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Least common denominator sin(2*pi+x)+((3*pi)/2-x)+cos(x+pi/2)

An expression to simplify:

The solution

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                3*pi          /    pi\
sin(2*pi + x) + ---- - x + cos|x + --|
                 2            \    2 /
$$\left(\left(- x + \frac{3 \pi}{2}\right) + \sin{\left(x + 2 \pi \right)}\right) + \cos{\left(x + \frac{\pi}{2} \right)}$$
sin(2*pi + x) + (3*pi)/2 - x + cos(x + pi/2)
General simplification [src]
     3*pi
-x + ----
      2  
$$- x + \frac{3 \pi}{2}$$
-x + 3*pi/2
Numerical answer [src]
4.71238898038469 - x + cos(x + pi/2) + sin(2*pi + x)
4.71238898038469 - x + cos(x + pi/2) + sin(2*pi + x)
Combinatorics [src]
     3*pi
-x + ----
      2  
$$- x + \frac{3 \pi}{2}$$
-x + 3*pi/2
Common denominator [src]
     3*pi
-x + ----
      2  
$$- x + \frac{3 \pi}{2}$$
-x + 3*pi/2
Rational denominator [src]
            /    pi\                  
-2*x + 2*cos|x + --| + 2*sin(x) + 3*pi
            \    2 /                  
--------------------------------------
                  2                   
$$\frac{- 2 x + 2 \sin{\left(x \right)} + 2 \cos{\left(x + \frac{\pi}{2} \right)} + 3 \pi}{2}$$
(-2*x + 2*cos(x + pi/2) + 2*sin(x) + 3*pi)/2
Powers [src]
     3*pi
-x + ----
      2  
$$- x + \frac{3 \pi}{2}$$
   /    pi\      /     pi\                                                  
 I*|x + --|    I*|-x - --|                                                  
   \    2 /      \     2 /                /   I*(-x - 2*pi)    I*(x + 2*pi)\
e             e                  3*pi   I*\- e              + e            /
----------- + ------------ - x + ---- - ------------------------------------
     2             2              2                      2                  
$$- x - \frac{i \left(- e^{i \left(- x - 2 \pi\right)} + e^{i \left(x + 2 \pi\right)}\right)}{2} + \frac{e^{i \left(- x - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(x + \frac{\pi}{2}\right)}}{2} + \frac{3 \pi}{2}$$
exp(i*(x + pi/2))/2 + exp(i*(-x - pi/2))/2 - x + 3*pi/2 - i*(-exp(i*(-x - 2*pi)) + exp(i*(x + 2*pi)))/2
Combining rational expressions [src]
            /pi + 2*x\                  
-2*x + 2*cos|--------| + 2*sin(x) + 3*pi
            \   2    /                  
----------------------------------------
                   2                    
$$\frac{- 2 x + 2 \sin{\left(x \right)} + 2 \cos{\left(\frac{2 x + \pi}{2} \right)} + 3 \pi}{2}$$
(-2*x + 2*cos((pi + 2*x)/2) + 2*sin(x) + 3*pi)/2
Expand expression [src]
     3*pi
-x + ----
      2  
$$- x + \frac{3 \pi}{2}$$
-x + 3*pi/2
Assemble expression [src]
     3*pi      /    pi\                
-x + ---- + cos|x + --| + sin(2*pi + x)
      2        \    2 /                
$$- x + \sin{\left(x + 2 \pi \right)} + \cos{\left(x + \frac{\pi}{2} \right)} + \frac{3 \pi}{2}$$
-x + 3*pi/2 + cos(x + pi/2) + sin(2*pi + x)
Trigonometric part [src]
     3*pi
-x + ----
      2  
$$- x + \frac{3 \pi}{2}$$
                                           /        2/x   pi\\
                                           |     sec |- - --||
                         /         1     \ |         \2   4 /|
                         |1 + -----------|*|-1 + ------------|
                         |       /    pi\| |        2/x   pi\|
                         |    sec|x - --|| |     sec |- + --||
     1            3*pi   \       \    2 // \         \2   4 //
----------- - x + ---- + -------------------------------------
   /    pi\        2                       2                  
sec|x - --|                                                   
   \    2 /                                                   
$$- x + \frac{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}} - 1\right)}{2} + \frac{3 \pi}{2} + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}$$
            /           /x\ \                                  
            |      2*tan|-| |                                  
            |           \2/ | /          1      \              
            |1 + -----------|*|-1 + ------------|              
            |           2/x\| |        2/x   pi\|          /x\ 
            |    1 + tan |-|| |     tan |- + --||     2*tan|-| 
     3*pi   \            \2// \         \2   4 //          \2/ 
-x + ---- + ------------------------------------- + -----------
      2                       2                            2/x\
                                                    1 + tan |-|
                                                            \2/
$$- x + \frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{2} + \frac{3 \pi}{2} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
     3*pi                       
-x + ---- + sin(x) + sin(pi + x)
      2                         
$$- x + \sin{\left(x \right)} + \sin{\left(x + \pi \right)} + \frac{3 \pi}{2}$$
              3*pi      /    3*pi\
-x - sin(x) + ---- + cos|x + ----|
               2        \     2  /
$$- x - \sin{\left(x \right)} + \cos{\left(x + \frac{3 \pi}{2} \right)} + \frac{3 \pi}{2}$$
                    2/x   pi\          /x\ 
            -1 + cot |- + --|     2*cot|-| 
     3*pi            \2   4 /          \2/ 
-x + ---- + ----------------- + -----------
      2             2/x   pi\          2/x\
             1 + cot |- + --|   1 + cot |-|
                     \2   4 /           \2/
$$- x + \frac{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + \frac{3 \pi}{2} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$
                         /          1      \         
            (1 + sin(x))*|-1 + ------------|         
                         |        2/x   pi\|         
                         |     tan |- + --||         
     3*pi                \         \2   4 //         
-x + ---- + -------------------------------- + sin(x)
      2                    2                         
$$- x + \frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right)}{2} + \sin{\left(x \right)} + \frac{3 \pi}{2}$$
                   2/x   pi\          /x\ 
            1 - tan |- + --|     2*tan|-| 
     3*pi           \2   4 /          \2/ 
-x + ---- + ---------------- + -----------
      2            2/x   pi\          2/x\
            1 + tan |- + --|   1 + tan |-|
                    \2   4 /           \2/
$$- x + \frac{1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + \frac{3 \pi}{2} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
                         /        2/x   pi\\         
            (1 + sin(x))*|-1 + cot |- + --||         
     3*pi                \         \2   4 //         
-x + ---- + -------------------------------- + sin(x)
      2                    2                         
$$- x + \frac{\left(\sin{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} + \sin{\left(x \right)} + \frac{3 \pi}{2}$$
                                 /         2/x   pi\ \
                                 |      csc |- + --| |
                    /      1   \ |          \2   4 / |
                    |1 + ------|*|-1 + --------------|
                    \    csc(x)/ |        2/  x   pi\|
                                 |     csc |- - + --||
  1          3*pi                \         \  2   4 //
------ - x + ---- + ----------------------------------
csc(x)        2                     2                 
$$- x + \frac{\left(-1 + \frac{\csc^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right)}{2} + \frac{3 \pi}{2} + \frac{1}{\csc{\left(x \right)}}$$
      1               1      3*pi
------------- - x - ------ + ----
   /    3*pi\       csc(x)    2  
sec|x + ----|                    
   \     2  /                    
$$- x + \frac{3 \pi}{2} + \frac{1}{\sec{\left(x + \frac{3 \pi}{2} \right)}} - \frac{1}{\csc{\left(x \right)}}$$
                         /           2       \         
                         |        cos (x)    |         
            (1 + sin(x))*|-1 + --------------|         
                         |          4/x   pi\|         
                         |     4*sin |- + --||         
     3*pi                \           \2   4 //         
-x + ---- + ---------------------------------- + sin(x)
      2                     2                          
$$- x + \frac{\left(-1 + \frac{\cos^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}\right) \left(\sin{\left(x \right)} + 1\right)}{2} + \sin{\left(x \right)} + \frac{3 \pi}{2}$$
  1               1        3*pi
------ - x - ----------- + ----
csc(x)          /pi    \    2  
             sec|-- - x|       
                \2     /       
$$- x + \frac{3 \pi}{2} - \frac{1}{\sec{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}$$
        /    pi\   3*pi         
-x - cos|x - --| + ---- + sin(x)
        \    2 /    2           
$$- x + \sin{\left(x \right)} - \cos{\left(x - \frac{\pi}{2} \right)} + \frac{3 \pi}{2}$$
            /           /x\ \                                  
            |      2*cot|-| |                                  
            |           \2/ | /        2/x   pi\\              
            |1 + -----------|*|-1 + cot |- + --||              
            |           2/x\| \         \2   4 //          /x\ 
            |    1 + cot |-||                         2*cot|-| 
     3*pi   \            \2//                              \2/ 
-x + ---- + ------------------------------------- + -----------
      2                       2                            2/x\
                                                    1 + cot |-|
                                                            \2/
$$- x + \frac{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{2} + \frac{3 \pi}{2} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$
                              /        2/x   pi\\              
                              |     cos |- + --||              
            /       /    pi\\ |         \2   4 /|              
            |1 + cos|x - --||*|-1 + ------------|              
            \       \    2 // |        2/x   pi\|              
                              |     cos |- - --||              
     3*pi                     \         \2   4 //      /    pi\
-x + ---- + ------------------------------------- + cos|x - --|
      2                       2                        \    2 /
$$- x + \frac{\left(-1 + \frac{\cos^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{4} \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}{2} + \cos{\left(x - \frac{\pi}{2} \right)} + \frac{3 \pi}{2}$$
-x + 3*pi/2 + (1 + cos(x - pi/2))*(-1 + cos(x/2 + pi/4)^2/cos(x/2 - pi/4)^2)/2 + cos(x - pi/2)
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