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10*(1+1)*cos(3*pi*x/4)/(pi^2*3^2)-2*(1+3)*sin(3*pi*x/4)/pi*3
Общий знаменатель 10*(1+1)*cos(3*pi*x/4)/(pi^2*3^2)-2*(1+3)*sin(3*pi*x/4)/pi*3
Решение
Вы ввели
[src]
/3*pi*x\
10*(1 + 1)*cos|------|
\ 4 / /3*pi*x\ 1
---------------------- - 2*(1 + 3)*sin|------|*--*3
2 2 \ 4 / pi
pi *3
$$- \frac{2 \cdot 3 \cdot \left(1 + 3\right) \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi} + \frac{10 \cdot \left(1 + 1\right) \cos{\left(\frac{3 \pi x}{4} \right)}}{3^{2} \pi^{2}}$$
10*(1 + 1)*cos(3*pi*x/4)/((pi^2*3^2)) - 2*(1 + 3)*sin(3*pi*x/4)*3/pi
Общее упрощение
[src]
/ /3*pi*x\ /3*pi*x\\
4*|5*cos|------| - 54*pi*sin|------||
\ \ 4 / \ 4 //
-------------------------------------
2
9*pi
$$\frac{4 \left(- 54 \pi \sin{\left(\frac{3 \pi x}{4} \right)} + 5 \cos{\left(\frac{3 \pi x}{4} \right)}\right)}{9 \pi^{2}}$$
4*(5*cos(3*pi*x/4) - 54*pi*sin(3*pi*x/4))/(9*pi^2)
Рациональный знаменатель
[src]
/3*pi*x\ /3*pi*x\
24*sin|------| 20*cos|------|
\ 4 / \ 4 /
- -------------- + --------------
pi 2
9*pi
$$- \frac{24 \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi} + \frac{20 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{2}}$$
2 /3*pi*x\ /3*pi*x\
- 216*pi *sin|------| + 20*pi*cos|------|
\ 4 / \ 4 /
-----------------------------------------
3
9*pi
$$\frac{- 216 \pi^{2} \sin{\left(\frac{3 \pi x}{4} \right)} + 20 \pi \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{3}}$$
(-216*pi^2*sin(3*pi*x/4) + 20*pi*cos(3*pi*x/4))/(9*pi^3)
Объединение рациональных выражений
[src]
/ /3*pi*x\ /3*pi*x\\
4*|5*cos|------| - 54*pi*sin|------||
\ \ 4 / \ 4 //
-------------------------------------
2
9*pi
$$\frac{4 \left(- 54 \pi \sin{\left(\frac{3 \pi x}{4} \right)} + 5 \cos{\left(\frac{3 \pi x}{4} \right)}\right)}{9 \pi^{2}}$$
4*(5*cos(3*pi*x/4) - 54*pi*sin(3*pi*x/4))/(9*pi^2)
Собрать выражение
[src]
/3*pi*x\ /3*pi*x\
24*sin|------| 20*cos|------|
\ 4 / \ 4 /
- -------------- + --------------
pi 2
9*pi
$$- \frac{24 \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi} + \frac{20 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{2}}$$
-24*sin(3*pi*x/4)/pi + 20*cos(3*pi*x/4)/(9*pi^2)
Тригонометрическая часть
[src]
24 20
- -------------- + -----------------
/3*pi*x\ 2 /3*pi*x\
pi*csc|------| 9*pi *sec|------|
\ 4 / \ 4 /
$$\frac{20}{9 \pi^{2} \sec{\left(\frac{3 \pi x}{4} \right)}} - \frac{24}{\pi \csc{\left(\frac{3 \pi x}{4} \right)}}$$
/3*pi*x\ /3*pi*x\
24*sin|------| 20*cos|------|
\ 4 / \ 4 /
- -------------- + --------------
pi 2
9*pi
$$- \frac{24 \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi} + \frac{20 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{2}}$$
/ /3*pi*x\\
| 5*cos|------||
| /3*pi*x\ \ 4 /|
4*|- 6*sin|------| + -------------|
\ \ 4 / 9*pi /
-----------------------------------
pi
$$\frac{4 \left(- 6 \sin{\left(\frac{3 \pi x}{4} \right)} + \frac{5 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi}\right)}{\pi}$$
/ pi 3*pi*x\ /3*pi*x\
24*cos|- -- + ------| 20*cos|------|
\ 2 4 / \ 4 /
- --------------------- + --------------
pi 2
9*pi
$$\frac{20 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{2}} - \frac{24 \cos{\left(\frac{3 \pi x}{4} - \frac{\pi}{2} \right)}}{\pi}$$
24 20
- -------------- + ----------------------
/3*pi*x\ 2 /pi 3*pi*x\
pi*csc|------| 9*pi *csc|-- - ------|
\ 4 / \2 4 /
$$\frac{20}{9 \pi^{2} \csc{\left(- \frac{3 \pi x}{4} + \frac{\pi}{2} \right)}} - \frac{24}{\pi \csc{\left(\frac{3 \pi x}{4} \right)}}$$
/3*pi*x\ /pi 3*pi*x\
24*sin|------| 20*sin|-- + ------|
\ 4 / \2 4 /
- -------------- + -------------------
pi 2
9*pi
$$- \frac{24 \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi} + \frac{20 \sin{\left(\frac{3 \pi x}{4} + \frac{\pi}{2} \right)}}{9 \pi^{2}}$$
24 20
- --------------------- + -----------------
/ pi 3*pi*x\ 2 /3*pi*x\
pi*sec|- -- + ------| 9*pi *sec|------|
\ 2 4 / \ 4 /
$$- \frac{24}{\pi \sec{\left(\frac{3 \pi x}{4} - \frac{\pi}{2} \right)}} + \frac{20}{9 \pi^{2} \sec{\left(\frac{3 \pi x}{4} \right)}}$$
24 20
- ------------------- + -----------------
/pi 3*pi*x\ 2 /3*pi*x\
pi*sec|-- - ------| 9*pi *sec|------|
\2 4 / \ 4 /
$$- \frac{24}{\pi \sec{\left(- \frac{3 \pi x}{4} + \frac{\pi}{2} \right)}} + \frac{20}{9 \pi^{2} \sec{\left(\frac{3 \pi x}{4} \right)}}$$
/ /pi 3*pi*x\\
| 5*sin|-- + ------||
| /3*pi*x\ \2 4 /|
4*|- 6*sin|------| + ------------------|
\ \ 4 / 9*pi /
----------------------------------------
pi
$$\frac{4 \left(- 6 \sin{\left(\frac{3 \pi x}{4} \right)} + \frac{5 \sin{\left(\frac{3 \pi x}{4} + \frac{\pi}{2} \right)}}{9 \pi}\right)}{\pi}$$
24 20
- ------------------- + ----------------------
/ 3*pi*x\ 2 /pi 3*pi*x\
pi*csc|pi - ------| 9*pi *csc|-- - ------|
\ 4 / \2 4 /
$$- \frac{24}{\pi \csc{\left(- \frac{3 \pi x}{4} + \pi \right)}} + \frac{20}{9 \pi^{2} \csc{\left(- \frac{3 \pi x}{4} + \frac{\pi}{2} \right)}}$$
/ /3*pi*x\\
| 5*cos|------||
| / pi 3*pi*x\ \ 4 /|
4*|- 6*cos|- -- + ------| + -------------|
\ \ 2 4 / 9*pi /
------------------------------------------
pi
$$\frac{4 \cdot \left(\frac{5 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi} - 6 \cos{\left(\frac{3 \pi x}{4} - \frac{\pi}{2} \right)}\right)}{\pi}$$
/ 2/3*pi*x\\
| 10*cos |------||
| /3*pi*x\ 5 \ 8 /|
4*|- 6*sin|------| - ---- + ---------------|
\ \ 4 / 9*pi 9*pi /
--------------------------------------------
pi
$$\frac{4 \left(\frac{10 \cos^{2}{\left(\frac{3 \pi x}{8} \right)}}{9 \pi} - 6 \sin{\left(\frac{3 \pi x}{4} \right)} - \frac{5}{9 \pi}\right)}{\pi}$$
/ 6 5 \
4*|- ------------------ + ----------------|
| / pi 3*pi*x\ /3*pi*x\|
| sec|- -- + ------| 9*pi*sec|------||
\ \ 2 4 / \ 4 //
-------------------------------------------
pi
$$\frac{4 \left(- \frac{6}{\sec{\left(\frac{3 \pi x}{4} - \frac{\pi}{2} \right)}} + \frac{5}{9 \pi \sec{\left(\frac{3 \pi x}{4} \right)}}\right)}{\pi}$$
/ 6 5 \
4*|- ----------- + ---------------------|
| /3*pi*x\ /pi 3*pi*x\|
| csc|------| 9*pi*csc|-- - ------||
\ \ 4 / \2 4 //
-----------------------------------------
pi
$$\frac{4 \cdot \left(\frac{5}{9 \pi \csc{\left(- \frac{3 \pi x}{4} + \frac{\pi}{2} \right)}} - \frac{6}{\csc{\left(\frac{3 \pi x}{4} \right)}}\right)}{\pi}$$
/3*pi*x\
6*(1 + 3)*sin|------|
\ 4 / 1 /3*pi*x\
- --------------------- + 10*-----*(1 + 1)*cos|------|
pi 2 \ 4 /
9*pi
$$- \frac{6 \cdot \left(1 + 3\right) \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi} + 10 \cdot \frac{1}{9 \pi^{2}} \cdot \left(1 + 1\right) \cos{\left(\frac{3 \pi x}{4} \right)}$$
/3*pi*x\ / 2/pi 3*pi*x\\ / /3*pi*x\\
20*cos|------| 12*|1 - cot |-- + ------||*|1 + sin|------||
\ 4 / \ \4 8 // \ \ 4 //
-------------- - --------------------------------------------
2 pi
9*pi
$$- \frac{12 \cdot \left(- \cot^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(\frac{3 \pi x}{4} \right)} + 1\right)}{\pi} + \frac{20 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{2}}$$
/3*pi*x\ / 2/3*pi*x\\
48*tan|------| 20*|1 - tan |------||
\ 8 / \ \ 8 //
- --------------------- + ------------------------
/ 2/3*pi*x\\ 2 / 2/3*pi*x\\
pi*|1 + tan |------|| 9*pi *|1 + tan |------||
\ \ 8 // \ \ 8 //
$$\frac{20 \cdot \left(- \tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)}{9 \pi^{2} \left(\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)} - \frac{48 \tan{\left(\frac{3 \pi x}{8} \right)}}{\pi \left(\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)}$$
/3*pi*x\ /pi 3*pi*x\
48*tan|------| 40*tan|-- + ------|
\ 8 / \4 8 /
- --------------------- + -----------------------------
/ 2/3*pi*x\\ 2 / 2/pi 3*pi*x\\
pi*|1 + tan |------|| 9*pi *|1 + tan |-- + ------||
\ \ 8 // \ \4 8 //
$$\frac{40 \tan{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)}}{9 \pi^{2} \left(\tan^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1\right)} - \frac{48 \tan{\left(\frac{3 \pi x}{8} \right)}}{\pi \left(\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)}$$
/3*pi*x\ /pi 3*pi*x\
48*cot|------| 40*tan|-- + ------|
\ 8 / \4 8 /
- --------------------- + -----------------------------
/ 2/3*pi*x\\ 2 / 2/pi 3*pi*x\\
pi*|1 + cot |------|| 9*pi *|1 + tan |-- + ------||
\ \ 8 // \ \4 8 //
$$- \frac{48 \cot{\left(\frac{3 \pi x}{8} \right)}}{\pi \left(\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)} + \frac{40 \tan{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)}}{9 \pi^{2} \left(\tan^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1\right)}$$
/ /3*pi*x\ / 2/3*pi*x\\ \
| 12*tan|------| 5*|1 - tan |------|| |
| \ 8 / \ \ 8 // |
4*|- ---------------- + -----------------------|
| 2/3*pi*x\ / 2/3*pi*x\\|
| 1 + tan |------| 9*pi*|1 + tan |------|||
\ \ 8 / \ \ 8 ///
------------------------------------------------
pi
$$\frac{4 \cdot \left(\frac{5 \cdot \left(- \tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)}{9 \pi \left(\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)} - \frac{12 \tan{\left(\frac{3 \pi x}{8} \right)}}{\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1}\right)}{\pi}$$
/ 1 \
20*|1 - ------------|
| 2/3*pi*x\|
| cot |------||
48 \ \ 8 //
- --------------------------------- + ------------------------
/ 1 \ /3*pi*x\ 2 / 1 \
pi*|1 + ------------|*cot|------| 9*pi *|1 + ------------|
| 2/3*pi*x\| \ 8 / | 2/3*pi*x\|
| cot |------|| | cot |------||
\ \ 8 // \ \ 8 //
$$\frac{20 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)}}\right)}{9 \pi^{2} \cdot \left(1 + \frac{1}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)}}\right)} - \frac{48}{\pi \left(1 + \frac{1}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)}}\right) \cot{\left(\frac{3 \pi x}{8} \right)}}$$
/ 2/pi 3*pi*x\\ / 2/3*pi*x\\
24*|-1 + tan |-- + ------|| 20*|-1 + cot |------||
\ \4 8 // \ \ 8 //
- --------------------------- + ------------------------
/ 2/pi 3*pi*x\\ 2 / 2/3*pi*x\\
pi*|1 + tan |-- + ------|| 9*pi *|1 + cot |------||
\ \4 8 // \ \ 8 //
$$- \frac{24 \left(\tan^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} - 1\right)}{\pi \left(\tan^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1\right)} + \frac{20 \left(\cot^{2}{\left(\frac{3 \pi x}{8} \right)} - 1\right)}{9 \pi^{2} \left(\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)}$$
/ 2/pi 3*pi*x\\ / 2/3*pi*x\\
24*|1 - cot |-- + ------|| 20*|1 - tan |------||
\ \4 8 // \ \ 8 //
- -------------------------- + ------------------------
/ 2/pi 3*pi*x\\ 2 / 2/3*pi*x\\
pi*|1 + cot |-- + ------|| 9*pi *|1 + tan |------||
\ \4 8 // \ \ 8 //
$$\frac{20 \cdot \left(- \tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)}{9 \pi^{2} \left(\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1\right)} - \frac{24 \cdot \left(- \cot^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1\right)}{\pi \left(\cot^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1\right)}$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
24*|< | 20*|< |
|| /3*pi*x\ | || /3*pi*x\ |
||sin|------| otherwise | ||cos|------| otherwise |
\\ \ 4 / / \\ \ 4 / /
- ----------------------------------------- + -----------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\sin{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\cos{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
/ // /3*x \ \\
| || 1 for pi*|--- mod 2| = 0||
| || \ 4 / ||
| 5*|< ||
| // /3*x \ \ || /3*pi*x\ ||
| || 0 for pi*|--- mod 1| = 0| ||cos|------| otherwise ||
| || \ 4 / | \\ \ 4 / /|
4*|- 6*|< | + ----------------------------------------|
| || /3*pi*x\ | 9*pi |
| ||sin|------| otherwise | |
\ \\ \ 4 / / /
-----------------------------------------------------------------------------------------
pi
$$\frac{4 \left(\left(- 6 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\sin{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{5 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\cos{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{9 \pi}\right)\right)}{\pi}$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
24*|< | 20*|< |
|| / pi 3*pi*x\ | || /3*pi*x\ |
||cos|- -- + ------| otherwise | ||cos|------| otherwise |
\\ \ 2 4 / / \\ \ 4 / /
- ------------------------------------------------ + -----------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\cos{\left(\frac{3 \pi x}{4} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\cos{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
24*|< | 20*|< |
|| /3*pi*x\ | || /pi 3*pi*x\ |
||sin|------| otherwise | ||sin|-- + ------| otherwise |
\\ \ 4 / / \\ \2 4 / /
- ----------------------------------------- + ----------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\sin{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\sin{\left(\frac{3 \pi x}{4} + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// //3 3*x\ \ \ // /3*x \ \
|| 1 for pi*||- + ---| mod 2| = 0| || 1 for pi*|--- mod 2| = 0|
|| \\2 4 / / | || \ 4 / |
24*|< | 20*|< |
|| /3*pi*x\ | || /3*pi*x\ |
||sin|------| otherwise | ||cos|------| otherwise |
\\ \ 4 / / \\ \ 4 / /
- ----------------------------------------------- + -----------------------------------------
pi 2
9*pi
$$\left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\cos{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right) - \left(\frac{24 \left(\begin{cases} 1 & \text{for}\: \pi \left(\left(\frac{3 x}{4} + \frac{3}{2}\right) \bmod 2\right) = 0 \\\sin{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{\pi}\right)$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
|| | || |
24*|< 1 | 20*|< 1 |
||------------------ otherwise | ||----------- otherwise |
|| / pi 3*pi*x\ | || /3*pi*x\ |
||sec|- -- + ------| | ||sec|------| |
\\ \ 2 4 / / \\ \ 4 / /
- ------------------------------------------------ + -----------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{1}{\sec{\left(\frac{3 \pi x}{4} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{1}{\sec{\left(\frac{3 \pi x}{4} \right)}} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
|| | || |
24*|< 1 | 20*|< 1 |
||----------- otherwise | ||---------------- otherwise |
|| /3*pi*x\ | || /pi 3*pi*x\ |
||csc|------| | ||csc|-- - ------| |
\\ \ 4 / / \\ \2 4 / /
- ----------------------------------------- + ----------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{1}{\csc{\left(\frac{3 \pi x}{4} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{1}{\csc{\left(- \frac{3 \pi x}{4} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
/ /3*pi*x\ /3*pi*x\\ 2/3*pi*x\ /3*pi*x\
40*|-1 - cos|------| + 2*cos|------|| 96*sin |------|*sin|------|
\ \ 2 / \ 4 // \ 8 / \ 4 /
---------------------------------------------- - ----------------------------------
/ 2\ / 2/3*pi*x\ 4/3*pi*x\\
2 | /3*pi*x\ / /3*pi*x\\ | pi*|sin |------| + 4*sin |------||
9*pi *|1 - cos|------| + 2*|1 - cos|------|| | \ \ 4 / \ 8 //
\ \ 2 / \ \ 4 // /
$$- \frac{96 \sin^{2}{\left(\frac{3 \pi x}{8} \right)} \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi \left(4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)} + \sin^{2}{\left(\frac{3 \pi x}{4} \right)}\right)} + \frac{40 \cdot \left(2 \cos{\left(\frac{3 \pi x}{4} \right)} - \cos{\left(\frac{3 \pi x}{2} \right)} - 1\right)}{9 \pi^{2} \cdot \left(2 \left(- \cos{\left(\frac{3 \pi x}{4} \right)} + 1\right)^{2} - \cos{\left(\frac{3 \pi x}{2} \right)} + 1\right)}$$
/ 4/3*pi*x\\
| 4*sin |------||
| \ 8 /|
20*|1 - --------------|
| 2/3*pi*x\ | 2/3*pi*x\
| sin |------| | 96*sin |------|
\ \ 4 / / \ 8 /
-------------------------- - -----------------------------------
/ 4/3*pi*x\\ / 4/3*pi*x\\
| 4*sin |------|| | 4*sin |------||
2 | \ 8 /| | \ 8 /| /3*pi*x\
9*pi *|1 + --------------| pi*|1 + --------------|*sin|------|
| 2/3*pi*x\ | | 2/3*pi*x\ | \ 4 /
| sin |------| | | sin |------| |
\ \ 4 / / \ \ 4 / /
$$\frac{20 \left(- \frac{4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)}}{\sin^{2}{\left(\frac{3 \pi x}{4} \right)}} + 1\right)}{9 \pi^{2} \cdot \left(\frac{4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)}}{\sin^{2}{\left(\frac{3 \pi x}{4} \right)}} + 1\right)} - \frac{96 \sin^{2}{\left(\frac{3 \pi x}{8} \right)}}{\pi \left(\frac{4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)}}{\sin^{2}{\left(\frac{3 \pi x}{4} \right)}} + 1\right) \sin{\left(\frac{3 \pi x}{4} \right)}}$$
// /3*x \ \ // //1 3*x\ \ \
|| 0 for pi*|--- mod 1| = 0| || 0 for pi*||- + ---| mod 1| = 0|
|| \ 4 / | || \\2 4 / / |
24*|< | 20*|< |
|| /3*pi*x\ | ||/ /3*pi*x\\ /pi 3*pi*x\ |
||sin|------| otherwise | |||1 + sin|------||*cot|-- + ------| otherwise |
\\ \ 4 / / \\\ \ 4 // \4 8 / /
- ----------------------------------------- + ----------------------------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\sin{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 0 & \text{for}\: \pi \left(\left(\frac{3 x}{4} + \frac{1}{2}\right) \bmod 1\right) = 0 \\\left(\sin{\left(\frac{3 \pi x}{4} \right)} + 1\right) \cot{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
|| | || |
|| /3*pi*x\ | || 2/3*pi*x\ |
24*|< 2*cot|------| | 20*|<-1 + cot |------| |
|| \ 8 / | || \ 8 / |
||---------------- otherwise | ||----------------- otherwise |
|| 2/3*pi*x\ | || 2/3*pi*x\ |
||1 + cot |------| | || 1 + cot |------| |
\\ \ 8 / / \\ \ 8 / /
- ---------------------------------------------- + -----------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{3 \pi x}{8} \right)}}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} - 1}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
|| | || |
|| /3*pi*x\ | || 2/3*pi*x\ |
24*|< 2*tan|------| | 20*|<1 - tan |------| |
|| \ 8 / | || \ 8 / |
||---------------- otherwise | ||---------------- otherwise |
|| 2/3*pi*x\ | || 2/3*pi*x\ |
||1 + tan |------| | ||1 + tan |------| |
\\ \ 8 / / \\ \ 8 / /
- ---------------------------------------------- + ----------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \tan{\left(\frac{3 \pi x}{8} \right)}}{\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{- \tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1}{\tan^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
/ 2/3*pi*x\ \
| sec |------| |
| \ 8 / |
20*|1 - -------------------|
| 2/ pi 3*pi*x\| /3*pi*x\
| sec |- -- + ------|| 48*sec|------|
\ \ 2 8 // \ 8 /
------------------------------- - -----------------------------------------------
/ 2/3*pi*x\ \ / 2/3*pi*x\ \
| sec |------| | | sec |------| |
2 | \ 8 / | | \ 8 / | / pi 3*pi*x\
9*pi *|1 + -------------------| pi*|1 + -------------------|*sec|- -- + ------|
| 2/ pi 3*pi*x\| | 2/ pi 3*pi*x\| \ 2 8 /
| sec |- -- + ------|| | sec |- -- + ------||
\ \ 2 8 // \ \ 2 8 //
$$\frac{20 \left(- \frac{\sec^{2}{\left(\frac{3 \pi x}{8} \right)}}{\sec^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}} + 1\right)}{9 \pi^{2} \left(\frac{\sec^{2}{\left(\frac{3 \pi x}{8} \right)}}{\sec^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}} + 1\right)} - \frac{48 \sec{\left(\frac{3 \pi x}{8} \right)}}{\pi \left(\frac{\sec^{2}{\left(\frac{3 \pi x}{8} \right)}}{\sec^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}$$
/ // /3*x \ \\
| || 1 for pi*|--- mod 2| = 0||
| || \ 4 / ||
| || ||
| || 2/3*pi*x\ ||
| 5*|<-1 + cot |------| ||
| // /3*x \ \ || \ 8 / ||
| || 0 for pi*|--- mod 1| = 0| ||----------------- otherwise ||
| || \ 4 / | || 2/3*pi*x\ ||
| || | || 1 + cot |------| ||
| || /3*pi*x\ | \\ \ 8 / /|
4*|- 6*|< 2*cot|------| | + ----------------------------------------------|
| || \ 8 / | 9*pi |
| ||---------------- otherwise | |
| || 2/3*pi*x\ | |
| ||1 + cot |------| | |
\ \\ \ 8 / / /
----------------------------------------------------------------------------------------------------
pi
$$\frac{4 \left(\left(- 6 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{3 \pi x}{8} \right)}}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{5 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} - 1}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi}\right)\right)}{\pi}$$
/ 2/ pi 3*pi*x\\
| cos |- -- + ------||
| \ 2 8 /|
20*|1 - -------------------|
| 2/3*pi*x\ | / pi 3*pi*x\
| cos |------| | 48*cos|- -- + ------|
\ \ 8 / / \ 2 8 /
------------------------------- - ----------------------------------------
/ 2/ pi 3*pi*x\\ / 2/ pi 3*pi*x\\
| cos |- -- + ------|| | cos |- -- + ------||
2 | \ 2 8 /| | \ 2 8 /| /3*pi*x\
9*pi *|1 + -------------------| pi*|1 + -------------------|*cos|------|
| 2/3*pi*x\ | | 2/3*pi*x\ | \ 8 /
| cos |------| | | cos |------| |
\ \ 8 / / \ \ 8 / /
$$\frac{20 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 \pi x}{8} \right)}}\right)}{9 \pi^{2} \cdot \left(1 + \frac{\cos^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 \pi x}{8} \right)}}\right)} - \frac{48 \cos{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\pi \left(1 + \frac{\cos^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 \pi x}{8} \right)}}\right) \cos{\left(\frac{3 \pi x}{8} \right)}}$$
/ 2/pi 3*pi*x\\
| csc |-- - ------||
| \2 8 /|
20*|1 - -----------------|
| 2/3*pi*x\ | /pi 3*pi*x\
| csc |------| | 48*csc|-- - ------|
\ \ 8 / / \2 8 /
----------------------------- - --------------------------------------
/ 2/pi 3*pi*x\\ / 2/pi 3*pi*x\\
| csc |-- - ------|| | csc |-- - ------||
2 | \2 8 /| | \2 8 /| /3*pi*x\
9*pi *|1 + -----------------| pi*|1 + -----------------|*csc|------|
| 2/3*pi*x\ | | 2/3*pi*x\ | \ 8 /
| csc |------| | | csc |------| |
\ \ 8 / / \ \ 8 / /
$$\frac{20 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 \pi x}{8} \right)}}\right)}{9 \pi^{2} \cdot \left(1 + \frac{\csc^{2}{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 \pi x}{8} \right)}}\right)} - \frac{48 \csc{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}}{\pi \left(1 + \frac{\csc^{2}{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 \pi x}{8} \right)}}\right) \csc{\left(\frac{3 \pi x}{8} \right)}}$$
// /3*x \ \
|| 1 for pi*|--- mod 2| = 0|
|| \ 4 / |
|| |
// /3*x \ \ || 1 |
|| 0 for pi*|--- mod 1| = 0| ||-1 + ------------ |
|| \ 4 / | || 2/3*pi*x\ |
|| | 20*|< tan |------| |
|| 2 | || \ 8 / |
24*|<------------------------------ otherwise | ||----------------- otherwise |
||/ 1 \ /3*pi*x\ | || 1 |
|||1 + ------------|*tan|------| | || 1 + ------------ |
||| 2/3*pi*x\| \ 8 / | || 2/3*pi*x\ |
||| tan |------|| | || tan |------| |
\\\ \ 8 // / \\ \ 8 / /
- ------------------------------------------------------------ + -----------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 \pi x}{8} \right)}}\right) \tan{\left(\frac{3 \pi x}{8} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 \pi x}{8} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 \pi x}{8} \right)}}} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \ // //1 3*x\ \ \
|| 0 for pi*|--- mod 1| = 0| || 0 for pi*||- + ---| mod 1| = 0|
|| \ 4 / | || \\2 4 / / |
|| | || |
|| /3*pi*x\ | || /pi 3*pi*x\ |
24*|< 2*cot|------| | 20*|< 2*cot|-- + ------| |
|| \ 8 / | || \4 8 / |
||---------------- otherwise | ||--------------------- otherwise |
|| 2/3*pi*x\ | || 2/pi 3*pi*x\ |
||1 + cot |------| | ||1 + cot |-- + ------| |
\\ \ 8 / / \\ \4 8 / /
- ---------------------------------------------- + ---------------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{3 \pi x}{8} \right)}}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 0 & \text{for}\: \pi \left(\left(\frac{3 x}{4} + \frac{1}{2}\right) \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// //3 3*x\ \ \ // /3*x \ \
|| 1 for pi*||- + ---| mod 2| = 0| || 1 for pi*|--- mod 2| = 0|
|| \\2 4 / / | || \ 4 / |
|| | || |
|| 2/pi 3*pi*x\ | || 2/3*pi*x\ |
24*|<-1 + tan |-- + ------| | 20*|<-1 + cot |------| |
|| \4 8 / | || \ 8 / |
||---------------------- otherwise | ||----------------- otherwise |
|| 2/pi 3*pi*x\ | || 2/3*pi*x\ |
||1 + tan |-- + ------| | || 1 + cot |------| |
\\ \4 8 / / \\ \ 8 / /
- ---------------------------------------------------------- + -----------------------------------------------
pi 2
9*pi
$$\left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} - 1}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right) - \left(\frac{24 \left(\begin{cases} 1 & \text{for}\: \pi \left(\left(\frac{3 x}{4} + \frac{3}{2}\right) \bmod 2\right) = 0 \\\frac{\tan^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{3 \pi x}{8} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\pi}\right)$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
|| | || |
||/ /3*x \ | ||/ /3*x \ |
24*|<| 0 for pi*|--- mod 1| = 0 | 20*|<| 1 for pi*|--- mod 2| = 0 |
||| \ 4 / | ||| \ 4 / |
||< otherwise | ||< otherwise |
||| /3*pi*x\ | ||| /3*pi*x\ |
|||sin|------| otherwise | |||cos|------| otherwise |
\\\ \ 4 / / \\\ \ 4 / /
- ------------------------------------------------------------------ + ------------------------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\sin{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\cos{\left(\frac{3 \pi x}{4} \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \
|| 0 for pi*|--- mod 1| = 0|
|| \ 4 / | // /3*x \ \
|| | || 1 for pi*|--- mod 2| = 0|
|| /3*pi*x\ | || \ 4 / |
|| sin|------| | || |
|| \ 4 / | || /3*pi*x\ /3*pi*x\ |
24*|<--------------------------------- otherwise | || -2 - 2*cos|------| + 4*cos|------| |
||/ 2/3*pi*x\ \ | 20*|< \ 2 / \ 4 / |
||| sin |------| | | ||-------------------------------------- otherwise |
||| \ 4 / | 2/3*pi*x\ | || 2 |
|||1 + --------------|*sin |------| | || /3*pi*x\ / /3*pi*x\\ |
||| 4/3*pi*x\| \ 8 / | ||1 - cos|------| + 2*|1 - cos|------|| |
||| 4*sin |------|| | || \ 2 / \ \ 4 // |
\\\ \ 8 // / \\ /
- --------------------------------------------------------------- + --------------------------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{\sin{\left(\frac{3 \pi x}{4} \right)}}{\left(1 + \frac{\sin^{2}{\left(\frac{3 \pi x}{4} \right)}}{4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)}}\right) \sin^{2}{\left(\frac{3 \pi x}{8} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{4 \cos{\left(\frac{3 \pi x}{4} \right)} - 2 \cos{\left(\frac{3 \pi x}{2} \right)} - 2}{2 \left(- \cos{\left(\frac{3 \pi x}{4} \right)} + 1\right)^{2} - \cos{\left(\frac{3 \pi x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \
|| 1 for pi*|--- mod 2| = 0|
|| \ 4 / |
|| |
// /3*x \ \ || 2/3*pi*x\ |
|| 0 for pi*|--- mod 1| = 0| || sin |------| |
|| \ 4 / | || \ 4 / |
|| | ||-1 + -------------- |
|| /3*pi*x\ | || 4/3*pi*x\ |
|| sin|------| | 20*|< 4*sin |------| |
|| \ 4 / | || \ 8 / |
24*|<--------------------------------- otherwise | ||------------------- otherwise |
||/ 2/3*pi*x\ \ | || 2/3*pi*x\ |
||| sin |------| | | || sin |------| |
||| \ 4 / | 2/3*pi*x\ | || \ 4 / |
|||1 + --------------|*sin |------| | || 1 + -------------- |
||| 4/3*pi*x\| \ 8 / | || 4/3*pi*x\ |
||| 4*sin |------|| | || 4*sin |------| |
\\\ \ 8 // / \\ \ 8 / /
- --------------------------------------------------------------- + -------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{\sin{\left(\frac{3 \pi x}{4} \right)}}{\left(1 + \frac{\sin^{2}{\left(\frac{3 \pi x}{4} \right)}}{4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)}}\right) \sin^{2}{\left(\frac{3 \pi x}{8} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{-1 + \frac{\sin^{2}{\left(\frac{3 \pi x}{4} \right)}}{4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)}}}{1 + \frac{\sin^{2}{\left(\frac{3 \pi x}{4} \right)}}{4 \sin^{4}{\left(\frac{3 \pi x}{8} \right)}}} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \ // /3*x \ \
|| 0 for pi*|--- mod 1| = 0| || 1 for pi*|--- mod 2| = 0|
|| \ 4 / | || \ 4 / |
|| | || |
||/ /3*x \ | ||/ /3*x \ |
||| 0 for pi*|--- mod 1| = 0 | ||| 1 for pi*|--- mod 2| = 0 |
||| \ 4 / | ||| \ 4 / |
24*|<| | 20*|<| |
||| /3*pi*x\ | ||| 2/3*pi*x\ |
||< 2*cot|------| otherwise | ||<-1 + cot |------| otherwise |
||| \ 8 / | ||| \ 8 / |
|||---------------- otherwise | |||----------------- otherwise |
||| 2/3*pi*x\ | ||| 2/3*pi*x\ |
|||1 + cot |------| | ||| 1 + cot |------| |
\\\ \ 8 / / \\\ \ 8 / /
- ----------------------------------------------------------------------- + ------------------------------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \cot{\left(\frac{3 \pi x}{8} \right)}}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} - 1}{\cot^{2}{\left(\frac{3 \pi x}{8} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \
|| 1 for pi*|--- mod 2| = 0|
|| \ 4 / |
|| |
// /3*x \ \ || 2/3*pi*x\ |
|| 0 for pi*|--- mod 1| = 0| || cos |------| |
|| \ 4 / | || \ 8 / |
|| | ||-1 + ------------------- |
|| /3*pi*x\ | || 2/ pi 3*pi*x\ |
|| 2*cos|------| | 20*|< cos |- -- + ------| |
|| \ 8 / | || \ 2 8 / |
24*|<-------------------------------------------- otherwise | ||------------------------ otherwise |
||/ 2/3*pi*x\ \ | || 2/3*pi*x\ |
||| cos |------| | | || cos |------| |
||| \ 8 / | / pi 3*pi*x\ | || \ 8 / |
|||1 + -------------------|*cos|- -- + ------| | ||1 + ------------------- |
||| 2/ pi 3*pi*x\| \ 2 8 / | || 2/ pi 3*pi*x\ |
||| cos |- -- + ------|| | || cos |- -- + ------| |
\\\ \ 2 8 // / \\ \ 2 8 / /
- -------------------------------------------------------------------------- + ------------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \cos{\left(\frac{3 \pi x}{8} \right)}}{\left(\frac{\cos^{2}{\left(\frac{3 \pi x}{8} \right)}}{\cos^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 \pi x}{8} \right)}}{\cos^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 \pi x}{8} \right)}}{\cos^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \
|| 1 for pi*|--- mod 2| = 0|
|| \ 4 / |
|| |
// /3*x \ \ || 2/ pi 3*pi*x\ |
|| 0 for pi*|--- mod 1| = 0| || sec |- -- + ------| |
|| \ 4 / | || \ 2 8 / |
|| | ||-1 + ------------------- |
|| / pi 3*pi*x\ | || 2/3*pi*x\ |
|| 2*sec|- -- + ------| | 20*|< sec |------| |
|| \ 2 8 / | || \ 8 / |
24*|<------------------------------------- otherwise | ||------------------------ otherwise |
||/ 2/ pi 3*pi*x\\ | || 2/ pi 3*pi*x\ |
||| sec |- -- + ------|| | || sec |- -- + ------| |
||| \ 2 8 /| /3*pi*x\ | || \ 2 8 / |
|||1 + -------------------|*sec|------| | ||1 + ------------------- |
||| 2/3*pi*x\ | \ 8 / | || 2/3*pi*x\ |
||| sec |------| | | || sec |------| |
\\\ \ 8 / / / \\ \ 8 / /
- ------------------------------------------------------------------- + ------------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \sec{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 \pi x}{8} \right)}}\right) \sec{\left(\frac{3 \pi x}{8} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 \pi x}{8} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 \pi x}{8} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 \pi x}{8} \right)}}} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
// /3*x \ \
|| 1 for pi*|--- mod 2| = 0|
|| \ 4 / |
|| |
// /3*x \ \ || 2/3*pi*x\ |
|| 0 for pi*|--- mod 1| = 0| || csc |------| |
|| \ 4 / | || \ 8 / |
|| | ||-1 + ----------------- |
|| /3*pi*x\ | || 2/pi 3*pi*x\ |
|| 2*csc|------| | 20*|< csc |-- - ------| |
|| \ 8 / | || \2 8 / |
24*|<---------------------------------------- otherwise | ||---------------------- otherwise |
||/ 2/3*pi*x\ \ | || 2/3*pi*x\ |
||| csc |------| | | || csc |------| |
||| \ 8 / | /pi 3*pi*x\ | || \ 8 / |
|||1 + -----------------|*csc|-- - ------| | ||1 + ----------------- |
||| 2/pi 3*pi*x\| \2 8 / | || 2/pi 3*pi*x\ |
||| csc |-- - ------|| | || csc |-- - ------| |
\\\ \2 8 // / \\ \2 8 / /
- ---------------------------------------------------------------------- + ----------------------------------------------------
pi 2
9*pi
$$\left(- \frac{24 \left(\begin{cases} 0 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 1\right) = 0 \\\frac{2 \csc{\left(\frac{3 \pi x}{8} \right)}}{\left(\frac{\csc^{2}{\left(\frac{3 \pi x}{8} \right)}}{\csc^{2}{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\pi}\right) + \left(\frac{20 \left(\begin{cases} 1 & \text{for}\: \pi \left(\frac{3 x}{4} \bmod 2\right) = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 \pi x}{8} \right)}}{\csc^{2}{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 \pi x}{8} \right)}}{\csc^{2}{\left(- \frac{3 \pi x}{8} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{9 \pi^{2}}\right)$$
-24*Piecewise((0, pi*(Mod(3*x/4 = 1), 0)), (2*csc(3*pi*x/8)/((1 + csc(3*pi*x/8)^2/csc(pi/2 - 3*pi*x/8)^2)*csc(pi/2 - 3*pi*x/8)), True))/pi + 20*Piecewise((1, pi*(Mod(3*x/4 = 2), 0)), ((-1 + csc(3*pi*x/8)^2/csc(pi/2 - 3*pi*x/8)^2)/(1 + csc(3*pi*x/8)^2/csc(pi/2 - 3*pi*x/8)^2), True))/(9*pi^2)
Комбинаторика
[src]
/ /3*pi*x\ /3*pi*x\\
4*|5*cos|------| - 54*pi*sin|------||
\ \ 4 / \ 4 //
-------------------------------------
2
9*pi
$$\frac{4 \left(- 54 \pi \sin{\left(\frac{3 \pi x}{4} \right)} + 5 \cos{\left(\frac{3 \pi x}{4} \right)}\right)}{9 \pi^{2}}$$
4*(5*cos(3*pi*x/4) - 54*pi*sin(3*pi*x/4))/(9*pi^2)
Численный ответ
[src]
0.225158185871862*cos(3*pi*x/4) - 7.63943726841098*sin(3*pi*x/4)
0.225158185871862*cos(3*pi*x/4) - 7.63943726841098*sin(3*pi*x/4)
Степени
[src]
/3*pi*x\ /3*pi*x\
24*sin|------| 20*cos|------|
\ 4 / \ 4 /
- -------------- + --------------
pi 2
9*pi
$$- \frac{24 \sin{\left(\frac{3 \pi x}{4} \right)}}{\pi} + \frac{20 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{2}}$$
/ -3*pi*I*x 3*pi*I*x\
| --------- --------|
| 4 4 | / -3*pi*I*x 3*pi*I*x\
|e e | | --------- --------|
20*|---------- + ---------| | 4 4 |
\ 2 2 / 12*I*\- e + e /
--------------------------- + -------------------------------
2 pi
9*pi
$$\frac{20 \left(\frac{e^{\frac{3 i \pi x}{4}}}{2} + \frac{e^{- \frac{3 i \pi x}{4}}}{2}\right)}{9 \pi^{2}} + \frac{12 i \left(e^{\frac{3 i \pi x}{4}} - e^{- \frac{3 i \pi x}{4}}\right)}{\pi}$$
20*(exp(-3*pi*i*x/4)/2 + exp(3*pi*i*x/4)/2)/(9*pi^2) + 12*i*(-exp(-3*pi*i*x/4) + exp(3*pi*i*x/4))/pi
Общий знаменатель
[src]
/3*pi*x\ /3*pi*x\
20*cos|------| - 216*pi*sin|------|
\ 4 / \ 4 /
-----------------------------------
2
9*pi
$$\frac{- 216 \pi \sin{\left(\frac{3 \pi x}{4} \right)} + 20 \cos{\left(\frac{3 \pi x}{4} \right)}}{9 \pi^{2}}$$
(20*cos(3*pi*x/4) - 216*pi*sin(3*pi*x/4))/(9*pi^2)