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Expression simplification/ Least common denominator/ pi*(1-cos(((2*pi)*x)/b))*sin((pi*x)/a)/a+2*pi*(1-cos((pi*x)/a))*sin(((2*pi)*x)/b)/b

Least common denominator pi*(1-cos(((2*pi)*x)/b))*sin((pi*x)/a)/a+2*pi*(1-cos((pi*x)/a))*sin(((2*pi)*x)/b)/b

An expression to simplify:

The solution

You have entered [src] 
   /       /2*pi*x\\    /pi*x\        /       /pi*x\\    /2*pi*x\
pi*|1 - cos|------||*sin|----|   2*pi*|1 - cos|----||*sin|------|
   \       \  b   //    \ a  /        \       \ a  //    \  b   /
------------------------------ + --------------------------------
              a                                 b
$$\frac{2 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \sin{\left(\frac{2 \pi x}{b} \right)}}{b} + \frac{\pi \left(1 - \cos{\left(\frac{2 \pi x}{b} \right)}\right) \sin{\left(\frac{\pi x}{a} \right)}}{a}$$ 
((pi*(1 - cos(((2*pi)*x)/b)))*sin((pi*x)/a))/a + (((2*pi)*(1 - cos((pi*x)/a)))*sin(((2*pi)*x)/b))/b
General simplification [src] 
    /  /        /2*pi*x\\    /pi*x\       /        /pi*x\\    /2*pi*x\\ 
-pi*|b*|-1 + cos|------||*sin|----| + 2*a*|-1 + cos|----||*sin|------|| 
    \  \        \  b   //    \ a  /       \        \ a  //    \  b   // 
------------------------------------------------------------------------
                                  a*b
$$- \frac{\pi \left(2 a \left(\cos{\left(\frac{\pi x}{a} \right)} - 1\right) \sin{\left(\frac{2 \pi x}{b} \right)} + b \left(\cos{\left(\frac{2 \pi x}{b} \right)} - 1\right) \sin{\left(\frac{\pi x}{a} \right)}\right)}{a b}$$ 
-pi*(b*(-1 + cos(2*pi*x/b))*sin(pi*x/a) + 2*a*(-1 + cos(pi*x/a))*sin(2*pi*x/b))/(a*b)
Combinatorics [src] 
    /       /pi*x\          /2*pi*x\        /2*pi*x\    /pi*x\          /pi*x\    /2*pi*x\\ 
-pi*|- b*sin|----| - 2*a*sin|------| + b*cos|------|*sin|----| + 2*a*cos|----|*sin|------|| 
    \       \ a  /          \  b   /        \  b   /    \ a  /          \ a  /    \  b   // 
--------------------------------------------------------------------------------------------
                                            a*b
$$- \frac{\pi \left(2 a \sin{\left(\frac{2 \pi x}{b} \right)} \cos{\left(\frac{\pi x}{a} \right)} - 2 a \sin{\left(\frac{2 \pi x}{b} \right)} + b \sin{\left(\frac{\pi x}{a} \right)} \cos{\left(\frac{2 \pi x}{b} \right)} - b \sin{\left(\frac{\pi x}{a} \right)}\right)}{a b}$$ 
-pi*(-b*sin(pi*x/a) - 2*a*sin(2*pi*x/b) + b*cos(2*pi*x/b)*sin(pi*x/a) + 2*a*cos(pi*x/a)*sin(2*pi*x/b))/(a*b)
Numerical answer [src] 
(3.14159265358979 - 3.14159265358979*cos(((2*pi)*x)/b))*sin((pi*x)/a)/a + (6.28318530717959 - 6.28318530717959*cos((pi*x)/a))*sin(((2*pi)*x)/b)/b
(3.14159265358979 - 3.14159265358979*cos(((2*pi)*x)/b))*sin((pi*x)/a)/a + (6.28318530717959 - 6.28318530717959*cos((pi*x)/a))*sin(((2*pi)*x)/b)/b
Powers [src] 
                                  /     pi*I*x    -pi*I*x \                                /     -2*pi*I*x    2*pi*I*x\
       /   -2*pi*I*x    2*pi*I*x\ |     ------    --------|        /   -pi*I*x     pi*I*x\ |     ---------    --------|
       |   ---------    --------| |       a          a    |        |   --------    ------| |         b           b    |
       |       b           b    | |    e         e        |        |      a          a   | |    e            e        |
  pi*I*\- e          + e        /*|1 - ------- - ---------|   pi*I*\- e         + e      /*|1 - ---------- - ---------|
                                  \       2          2    /                                \        2            2    /
- --------------------------------------------------------- - ---------------------------------------------------------
                              b                                                          2*a
$$- \frac{i \pi \left(e^{\frac{2 i \pi x}{b}} - e^{- \frac{2 i \pi x}{b}}\right) \left(- \frac{e^{\frac{i \pi x}{a}}}{2} + 1 - \frac{e^{- \frac{i \pi x}{a}}}{2}\right)}{b} - \frac{i \pi \left(e^{\frac{i \pi x}{a}} - e^{- \frac{i \pi x}{a}}\right) \left(- \frac{e^{\frac{2 i \pi x}{b}}}{2} + 1 - \frac{e^{- \frac{2 i \pi x}{b}}}{2}\right)}{2 a}$$ 
-pi*i*(-exp(-2*pi*i*x/b) + exp(2*pi*i*x/b))*(1 - exp(pi*i*x/a)/2 - exp(-pi*i*x/a)/2)/b - pi*i*(-exp(-pi*i*x/a) + exp(pi*i*x/a))*(1 - exp(-2*pi*i*x/b)/2 - exp(2*pi*i*x/b)/2)/(2*a)
Common denominator [src] 
 /          /pi*x\             /2*pi*x\           /2*pi*x\    /pi*x\             /pi*x\    /2*pi*x\\ 
-|- pi*b*sin|----| - 2*pi*a*sin|------| + pi*b*cos|------|*sin|----| + 2*pi*a*cos|----|*sin|------|| 
 \          \ a  /             \  b   /           \  b   /    \ a  /             \ a  /    \  b   // 
-----------------------------------------------------------------------------------------------------
                                                 a*b
$$- \frac{2 \pi a \sin{\left(\frac{2 \pi x}{b} \right)} \cos{\left(\frac{\pi x}{a} \right)} - 2 \pi a \sin{\left(\frac{2 \pi x}{b} \right)} + \pi b \sin{\left(\frac{\pi x}{a} \right)} \cos{\left(\frac{2 \pi x}{b} \right)} - \pi b \sin{\left(\frac{\pi x}{a} \right)}}{a b}$$ 
-(-pi*b*sin(pi*x/a) - 2*pi*a*sin(2*pi*x/b) + pi*b*cos(2*pi*x/b)*sin(pi*x/a) + 2*pi*a*cos(pi*x/a)*sin(2*pi*x/b))/(a*b)
Combining rational expressions [src] 
   /  /       /2*pi*x\\    /pi*x\       /       /pi*x\\    /2*pi*x\\
pi*|b*|1 - cos|------||*sin|----| + 2*a*|1 - cos|----||*sin|------||
   \  \       \  b   //    \ a  /       \       \ a  //    \  b   //
--------------------------------------------------------------------
                                a*b
$$\frac{\pi \left(2 a \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \sin{\left(\frac{2 \pi x}{b} \right)} + b \left(1 - \cos{\left(\frac{2 \pi x}{b} \right)}\right) \sin{\left(\frac{\pi x}{a} \right)}\right)}{a b}$$ 
pi*(b*(1 - cos(2*pi*x/b))*sin(pi*x/a) + 2*a*(1 - cos(pi*x/a))*sin(2*pi*x/b))/(a*b)
Rational denominator [src] 
        /pi*x\             /2*pi*x\           /2*pi*x\    /pi*x\             /pi*x\    /2*pi*x\
pi*b*sin|----| + 2*pi*a*sin|------| - pi*b*cos|------|*sin|----| - 2*pi*a*cos|----|*sin|------|
        \ a  /             \  b   /           \  b   /    \ a  /             \ a  /    \  b   /
-----------------------------------------------------------------------------------------------
                                              a*b
$$\frac{- 2 \pi a \sin{\left(\frac{2 \pi x}{b} \right)} \cos{\left(\frac{\pi x}{a} \right)} + 2 \pi a \sin{\left(\frac{2 \pi x}{b} \right)} - \pi b \sin{\left(\frac{\pi x}{a} \right)} \cos{\left(\frac{2 \pi x}{b} \right)} + \pi b \sin{\left(\frac{\pi x}{a} \right)}}{a b}$$ 
(pi*b*sin(pi*x/a) + 2*pi*a*sin(2*pi*x/b) - pi*b*cos(2*pi*x/b)*sin(pi*x/a) - 2*pi*a*cos(pi*x/a)*sin(2*pi*x/b))/(a*b)
Expand expression [src] 
      /pi*x\           /2*pi*x\         /2*pi*x\    /pi*x\           /pi*x\    /2*pi*x\
pi*sin|----|   2*pi*sin|------|   pi*cos|------|*sin|----|   2*pi*cos|----|*sin|------|
      \ a  /           \  b   /         \  b   /    \ a  /           \ a  /    \  b   /
------------ + ---------------- - ------------------------ - --------------------------
     a                b                      a                           b
$$- \frac{2 \pi \sin{\left(\frac{2 \pi x}{b} \right)} \cos{\left(\frac{\pi x}{a} \right)}}{b} + \frac{2 \pi \sin{\left(\frac{2 \pi x}{b} \right)}}{b} - \frac{\pi \sin{\left(\frac{\pi x}{a} \right)} \cos{\left(\frac{2 \pi x}{b} \right)}}{a} + \frac{\pi \sin{\left(\frac{\pi x}{a} \right)}}{a}$$ 
pi*sin((pi*x)/a)/a + 2*pi*sin(((2*pi)*x)/b)/b - pi*cos(((2*pi)*x)/b)*sin((pi*x)/a)/a - 2*pi*cos((pi*x)/a)*sin(((2*pi)*x)/b)/b
Trigonometric part [src] 
                                            /        1    \
                                       2*pi*|1 - ---------|
                                            |       /pi*x\|
                                            |    sec|----||
                2*pi                        \       \ a  //
------------------------------------ + --------------------
     /  pi   pi*x\    2/  pi   pi*x\        /  pi   2*pi*x\
a*sec|- -- + ----|*sec |- -- + ----|   b*sec|- -- + ------|
     \  2     a  /     \  2     b  /        \  2      b   /
$$\frac{2 \pi \left(1 - \frac{1}{\sec{\left(\frac{\pi x}{a} \right)}}\right)}{b \sec{\left(- \frac{\pi}{2} + \frac{2 \pi x}{b} \right)}} + \frac{2 \pi}{a \sec{\left(- \frac{\pi}{2} + \frac{\pi x}{a} \right)} \sec^{2}{\left(- \frac{\pi}{2} + \frac{\pi x}{b} \right)}}$$ 
   /         1     \        /        1    \
pi*|1 - -----------|   2*pi*|1 - ---------|
   |       /2*pi*x\|        |       /pi*x\|
   |    sec|------||        |    sec|----||
   \       \  b   //        \       \ a  //
-------------------- + --------------------
         /pi*x\                /2*pi*x\    
    a*csc|----|           b*csc|------|    
         \ a  /                \  b   /
$$\frac{2 \pi \left(1 - \frac{1}{\sec{\left(\frac{\pi x}{a} \right)}}\right)}{b \csc{\left(\frac{2 \pi x}{b} \right)}} + \frac{\pi \left(1 - \frac{1}{\sec{\left(\frac{2 \pi x}{b} \right)}}\right)}{a \csc{\left(\frac{\pi x}{a} \right)}}$$ 
                                                          /        1    \    
                                                     4*pi*|1 - ---------|    
                                                          |       /pi*x\|    
                                                          |    sec|----||    
                     4*pi                                 \       \ a  //    
---------------------------------------------- + ----------------------------
     /pi*x\    2/  pi   pi*x\    /  pi   pi*x\        /pi*x\    /  pi   pi*x\
a*sec|----|*sec |- -- + ----|*sec|- -- + ----|   b*sec|----|*sec|- -- + ----|
     \2*a /     \  2     b  /    \  2    2*a /        \ b  /    \  2     b  /
$$\frac{4 \pi \left(1 - \frac{1}{\sec{\left(\frac{\pi x}{a} \right)}}\right)}{b \sec{\left(\frac{\pi x}{b} \right)} \sec{\left(- \frac{\pi}{2} + \frac{\pi x}{b} \right)}} + \frac{4 \pi}{a \sec{\left(\frac{\pi x}{2 a} \right)} \sec{\left(- \frac{\pi}{2} + \frac{\pi x}{2 a} \right)} \sec^{2}{\left(- \frac{\pi}{2} + \frac{\pi x}{b} \right)}}$$ 
        4/pi*x\ /       2/pi*x\\ /       /pi*x\\    /pi*x\            4/pi*x\    4/pi*x\    2/pi*x\ /       2/pi*x\\    /pi*x\
8*pi*cos |----|*|1 - tan |----||*|1 - cos|----||*tan|----|   32*pi*cos |----|*cos |----|*tan |----|*|1 - tan |----||*tan|----|
         \2*b / \        \2*b // \       \ a  //    \2*b /             \2*b /     \4*a /     \2*b / \        \4*a //    \4*a /
---------------------------------------------------------- + -----------------------------------------------------------------
                            b                                                                a
$$\frac{8 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \left(1 - \tan^{2}{\left(\frac{\pi x}{2 b} \right)}\right) \cos^{4}{\left(\frac{\pi x}{2 b} \right)} \tan{\left(\frac{\pi x}{2 b} \right)}}{b} + \frac{32 \pi \left(1 - \tan^{2}{\left(\frac{\pi x}{4 a} \right)}\right) \cos^{4}{\left(\frac{\pi x}{4 a} \right)} \cos^{4}{\left(\frac{\pi x}{2 b} \right)} \tan{\left(\frac{\pi x}{4 a} \right)} \tan^{2}{\left(\frac{\pi x}{2 b} \right)}}{a}$$ 
     /            2/pi*x\\                                                 
     |    -1 + cot |----||                                                 
     |             \2*a /|    /pi*x\                                       
4*pi*|1 - ---------------|*cot|----|                                       
     |            2/pi*x\|    \ b  /                 2/pi*x\    /pi*x\     
     |     1 + cot |----||                  16*pi*cot |----|*cot|----|     
     \             \2*a //                            \2*b /    \2*a /     
------------------------------------ + ------------------------------------
           /       2/pi*x\\                                               2
         b*|1 + cot |----||              /       2/pi*x\\ /       2/pi*x\\ 
           \        \ b  //            a*|1 + cot |----||*|1 + cot |----|| 
                                         \        \2*a // \        \2*b //
$$\frac{4 \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{2 a} \right)} - 1}{\cot^{2}{\left(\frac{\pi x}{2 a} \right)} + 1} + 1\right) \cot{\left(\frac{\pi x}{b} \right)}}{b \left(\cot^{2}{\left(\frac{\pi x}{b} \right)} + 1\right)} + \frac{16 \pi \cot{\left(\frac{\pi x}{2 a} \right)} \cot^{2}{\left(\frac{\pi x}{2 b} \right)}}{a \left(\cot^{2}{\left(\frac{\pi x}{2 a} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi x}{2 b} \right)} + 1\right)^{2}}$$ 
        2/  pi   pi*x\    /  pi   pi*x\        /       /pi*x\\    /  pi   2*pi*x\
2*pi*cos |- -- + ----|*cos|- -- + ----|   2*pi*|1 - cos|----||*cos|- -- + ------|
         \  2     b  /    \  2     a  /        \       \ a  //    \  2      b   /
--------------------------------------- + ---------------------------------------
                   a                                         b
$$\frac{2 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \cos{\left(- \frac{\pi}{2} + \frac{2 \pi x}{b} \right)}}{b} + \frac{2 \pi \cos{\left(- \frac{\pi}{2} + \frac{\pi x}{a} \right)} \cos^{2}{\left(- \frac{\pi}{2} + \frac{\pi x}{b} \right)}}{a}$$ 
   /       /pi   2*pi*x\\    /pi*x\        /       /pi   pi*x\\    /2*pi*x\
pi*|1 - sin|-- + ------||*sin|----|   2*pi*|1 - sin|-- + ----||*sin|------|
   \       \2      b   //    \ a  /        \       \2     a  //    \  b   /
----------------------------------- + -------------------------------------
                 a                                      b
$$\frac{2 \pi \left(1 - \sin{\left(\frac{\pi}{2} + \frac{\pi x}{a} \right)}\right) \sin{\left(\frac{2 \pi x}{b} \right)}}{b} + \frac{\pi \left(1 - \sin{\left(\frac{\pi}{2} + \frac{2 \pi x}{b} \right)}\right) \sin{\left(\frac{\pi x}{a} \right)}}{a}$$ 
        2/  pi   pi*x\    /pi*x\    /  pi   pi*x\        /       /pi*x\\    /pi*x\    /  pi   pi*x\
4*pi*cos |- -- + ----|*cos|----|*cos|- -- + ----|   4*pi*|1 - cos|----||*cos|----|*cos|- -- + ----|
         \  2     b  /    \2*a /    \  2    2*a /        \       \ a  //    \ b  /    \  2     b  /
------------------------------------------------- + -----------------------------------------------
                        a                                                  b
$$\frac{4 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \cos{\left(\frac{\pi x}{b} \right)} \cos{\left(- \frac{\pi}{2} + \frac{\pi x}{b} \right)}}{b} + \frac{4 \pi \cos{\left(\frac{\pi x}{2 a} \right)} \cos{\left(- \frac{\pi}{2} + \frac{\pi x}{2 a} \right)} \cos^{2}{\left(- \frac{\pi}{2} + \frac{\pi x}{b} \right)}}{a}$$ 
   /           1        \        /          1       \
pi*|1 - ----------------|   2*pi*|1 - --------------|
   |       /pi   2*pi*x\|        |       /pi   pi*x\|
   |    csc|-- - ------||        |    csc|-- - ----||
   \       \2      b   //        \       \2     a  //
------------------------- + -------------------------
            /pi*x\                     /2*pi*x\      
       a*csc|----|                b*csc|------|      
            \ a  /                     \  b   /
$$\frac{2 \pi \left(1 - \frac{1}{\csc{\left(\frac{\pi}{2} - \frac{\pi x}{a} \right)}}\right)}{b \csc{\left(\frac{2 \pi x}{b} \right)}} + \frac{\pi \left(1 - \frac{1}{\csc{\left(\frac{\pi}{2} - \frac{2 \pi x}{b} \right)}}\right)}{a \csc{\left(\frac{\pi x}{a} \right)}}$$ 
        2/pi*x\    /pi*x\        /       /pi   pi*x\\    /2*pi*x\
2*pi*sin |----|*sin|----|   2*pi*|1 - sin|-- + ----||*sin|------|
         \ b  /    \ a  /        \       \2     a  //    \  b   /
------------------------- + -------------------------------------
            a                                 b
$$\frac{2 \pi \left(1 - \sin{\left(\frac{\pi}{2} + \frac{\pi x}{a} \right)}\right) \sin{\left(\frac{2 \pi x}{b} \right)}}{b} + \frac{2 \pi \sin{\left(\frac{\pi x}{a} \right)} \sin^{2}{\left(\frac{\pi x}{b} \right)}}{a}$$ 
        2/pi*x\    /pi*x\    /pi   pi*x\        /       /pi   pi*x\\    /pi*x\    /pi   pi*x\
4*pi*sin |----|*sin|----|*sin|-- + ----|   4*pi*|1 - sin|-- + ----||*sin|----|*sin|-- + ----|
         \ b  /    \2*a /    \2    2*a /        \       \2     a  //    \ b  /    \2     b  /
---------------------------------------- + --------------------------------------------------
                   a                                               b
$$\frac{4 \pi \left(1 - \sin{\left(\frac{\pi}{2} + \frac{\pi x}{a} \right)}\right) \sin{\left(\frac{\pi x}{b} \right)} \sin{\left(\frac{\pi}{2} + \frac{\pi x}{b} \right)}}{b} + \frac{4 \pi \sin{\left(\frac{\pi x}{2 a} \right)} \sin^{2}{\left(\frac{\pi x}{b} \right)} \sin{\left(\frac{\pi}{2} + \frac{\pi x}{2 a} \right)}}{a}$$ 
        2/pi*x\    /pi*x\        /       /pi*x\\    /2*pi*x\
2*pi*sin |----|*sin|----|   2*pi*|1 - cos|----||*sin|------|
         \ b  /    \ a  /        \       \ a  //    \  b   /
------------------------- + --------------------------------
            a                              b
$$\frac{2 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \sin{\left(\frac{2 \pi x}{b} \right)}}{b} + \frac{2 \pi \sin{\left(\frac{\pi x}{a} \right)} \sin^{2}{\left(\frac{\pi x}{b} \right)}}{a}$$ 
        2/pi*x\    /pi*x\    /pi*x\        /       /pi*x\\    /pi*x\    /pi*x\
4*pi*sin |----|*cos|----|*sin|----|   4*pi*|1 - cos|----||*cos|----|*sin|----|
         \ b  /    \2*a /    \2*a /        \       \ a  //    \ b  /    \ b  /
----------------------------------- + ----------------------------------------
                 a                                       b
$$\frac{4 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \sin{\left(\frac{\pi x}{b} \right)} \cos{\left(\frac{\pi x}{b} \right)}}{b} + \frac{4 \pi \sin{\left(\frac{\pi x}{2 a} \right)} \sin^{2}{\left(\frac{\pi x}{b} \right)} \cos{\left(\frac{\pi x}{2 a} \right)}}{a}$$ 
     /            2/pi*x\\                  /            2/pi*x\\          
     |    -1 + cot |----||                  |    -1 + cot |----||          
     |             \ b  /|    /pi*x\        |             \2*a /|    /pi*x\
2*pi*|1 - ---------------|*cot|----|   4*pi*|1 - ---------------|*cot|----|
     |            2/pi*x\|    \2*a /        |            2/pi*x\|    \ b  /
     |     1 + cot |----||                  |     1 + cot |----||          
     \             \ b  //                  \             \2*a //          
------------------------------------ + ------------------------------------
           /       2/pi*x\\                       /       2/pi*x\\         
         a*|1 + cot |----||                     b*|1 + cot |----||         
           \        \2*a //                       \        \ b  //
$$\frac{4 \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{2 a} \right)} - 1}{\cot^{2}{\left(\frac{\pi x}{2 a} \right)} + 1} + 1\right) \cot{\left(\frac{\pi x}{b} \right)}}{b \left(\cot^{2}{\left(\frac{\pi x}{b} \right)} + 1\right)} + \frac{2 \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{b} \right)} - 1}{\cot^{2}{\left(\frac{\pi x}{b} \right)} + 1} + 1\right) \cot{\left(\frac{\pi x}{2 a} \right)}}{a \left(\cot^{2}{\left(\frac{\pi x}{2 a} \right)} + 1\right)}$$ 
     /            2/pi*x\\                                                                           
     |    -1 + cot |----||                                                                           
     |             \2*a /| /        2/pi*x\\    /pi*x\                                               
8*pi*|1 - ---------------|*|-1 + cot |----||*cot|----|                                               
     |            2/pi*x\| \         \2*b //    \2*b /            2/pi*x\ /        2/pi*x\\    /pi*x\
     |     1 + cot |----||                               32*pi*cot |----|*|-1 + cot |----||*cot|----|
     \             \2*a //                                         \2*b / \         \4*a //    \4*a /
------------------------------------------------------ + --------------------------------------------
                                   2                                          2                 2    
                   /       2/pi*x\\                           /       2/pi*x\\  /       2/pi*x\\     
                 b*|1 + cot |----||                         a*|1 + cot |----|| *|1 + cot |----||     
                   \        \2*b //                           \        \2*b //  \        \4*a //
$$\frac{8 \pi \left(- \frac{\cot^{2}{\left(\frac{\pi x}{2 a} \right)} - 1}{\cot^{2}{\left(\frac{\pi x}{2 a} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{\pi x}{2 b} \right)} - 1\right) \cot{\left(\frac{\pi x}{2 b} \right)}}{b \left(\cot^{2}{\left(\frac{\pi x}{2 b} \right)} + 1\right)^{2}} + \frac{32 \pi \left(\cot^{2}{\left(\frac{\pi x}{4 a} \right)} - 1\right) \cot{\left(\frac{\pi x}{4 a} \right)} \cot^{2}{\left(\frac{\pi x}{2 b} \right)}}{a \left(\cot^{2}{\left(\frac{\pi x}{4 a} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{\pi x}{2 b} \right)} + 1\right)^{2}}$$ 
   /         1     \        /        1    \
pi*|1 - -----------|   2*pi*|1 - ---------|
   |       /2*pi*x\|        |       /pi*x\|
   |    sec|------||        |    sec|----||
   \       \  b   //        \       \ a  //
-------------------- + --------------------
      /  pi   pi*x\         /  pi   2*pi*x\
 a*sec|- -- + ----|    b*sec|- -- + ------|
      \  2     a  /         \  2      b   /
$$\frac{2 \pi \left(1 - \frac{1}{\sec{\left(\frac{\pi x}{a} \right)}}\right)}{b \sec{\left(- \frac{\pi}{2} + \frac{2 \pi x}{b} \right)}} + \frac{\pi \left(1 - \frac{1}{\sec{\left(\frac{2 \pi x}{b} \right)}}\right)}{a \sec{\left(- \frac{\pi}{2} + \frac{\pi x}{a} \right)}}$$ 
   /       /2*pi*x\\    /  pi   pi*x\        /       /pi*x\\    /  pi   2*pi*x\
pi*|1 - cos|------||*cos|- -- + ----|   2*pi*|1 - cos|----||*cos|- -- + ------|
   \       \  b   //    \  2     a  /        \       \ a  //    \  2      b   /
------------------------------------- + ---------------------------------------
                  a                                        b
$$\frac{2 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \cos{\left(- \frac{\pi}{2} + \frac{2 \pi x}{b} \right)}}{b} + \frac{\pi \left(1 - \cos{\left(\frac{2 \pi x}{b} \right)}\right) \cos{\left(- \frac{\pi}{2} + \frac{\pi x}{a} \right)}}{a}$$ 
     /           2/pi*x\\                                                 
     |    1 - tan |----||                                                 
     |            \2*a /|    /pi*x\                                       
4*pi*|1 - --------------|*tan|----|                                       
     |           2/pi*x\|    \ b  /                 2/pi*x\    /pi*x\     
     |    1 + tan |----||                  16*pi*tan |----|*tan|----|     
     \            \2*a //                            \2*b /    \2*a /     
----------------------------------- + ------------------------------------
           /       2/pi*x\\                                              2
         b*|1 + tan |----||             /       2/pi*x\\ /       2/pi*x\\ 
           \        \ b  //           a*|1 + tan |----||*|1 + tan |----|| 
                                        \        \2*a // \        \2*b //
$$\frac{4 \pi \left(- \frac{1 - \tan^{2}{\left(\frac{\pi x}{2 a} \right)}}{\tan^{2}{\left(\frac{\pi x}{2 a} \right)} + 1} + 1\right) \tan{\left(\frac{\pi x}{b} \right)}}{b \left(\tan^{2}{\left(\frac{\pi x}{b} \right)} + 1\right)} + \frac{16 \pi \tan{\left(\frac{\pi x}{2 a} \right)} \tan^{2}{\left(\frac{\pi x}{2 b} \right)}}{a \left(\tan^{2}{\left(\frac{\pi x}{2 a} \right)} + 1\right) \left(\tan^{2}{\left(\frac{\pi x}{2 b} \right)} + 1\right)^{2}}$$ 
                                             /          1       \ 
                                        4*pi*|1 - --------------| 
                                             |       /pi   pi*x\| 
                                             |    csc|-- - ----|| 
                 4*pi                        \       \2     a  // 
------------------------------------- + --------------------------
     2/pi*x\    /pi*x\    /pi   pi*x\        /pi*x\    /pi   pi*x\
a*csc |----|*csc|----|*csc|-- - ----|   b*csc|----|*csc|-- - ----|
      \ b  /    \2*a /    \2    2*a /        \ b  /    \2     b  /
$$\frac{4 \pi \left(1 - \frac{1}{\csc{\left(\frac{\pi}{2} - \frac{\pi x}{a} \right)}}\right)}{b \csc{\left(\frac{\pi x}{b} \right)} \csc{\left(\frac{\pi}{2} - \frac{\pi x}{b} \right)}} + \frac{4 \pi}{a \csc{\left(\frac{\pi x}{2 a} \right)} \csc^{2}{\left(\frac{\pi x}{b} \right)} \csc{\left(\frac{\pi}{2} - \frac{\pi x}{2 a} \right)}}$$ 
     /     /pi*x\      /pi*x\    /pi*x\        /pi*x\    /pi*x\\          
     |2*cos|----|   sin|----|*sin|----|   2*cos|----|*cos|----||          
     |     \ b  /      \ a  /    \ b  /        \ a  /    \ b  /|    /pi*x\
2*pi*|----------- + ------------------- - ---------------------|*sin|----|
     \     b                 a                      b          /    \ b  /
$$2 \pi \left(- \frac{2 \cos{\left(\frac{\pi x}{a} \right)} \cos{\left(\frac{\pi x}{b} \right)}}{b} + \frac{2 \cos{\left(\frac{\pi x}{b} \right)}}{b} + \frac{\sin{\left(\frac{\pi x}{a} \right)} \sin{\left(\frac{\pi x}{b} \right)}}{a}\right) \sin{\left(\frac{\pi x}{b} \right)}$$ 
                              /          1       \
                         2*pi*|1 - --------------|
                              |       /pi   pi*x\|
                              |    csc|-- - ----||
         2*pi                 \       \2     a  //
---------------------- + -------------------------
     /pi*x\    2/pi*x\              /2*pi*x\      
a*csc|----|*csc |----|         b*csc|------|      
     \ a  /     \ b  /              \  b   /
$$\frac{2 \pi \left(1 - \frac{1}{\csc{\left(\frac{\pi}{2} - \frac{\pi x}{a} \right)}}\right)}{b \csc{\left(\frac{2 \pi x}{b} \right)}} + \frac{2 \pi}{a \csc{\left(\frac{\pi x}{a} \right)} \csc^{2}{\left(\frac{\pi x}{b} \right)}}$$ 
                      /           2/pi*x\\                                                        
                      |    1 - tan |----||                                                        
     /       2/pi*x\\ |            \2*a /|    /pi*x\                                              
8*pi*|1 - tan |----||*|1 - --------------|*tan|----|                                              
     \        \2*b // |           2/pi*x\|    \2*b /            2/pi*x\ /       2/pi*x\\    /pi*x\
                      |    1 + tan |----||             32*pi*tan |----|*|1 - tan |----||*tan|----|
                      \            \2*a //                       \2*b / \        \4*a //    \4*a /
---------------------------------------------------- + -------------------------------------------
                                  2                                         2                 2   
                  /       2/pi*x\\                          /       2/pi*x\\  /       2/pi*x\\    
                b*|1 + tan |----||                        a*|1 + tan |----|| *|1 + tan |----||    
                  \        \2*b //                          \        \2*b //  \        \4*a //
$$\frac{8 \pi \left(1 - \tan^{2}{\left(\frac{\pi x}{2 b} \right)}\right) \left(- \frac{1 - \tan^{2}{\left(\frac{\pi x}{2 a} \right)}}{\tan^{2}{\left(\frac{\pi x}{2 a} \right)} + 1} + 1\right) \tan{\left(\frac{\pi x}{2 b} \right)}}{b \left(\tan^{2}{\left(\frac{\pi x}{2 b} \right)} + 1\right)^{2}} + \frac{32 \pi \left(1 - \tan^{2}{\left(\frac{\pi x}{4 a} \right)}\right) \tan{\left(\frac{\pi x}{4 a} \right)} \tan^{2}{\left(\frac{\pi x}{2 b} \right)}}{a \left(\tan^{2}{\left(\frac{\pi x}{4 a} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{\pi x}{2 b} \right)} + 1\right)^{2}}$$ 
        4/pi*x\ /       /pi*x\\ /        2/pi*x\\    /pi*x\            2/pi*x\    4/pi*x\    4/pi*x\ /        2/pi*x\\    /pi*x\
8*pi*sin |----|*|1 - cos|----||*|-1 + cot |----||*cot|----|   32*pi*cot |----|*sin |----|*sin |----|*|-1 + cot |----||*cot|----|
         \2*b / \       \ a  // \         \2*b //    \2*b /             \2*b /     \2*b /     \4*a / \         \4*a //    \4*a /
----------------------------------------------------------- + ------------------------------------------------------------------
                             b                                                                a
$$\frac{8 \pi \left(1 - \cos{\left(\frac{\pi x}{a} \right)}\right) \left(\cot^{2}{\left(\frac{\pi x}{2 b} \right)} - 1\right) \sin^{4}{\left(\frac{\pi x}{2 b} \right)} \cot{\left(\frac{\pi x}{2 b} \right)}}{b} + \frac{32 \pi \left(\cot^{2}{\left(\frac{\pi x}{4 a} \right)} - 1\right) \sin^{4}{\left(\frac{\pi x}{4 a} \right)} \sin^{4}{\left(\frac{\pi x}{2 b} \right)} \cot{\left(\frac{\pi x}{4 a} \right)} \cot^{2}{\left(\frac{\pi x}{2 b} \right)}}{a}$$ 
     /           2/pi*x\\                  /           2/pi*x\\          
     |    1 - tan |----||                  |    1 - tan |----||          
     |            \ b  /|    /pi*x\        |            \2*a /|    /pi*x\
2*pi*|1 - --------------|*tan|----|   4*pi*|1 - --------------|*tan|----|
     |           2/pi*x\|    \2*a /        |           2/pi*x\|    \ b  /
     |    1 + tan |----||                  |    1 + tan |----||          
     \            \ b  //                  \            \2*a //          
----------------------------------- + -----------------------------------
           /       2/pi*x\\                      /       2/pi*x\\        
         a*|1 + tan |----||                    b*|1 + tan |----||        
           \        \2*a //                      \        \ b  //
$$\frac{4 \pi \left(- \frac{1 - \tan^{2}{\left(\frac{\pi x}{2 a} \right)}}{\tan^{2}{\left(\frac{\pi x}{2 a} \right)} + 1} + 1\right) \tan{\left(\frac{\pi x}{b} \right)}}{b \left(\tan^{2}{\left(\frac{\pi x}{b} \right)} + 1\right)} + \frac{2 \pi \left(- \frac{1 - \tan^{2}{\left(\frac{\pi x}{b} \right)}}{\tan^{2}{\left(\frac{\pi x}{b} \right)} + 1} + 1\right) \tan{\left(\frac{\pi x}{2 a} \right)}}{a \left(\tan^{2}{\left(\frac{\pi x}{2 a} \right)} + 1\right)}$$ 
2*pi*(1 - (1 - tan(pi*x/b)^2)/(1 + tan(pi*x/b)^2))*tan(pi*x/(2*a))/(a*(1 + tan(pi*x/(2*a))^2)) + 4*pi*(1 - (1 - tan(pi*x/(2*a))^2)/(1 + tan(pi*x/(2*a))^2))*tan(pi*x/b)/(b*(1 + tan(pi*x/b)^2))
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