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Least common denominator pi*sqrt(2)*(4*sin(pi*x*(3-x)/8)-pi*(-3+2*x)^2*cos(pi*x*(3-x)/8)/4)/160

An expression to simplify:

The solution

You have entered [src]
         /                                   2    /pi*x*(3 - x)\\
         |                      pi*(-3 + 2*x) *cos|------------||
     ___ |     /pi*x*(3 - x)\                     \     8      /|
pi*\/ 2 *|4*sin|------------| - --------------------------------|
         \     \     8      /                  4                /
-----------------------------------------------------------------
                               160                               
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}}{4} + 4 \sin{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}\right)}{160}$$
((pi*sqrt(2))*(4*sin(((pi*x)*(3 - x))/8) - (pi*(-3 + 2*x)^2)*cos(((pi*x)*(3 - x))/8)/4))/160
General simplification [src]
      ___ /      /pi*x*(-3 + x)\                2    /pi*x*(-3 + x)\\ 
-pi*\/ 2 *|16*sin|-------------| + pi*(-3 + 2*x) *cos|-------------|| 
          \      \      8      /                     \      8      // 
----------------------------------------------------------------------
                                 640                                  
$$- \frac{\sqrt{2} \pi \left(\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(x - 3\right)}{8} \right)} + 16 \sin{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}\right)}{640}$$
-pi*sqrt(2)*(16*sin(pi*x*(-3 + x)/8) + pi*(-3 + 2*x)^2*cos(pi*x*(-3 + x)/8))/640
Numerical answer [src]
0.111072073453959*sin(((pi*x)*(3 - x))/8) - 0.196280555614843*(-1 + 0.666666666666667*x)^2*cos(((pi*x)*(3 - x))/8)
0.111072073453959*sin(((pi*x)*(3 - x))/8) - 0.196280555614843*(-1 + 0.666666666666667*x)^2*cos(((pi*x)*(3 - x))/8)
Common denominator [src]
                 /               2\               /               2\                   /               2\                    /               2\
      ___   2    |  3*pi*x   pi*x |        ___    |  3*pi*x   pi*x |     ___   2  2    |  3*pi*x   pi*x |         ___   2    |  3*pi*x   pi*x |
  9*\/ 2 *pi *cos|- ------ + -----|   pi*\/ 2 *sin|- ------ + -----|   \/ 2 *pi *x *cos|- ------ + -----|   3*x*\/ 2 *pi *cos|- ------ + -----|
                 \    8        8  /               \    8        8  /                   \    8        8  /                    \    8        8  /
- --------------------------------- - ------------------------------ - ---------------------------------- + -----------------------------------
                 640                                40                                160                                   160                
$$- \frac{\sqrt{2} \pi^{2} x^{2} \cos{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)}}{160} + \frac{3 \sqrt{2} \pi^{2} x \cos{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)}}{160} - \frac{\sqrt{2} \pi \sin{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)}}{40} - \frac{9 \sqrt{2} \pi^{2} \cos{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)}}{640}$$
-9*sqrt(2)*pi^2*cos(-3*pi*x/8 + pi*x^2/8)/640 - pi*sqrt(2)*sin(-3*pi*x/8 + pi*x^2/8)/40 - sqrt(2)*pi^2*x^2*cos(-3*pi*x/8 + pi*x^2/8)/160 + 3*x*sqrt(2)*pi^2*cos(-3*pi*x/8 + pi*x^2/8)/160
Rational denominator [src]
          /      /               2\                     /               2\\ 
      ___ |      |  3*pi*x   pi*x |                2    |  3*pi*x   pi*x || 
-pi*\/ 2 *|16*sin|- ------ + -----| + pi*(-3 + 2*x) *cos|- ------ + -----|| 
          \      \    8        8  /                     \    8        8  // 
----------------------------------------------------------------------------
                                    640                                     
$$- \frac{\sqrt{2} \pi \left(\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)} + 16 \sin{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)}\right)}{640}$$
-pi*sqrt(2)*(16*sin(-3*pi*x/8 + pi*x^2/8) + pi*(-3 + 2*x)^2*cos(-3*pi*x/8 + pi*x^2/8))/640
Combining rational expressions [src]
     ___ /      /pi*x*(3 - x)\                2    /pi*x*(3 - x)\\
pi*\/ 2 *|16*sin|------------| - pi*(-3 + 2*x) *cos|------------||
         \      \     8      /                     \     8      //
------------------------------------------------------------------
                               640                                
$$\frac{\sqrt{2} \pi \left(- \pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(3 - x\right)}{8} \right)} + 16 \sin{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}\right)}{640}$$
pi*sqrt(2)*(16*sin(pi*x*(3 - x)/8) - pi*(-3 + 2*x)^2*cos(pi*x*(3 - x)/8))/640
Powers [src]
         /                                                               / -pi*I*x*(3 - x)     pi*I*x*(3 - x)\\
         |                                                               | ----------------    --------------||
         |                                                               |        8                  8       ||
         |      /   -pi*I*x*(3 - x)     pi*I*x*(3 - x)\                2 |e                   e              ||
         |      |   ----------------    --------------|   pi*(-3 + 2*x) *|----------------- + ---------------||
     ___ |      |          8                  8       |                  \        2                  2       /|
pi*\/ 2 *|- 2*I*\- e                 + e              / - ----------------------------------------------------|
         \                                                                         4                          /
---------------------------------------------------------------------------------------------------------------
                                                      160                                                      
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \left(\frac{e^{\frac{i \pi x \left(3 - x\right)}{8}}}{2} + \frac{e^{- \frac{i \pi x \left(3 - x\right)}{8}}}{2}\right)}{4} - 2 i \left(e^{\frac{i \pi x \left(3 - x\right)}{8}} - e^{- \frac{i \pi x \left(3 - x\right)}{8}}\right)\right)}{160}$$
pi*sqrt(2)*(-2*i*(-exp(-pi*i*x*(3 - x)/8) + exp(pi*i*x*(3 - x)/8)) - pi*(-3 + 2*x)^2*(exp(-pi*i*x*(3 - x)/8)/2 + exp(pi*i*x*(3 - x)/8)/2)/4)/160
Combinatorics [src]
          /      /               2\           /               2\              /               2\              /               2\\ 
      ___ |      |  3*pi*x   pi*x |           |  3*pi*x   pi*x |              |  3*pi*x   pi*x |         2    |  3*pi*x   pi*x || 
-pi*\/ 2 *|16*sin|- ------ + -----| + 9*pi*cos|- ------ + -----| - 12*pi*x*cos|- ------ + -----| + 4*pi*x *cos|- ------ + -----|| 
          \      \    8        8  /           \    8        8  /              \    8        8  /              \    8        8  // 
----------------------------------------------------------------------------------------------------------------------------------
                                                               640                                                                
$$- \frac{\sqrt{2} \pi \left(4 \pi x^{2} \cos{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)} - 12 \pi x \cos{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)} + 16 \sin{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)} + 9 \pi \cos{\left(\frac{\pi x^{2}}{8} - \frac{3 \pi x}{8} \right)}\right)}{640}$$
-pi*sqrt(2)*(16*sin(-3*pi*x/8 + pi*x^2/8) + 9*pi*cos(-3*pi*x/8 + pi*x^2/8) - 12*pi*x*cos(-3*pi*x/8 + pi*x^2/8) + 4*pi*x^2*cos(-3*pi*x/8 + pi*x^2/8))/640
Expand expression [src]
                 /    2\                              /    2\                                       /    2\               /    2\                               /    2\                               /    2\                                /    2\                                /    2\            
      ___   2    |pi*x |    /3*pi*x\       ___   2    |pi*x |    /3*pi*x\        ___    /3*pi*x\    |pi*x |        ___    |pi*x |    /3*pi*x\     ___   2  2    |pi*x |    /3*pi*x\     ___   2  2    |pi*x |    /3*pi*x\         ___   2    |pi*x |    /3*pi*x\         ___   2    |pi*x |    /3*pi*x\
  9*\/ 2 *pi *cos|-----|*cos|------|   9*\/ 2 *pi *sin|-----|*sin|------|   pi*\/ 2 *cos|------|*sin|-----|   pi*\/ 2 *cos|-----|*sin|------|   \/ 2 *pi *x *cos|-----|*cos|------|   \/ 2 *pi *x *sin|-----|*sin|------|   3*x*\/ 2 *pi *cos|-----|*cos|------|   3*x*\/ 2 *pi *sin|-----|*sin|------|
                 \  8  /    \  8   /                  \  8  /    \  8   /               \  8   /    \  8  /               \  8  /    \  8   /                   \  8  /    \  8   /                   \  8  /    \  8   /                    \  8  /    \  8   /                    \  8  /    \  8   /
- ---------------------------------- - ---------------------------------- - ------------------------------- + ------------------------------- - ----------------------------------- - ----------------------------------- + ------------------------------------ + ------------------------------------
                 640                                  640                                  40                                40                                 160                                   160                                   160                                    160                 
$$- \frac{\sqrt{2} \pi^{2} x^{2} \sin{\left(\frac{3 \pi x}{8} \right)} \sin{\left(\frac{\pi x^{2}}{8} \right)}}{160} - \frac{\sqrt{2} \pi^{2} x^{2} \cos{\left(\frac{3 \pi x}{8} \right)} \cos{\left(\frac{\pi x^{2}}{8} \right)}}{160} + \frac{3 \sqrt{2} \pi^{2} x \sin{\left(\frac{3 \pi x}{8} \right)} \sin{\left(\frac{\pi x^{2}}{8} \right)}}{160} + \frac{3 \sqrt{2} \pi^{2} x \cos{\left(\frac{3 \pi x}{8} \right)} \cos{\left(\frac{\pi x^{2}}{8} \right)}}{160} - \frac{9 \sqrt{2} \pi^{2} \sin{\left(\frac{3 \pi x}{8} \right)} \sin{\left(\frac{\pi x^{2}}{8} \right)}}{640} + \frac{\sqrt{2} \pi \sin{\left(\frac{3 \pi x}{8} \right)} \cos{\left(\frac{\pi x^{2}}{8} \right)}}{40} - \frac{\sqrt{2} \pi \sin{\left(\frac{\pi x^{2}}{8} \right)} \cos{\left(\frac{3 \pi x}{8} \right)}}{40} - \frac{9 \sqrt{2} \pi^{2} \cos{\left(\frac{3 \pi x}{8} \right)} \cos{\left(\frac{\pi x^{2}}{8} \right)}}{640}$$
-9*sqrt(2)*pi^2*cos(pi*x^2/8)*cos(3*pi*x/8)/640 - 9*sqrt(2)*pi^2*sin(pi*x^2/8)*sin(3*pi*x/8)/640 - pi*sqrt(2)*cos(3*pi*x/8)*sin(pi*x^2/8)/40 + pi*sqrt(2)*cos(pi*x^2/8)*sin(3*pi*x/8)/40 - sqrt(2)*pi^2*x^2*cos(pi*x^2/8)*cos(3*pi*x/8)/160 - sqrt(2)*pi^2*x^2*sin(pi*x^2/8)*sin(3*pi*x/8)/160 + 3*x*sqrt(2)*pi^2*cos(pi*x^2/8)*cos(3*pi*x/8)/160 + 3*x*sqrt(2)*pi^2*sin(pi*x^2/8)*sin(3*pi*x/8)/160
Trigonometric part [src]
         /      /pi*x*(3 - x)\                  2 /       2/pi*x*(3 - x)\\\
         | 8*tan|------------|     pi*(-3 + 2*x) *|1 - tan |------------|||
     ___ |      \     16     /                    \        \     16     //|
pi*\/ 2 *|---------------------- - ---------------------------------------|
         |       2/pi*x*(3 - x)\            /       2/pi*x*(3 - x)\\      |
         |1 + tan |------------|          4*|1 + tan |------------||      |
         \        \     16     /            \        \     16     //      /
---------------------------------------------------------------------------
                                    160                                    
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)}\right) \left(2 x - 3\right)^{2}}{4 \left(\tan^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)} + 1\right)} + \frac{8 \tan{\left(\frac{\pi x \left(3 - x\right)}{16} \right)}}{\tan^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)} + 1}\right)}{160}$$
         /                                           2   \
     ___ |           4                  pi*(-3 + 2*x)    |
pi*\/ 2 *|------------------------ - --------------------|
         |   /  pi   pi*x*(3 - x)\        /pi*x*(-3 + x)\|
         |sec|- -- + ------------|   4*sec|-------------||
         \   \  2         8      /        \      8      //
----------------------------------------------------------
                           160                            
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2}}{4 \sec{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}} + \frac{4}{\sec{\left(\frac{\pi x \left(3 - x\right)}{8} - \frac{\pi}{2} \right)}}\right)}{160}$$
         /                                      2     \
     ___ |        4                pi*(-3 + 2*x)      |
pi*\/ 2 *|----------------- - ------------------------|
         |   /pi*x*(3 - x)\        /pi   pi*x*(3 - x)\|
         |csc|------------|   4*csc|-- - ------------||
         \   \     8      /        \2         8      //
-------------------------------------------------------
                          160                          
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2}}{4 \csc{\left(- \frac{\pi x \left(3 - x\right)}{8} + \frac{\pi}{2} \right)}} + \frac{4}{\csc{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}}\right)}{160}$$
         /         /pi*x*(-3 + x)\                 2 /       2/pi*x*(-3 + x)\\\
         |    8*tan|-------------|    pi*(-3 + 2*x) *|1 - tan |-------------|||
     ___ |         \      16     /                   \        \      16     //|
pi*\/ 2 *|- ----------------------- - ----------------------------------------|
         |         2/pi*x*(-3 + x)\           /       2/pi*x*(-3 + x)\\       |
         |  1 + tan |-------------|         4*|1 + tan |-------------||       |
         \          \      16     /           \        \      16     //       /
-------------------------------------------------------------------------------
                                      160                                      
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)}\right) \left(2 x - 3\right)^{2}}{4 \left(\tan^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} + 1\right)} - \frac{8 \tan{\left(\frac{\pi x \left(x - 3\right)}{16} \right)}}{\tan^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} + 1}\right)}{160}$$
         /                                   2    /pi   pi*x*(-3 + x)\\
         |                      pi*(-3 + 2*x) *sin|-- + -------------||
     ___ |     /pi*x*(3 - x)\                     \2          8      /|
pi*\/ 2 *|4*sin|------------| - --------------------------------------|
         \     \     8      /                     4                   /
-----------------------------------------------------------------------
                                  160                                  
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \sin{\left(\frac{\pi x \left(x - 3\right)}{8} + \frac{\pi}{2} \right)}}{4} + 4 \sin{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}\right)}{160}$$
         /                                   2    /pi*x*(-3 + x)\\
         |                      pi*(-3 + 2*x) *cos|-------------||
     ___ |     /pi*x*(3 - x)\                     \      8      /|
pi*\/ 2 *|4*sin|------------| - ---------------------------------|
         \     \     8      /                   4                /
------------------------------------------------------------------
                               160                                
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}}{4} + 4 \sin{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}\right)}{160}$$
         /                                   2    /pi   pi*x*(3 - x)\\
         |                      pi*(-3 + 2*x) *sin|-- + ------------||
     ___ |     /pi*x*(3 - x)\                     \2         8      /|
pi*\/ 2 *|4*sin|------------| - -------------------------------------|
         \     \     8      /                     4                  /
----------------------------------------------------------------------
                                 160                                  
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \sin{\left(\frac{\pi x \left(3 - x\right)}{8} + \frac{\pi}{2} \right)}}{4} + 4 \sin{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}\right)}{160}$$
         /      /pi*x*(3 - x)\                  2 /        2/pi*x*(3 - x)\\\
         | 8*cot|------------|     pi*(-3 + 2*x) *|-1 + cot |------------|||
     ___ |      \     16     /                    \         \     16     //|
pi*\/ 2 *|---------------------- - ----------------------------------------|
         |       2/pi*x*(3 - x)\            /       2/pi*x*(3 - x)\\       |
         |1 + cot |------------|          4*|1 + cot |------------||       |
         \        \     16     /            \        \     16     //       /
----------------------------------------------------------------------------
                                    160                                     
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \left(\cot^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)} - 1\right)}{4 \left(\cot^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)} + 1\right)} + \frac{8 \cot{\left(\frac{\pi x \left(3 - x\right)}{16} \right)}}{\cot^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)} + 1}\right)}{160}$$
         /                                          2    /pi*x*(3 - x)\\
         |                             pi*(-3 + 2*x) *cos|------------||
     ___ |     /  pi   pi*x*(3 - x)\                     \     8      /|
pi*\/ 2 *|4*cos|- -- + ------------| - --------------------------------|
         \     \  2         8      /                  4                /
------------------------------------------------------------------------
                                  160                                   
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}}{4} + 4 \cos{\left(\frac{\pi x \left(3 - x\right)}{8} - \frac{\pi}{2} \right)}\right)}{160}$$
         /                                             2    /pi*x*(-3 + x)\\
         |                                pi*(-3 + 2*x) *cos|-------------||
     ___ |       /  pi   pi*x*(-3 + x)\                     \      8      /|
pi*\/ 2 *|- 4*cos|- -- + -------------| - ---------------------------------|
         \       \  2          8      /                   4                /
----------------------------------------------------------------------------
                                    160                                     
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}}{4} - 4 \cos{\left(\frac{\pi x \left(x - 3\right)}{8} - \frac{\pi}{2} \right)}\right)}{160}$$
         /         /pi*x*(-3 + x)\                 2 /        2/pi*x*(-3 + x)\\\
         |    8*cot|-------------|    pi*(-3 + 2*x) *|-1 + cot |-------------|||
     ___ |         \      16     /                   \         \      16     //|
pi*\/ 2 *|- ----------------------- - -----------------------------------------|
         |         2/pi*x*(-3 + x)\            /       2/pi*x*(-3 + x)\\       |
         |  1 + cot |-------------|          4*|1 + cot |-------------||       |
         \          \      16     /            \        \      16     //       /
--------------------------------------------------------------------------------
                                      160                                       
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \left(\cot^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} - 1\right)}{4 \left(\cot^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} + 1\right)} - \frac{8 \cot{\left(\frac{\pi x \left(x - 3\right)}{16} \right)}}{\cot^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} + 1}\right)}{160}$$
         /                                      2    /pi*x*(-3 + x)\\
         |                         pi*(-3 + 2*x) *cos|-------------||
     ___ |       /pi*x*(-3 + x)\                     \      8      /|
pi*\/ 2 *|- 4*sin|-------------| - ---------------------------------|
         \       \      8      /                   4                /
---------------------------------------------------------------------
                                 160                                 
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}}{4} - 4 \sin{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}\right)}{160}$$
         /                                      2    /pi   pi*x*(-3 + x)\\
         |                         pi*(-3 + 2*x) *sin|-- + -------------||
     ___ |       /pi*x*(-3 + x)\                     \2          8      /|
pi*\/ 2 *|- 4*sin|-------------| - --------------------------------------|
         \       \      8      /                     4                   /
--------------------------------------------------------------------------
                                   160                                    
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \sin{\left(\frac{\pi x \left(x - 3\right)}{8} + \frac{\pi}{2} \right)}}{4} - 4 \sin{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}\right)}{160}$$
         /      /pi*x*(3 - x)\                  2 /       2/pi*x*(-3 + x)\\\
         | 8*tan|------------|     pi*(-3 + 2*x) *|1 - tan |-------------|||
     ___ |      \     16     /                    \        \      16     //|
pi*\/ 2 *|---------------------- - ----------------------------------------|
         |       2/pi*x*(3 - x)\           /       2/pi*x*(-3 + x)\\       |
         |1 + tan |------------|         4*|1 + tan |-------------||       |
         \        \     16     /           \        \      16     //       /
----------------------------------------------------------------------------
                                    160                                     
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(1 - \tan^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)}\right) \left(2 x - 3\right)^{2}}{4 \left(\tan^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} + 1\right)} + \frac{8 \tan{\left(\frac{\pi x \left(3 - x\right)}{16} \right)}}{\tan^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)} + 1}\right)}{160}$$
         /                                          2    /pi*x*(-3 + x)\\
         |                             pi*(-3 + 2*x) *cos|-------------||
     ___ |     /  pi   pi*x*(3 - x)\                     \      8      /|
pi*\/ 2 *|4*cos|- -- + ------------| - ---------------------------------|
         \     \  2         8      /                   4                /
-------------------------------------------------------------------------
                                   160                                   
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}}{4} + 4 \cos{\left(\frac{\pi x \left(3 - x\right)}{8} - \frac{\pi}{2} \right)}\right)}{160}$$
         /                                           2  \
     ___ |           4                  pi*(-3 + 2*x)   |
pi*\/ 2 *|------------------------ - -------------------|
         |   /  pi   pi*x*(3 - x)\        /pi*x*(3 - x)\|
         |sec|- -- + ------------|   4*sec|------------||
         \   \  2         8      /        \     8      //
---------------------------------------------------------
                           160                           
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2}}{4 \sec{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}} + \frac{4}{\sec{\left(\frac{\pi x \left(3 - x\right)}{8} - \frac{\pi}{2} \right)}}\right)}{160}$$
         /                                   2    /pi*x*(-3 + x)\\
         |                      pi*(-3 + 2*x) *cos|-------------||
     ___ |     /pi*x*(3 - x)\                     \      8      /|
pi*\/ 2 *|4*sin|------------| - ---------------------------------|
         \     \     8      /                   4                /
------------------------------------------------------------------
                               160                                
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \cos{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}}{4} + 4 \sin{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}\right)}{160}$$
         /                                       2     \
     ___ |        4                 pi*(-3 + 2*x)      |
pi*\/ 2 *|----------------- - -------------------------|
         |   /pi*x*(3 - x)\        /pi   pi*x*(-3 + x)\|
         |csc|------------|   4*csc|-- - -------------||
         \   \     8      /        \2          8      //
--------------------------------------------------------
                          160                           
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2}}{4 \csc{\left(- \frac{\pi x \left(x - 3\right)}{8} + \frac{\pi}{2} \right)}} + \frac{4}{\csc{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}}\right)}{160}$$
         /                                          2     \
     ___ |          4                  pi*(-3 + 2*x)      |
pi*\/ 2 *|- ------------------ - -------------------------|
         |     /pi*x*(-3 + x)\        /pi   pi*x*(-3 + x)\|
         |  csc|-------------|   4*csc|-- - -------------||
         \     \      8      /        \2          8      //
-----------------------------------------------------------
                            160                            
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2}}{4 \csc{\left(- \frac{\pi x \left(x - 3\right)}{8} + \frac{\pi}{2} \right)}} - \frac{4}{\csc{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}}\right)}{160}$$
         /                                    2  \
     ___ |        4              pi*(-3 + 2*x)   |
pi*\/ 2 *|----------------- - -------------------|
         |   /pi*x*(3 - x)\        /pi*x*(3 - x)\|
         |csc|------------|   4*sec|------------||
         \   \     8      /        \     8      //
--------------------------------------------------
                       160                        
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2}}{4 \sec{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}} + \frac{4}{\csc{\left(\frac{\pi x \left(3 - x\right)}{8} \right)}}\right)}{160}$$
         /      /pi*x*(3 - x)\                  2 /        2/pi*x*(-3 + x)\\\
         | 8*cot|------------|     pi*(-3 + 2*x) *|-1 + cot |-------------|||
     ___ |      \     16     /                    \         \      16     //|
pi*\/ 2 *|---------------------- - -----------------------------------------|
         |       2/pi*x*(3 - x)\            /       2/pi*x*(-3 + x)\\       |
         |1 + cot |------------|          4*|1 + cot |-------------||       |
         \        \     16     /            \        \      16     //       /
-----------------------------------------------------------------------------
                                     160                                     
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2} \left(\cot^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} - 1\right)}{4 \left(\cot^{2}{\left(\frac{\pi x \left(x - 3\right)}{16} \right)} + 1\right)} + \frac{8 \cot{\left(\frac{\pi x \left(3 - x\right)}{16} \right)}}{\cot^{2}{\left(\frac{\pi x \left(3 - x\right)}{16} \right)} + 1}\right)}{160}$$
         /                                              2   \
     ___ |              4                  pi*(-3 + 2*x)    |
pi*\/ 2 *|- ------------------------- - --------------------|
         |     /  pi   pi*x*(-3 + x)\        /pi*x*(-3 + x)\|
         |  sec|- -- + -------------|   4*sec|-------------||
         \     \  2          8      /        \      8      //
-------------------------------------------------------------
                             160                             
$$\frac{\sqrt{2} \pi \left(- \frac{\pi \left(2 x - 3\right)^{2}}{4 \sec{\left(\frac{\pi x \left(x - 3\right)}{8} \right)}} - \frac{4}{\sec{\left(\frac{\pi x \left(x - 3\right)}{8} - \frac{\pi}{2} \right)}}\right)}{160}$$
pi*sqrt(2)*(-4/sec(-pi/2 + pi*x*(-3 + x)/8) - pi*(-3 + 2*x)^2/(4*sec(pi*x*(-3 + x)/8)))/160