)} - 2 \cos{\left(6 x \right)} - 2}{2 \left(- \cos{\left(3 x \right)} + 1\right)^{2} - \cos{\left(6 x \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*x mod 2*pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || |
// 0 for 5*x mod pi = 0\ || 2 | // 0 for 3*x mod pi = 0\ || 2 |
|| | || sin (3*x) | || | || sin (5*x) |
|| sin(5*x) | ||-1 + ----------- | || sin(3*x) | ||-1 + ----------- |
||--------------------------- otherwise | || 4/3*x\ | ||--------------------------- otherwise | || 4/5*x\ |
||/ 2 \ | || 4*sin |---| | ||/ 2 \ | || 4*sin |---| |
- 2*|<| sin (5*x) | 2/5*x\ |*|< \ 2 / | + 2*|<| sin (3*x) | 2/3*x\ |*|< \ 2 / |
|||1 + -----------|*sin |---| | ||---------------- otherwise | |||1 + -----------|*sin |---| | ||---------------- otherwise |
||| 4/5*x\| \ 2 / | || 2 | ||| 4/3*x\| \ 2 / | || 2 |
||| 4*sin |---|| | || sin (3*x) | ||| 4*sin |---|| | || sin (5*x) |
||\ \ 2 // | ||1 + ----------- | ||\ \ 2 // | ||1 + ----------- |
\\ / || 4/3*x\ | \\ / || 4/5*x\ |
|| 4*sin |---| | || 4*sin |---| |
\\ \ 2 / / \\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{\sin{\left(3 x \right)}}{\left(1 + \frac{\sin^{2}{\left(3 x \right)}}{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}\right) \sin^{2}{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(5 x \right)}}{4 \sin^{4}{\left(\frac{5 x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(5 x \right)}}{4 \sin^{4}{\left(\frac{5 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{\sin{\left(5 x \right)}}{\left(1 + \frac{\sin^{2}{\left(5 x \right)}}{4 \sin^{4}{\left(\frac{5 x}{2} \right)}}\right) \sin^{2}{\left(\frac{5 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(3 x \right)}}{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(3 x \right)}}{4 \sin^{4}{\left(\frac{3 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 5*x mod pi = 0\ // 1 for 3*x mod 2*pi = 0\ // 0 for 3*x mod pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || | || | || |
||/ 0 for 5*x mod pi = 0 | ||/ 1 for 3*x mod 2*pi = 0 | ||/ 0 for 3*x mod pi = 0 | ||/ 1 for 5*x mod 2*pi = 0 |
||| | ||| | ||| | ||| |
||| /5*x\ | ||| 2/3*x\ | ||| /3*x\ | ||| 2/5*x\ |
- 2*|<| 2*cot|---| |*|<|-1 + cot |---| | + 2*|<| 2*cot|---| |*|<|-1 + cot |---| |
||< \ 2 / otherwise | ||< \ 2 / otherwise | ||< \ 2 / otherwise | ||< \ 2 / otherwise |
|||------------- otherwise | |||-------------- otherwise | |||------------- otherwise | |||-------------- otherwise |
||| 2/5*x\ | ||| 2/3*x\ | ||| 2/3*x\ | ||| 2/5*x\ |
|||1 + cot |---| | |||1 + cot |---| | |||1 + cot |---| | |||1 + cot |---| |
\\\ \ 2 / / \\\ \ 2 / / \\\ \ 2 / / \\\ \ 2 / /
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*x mod 2*pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || |
// 0 for 5*x mod pi = 0\ || 2/3*x\ | // 0 for 3*x mod pi = 0\ || 2/5*x\ |
|| | || cos |---| | || | || cos |---| |
|| /5*x\ | || \ 2 / | || /3*x\ | || \ 2 / |
|| 2*cos|---| | ||-1 + ---------------- | || 2*cos|---| | ||-1 + ---------------- |
|| \ 2 / | || 2/ pi 3*x\ | || \ 2 / | || 2/ pi 5*x\ |
||-------------------------------------- otherwise | || cos |- -- + ---| | ||-------------------------------------- otherwise | || cos |- -- + ---| |
- 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{3 x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{3 x}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{5 x}{2} \right)}}{\cos^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{5 x}{2} \right)}}{\cos^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{5 x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{5 x}{2} \right)}}{\cos^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 x}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 x}{2} \right)}}{\cos^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*x mod 2*pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || |
// 0 for 5*x mod pi = 0\ || 2/ pi 3*x\ | // 0 for 3*x mod pi = 0\ || 2/ pi 5*x\ |
|| | || sec |- -- + ---| | || | || sec |- -- + ---| |
|| / pi 5*x\ | || \ 2 2 / | || / pi 3*x\ | || \ 2 2 / |
|| 2*sec|- -- + ---| | ||-1 + ---------------- | || 2*sec|- -- + ---| | ||-1 + ---------------- |
|| \ 2 2 / | || 2/3*x\ | || \ 2 2 / | || 2/5*x\ |
||------------------------------- otherwise | || sec |---| | ||------------------------------- otherwise | || sec |---| |
- 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} \right)}}\right) \sec{\left(\frac{3 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{5 x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{5 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{5 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{5 x}{2} \right)}}\right) \sec{\left(\frac{5 x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*x mod 2*pi = 0\ // 1 for 5*x mod 2*pi = 0\
|| | || |
// 0 for 5*x mod pi = 0\ || 2/3*x\ | // 0 for 3*x mod pi = 0\ || 2/5*x\ |
|| | || csc |---| | || | || csc |---| |
|| /5*x\ | || \ 2 / | || /3*x\ | || \ 2 / |
|| 2*csc|---| | ||-1 + -------------- | || 2*csc|---| | ||-1 + -------------- |
|| \ 2 / | || 2/pi 3*x\ | || \ 2 / | || 2/pi 5*x\ |
||---------------------------------- otherwise | || csc |-- - ---| | ||---------------------------------- otherwise | || csc |-- - ---| |
- 2*|
$$\left(2 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{3 x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{3 x}{2} \right)}}{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{5 x}{2} \right)}}{\csc^{2}{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{5 x}{2} \right)}}{\csc^{2}{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(2 \left(\begin{cases} 0 & \text{for}\: 5 x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{5 x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{5 x}{2} \right)}}{\csc^{2}{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{5 x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 3 x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 x}{2} \right)}}{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 x}{2} \right)}}{\csc^{2}{\left(- \frac{3 x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
-2*Piecewise((0, Mod(5*x = pi, 0)), (2*csc(5*x/2)/((1 + csc(5*x/2)^2/csc(pi/2 - 5*x/2)^2)*csc(pi/2 - 5*x/2)), True))*Piecewise((1, Mod(3*x = 2*pi, 0)), ((-1 + csc(3*x/2)^2/csc(pi/2 - 3*x/2)^2)/(1 + csc(3*x/2)^2/csc(pi/2 - 3*x/2)^2), True)) + 2*Piecewise((0, Mod(3*x = pi, 0)), (2*csc(3*x/2)/((1 + csc(3*x/2)^2/csc(pi/2 - 3*x/2)^2)*csc(pi/2 - 3*x/2)), True))*Piecewise((1, Mod(5*x = 2*pi, 0)), ((-1 + csc(5*x/2)^2/csc(pi/2 - 5*x/2)^2)/(1 + csc(5*x/2)^2/csc(pi/2 - 5*x/2)^2), True))