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Other calculators

How to use it?
Least common denominator:
(p/(p^2-4)+2/(2-p)+1/(p+2))/(p-2+(10-p^2)/(p+2))
((p^i)*sin((i+1)*x)/i+1)*i/((p^i)*sin(i*x))
pi/9-pi/5
((n/(b+1/(p*s))+1)*(d+p*l)+n)/c+n/(b+1/(p*s))+1
Factor polynomial:
x^4-6*x^3-7*x^2+36*x+36
x^4-6*x^3+4*x^2-8*x-96
x^4+5*x^3+2*x^2+5*x+1
x^4+5*x^2/2
Factor squared:
y^2+4*y+10
y^2-4*y-1
-y^2-4*y-1
y^2-4*y+1
How do you in partial fractions?:
log(((6*sin(x))/(1+cos(x))-4*sqrt(3)+6)/(6+4*sqrt(3)+(6*sin(x))/(1+cos(x))))/(2*sqrt(3))
log((4*x-2*sqrt(34)+12)/(12+2*sqrt(34)+4*x))/(2*sqrt(34))
log((2*sqrt(64*n+3)-2*sqrt(3))/(2*sqrt(3)+2*sqrt(64*n+3)))/sqrt(3)
exp(x)/(E^x-E^(-x))+(E^x+1)*(-E^x-exp(-x))/(E^x-E^(-x))^2
Identical expressions
cos(x/ three)^ two *exp(x/ three)/ three - two *cos(x/ three)*exp(x/ three)*sin(x/ three)/ three
co sinus of e of (x divide by 3) squared multiply by exponent of (x divide by 3) divide by 3 minus 2 multiply by co sinus of e of (x divide by 3) multiply by exponent of (x divide by 3) multiply by sinus of (x divide by 3) divide by 3
co sinus of e of (x divide by three) to the power of two multiply by exponent of (x divide by three) divide by three minus two multiply by co sinus of e of (x divide by three) multiply by exponent of (x divide by three) multiply by sinus of (x divide by three) divide by three
cos(x/3)2*exp(x/3)/3-2*cos(x/3)*exp(x/3)*sin(x/3)/3
cosx/32*expx/3/3-2*cosx/3*expx/3*sinx/3/3
cos(x/3)²*exp(x/3)/3-2*cos(x/3)*exp(x/3)*sin(x/3)/3
cos(x/3) to the power of 2*exp(x/3)/3-2*cos(x/3)*exp(x/3)*sin(x/3)/3
cos(x/3)^2exp(x/3)/3-2cos(x/3)exp(x/3)sin(x/3)/3
cos(x/3)2exp(x/3)/3-2cos(x/3)exp(x/3)sin(x/3)/3
cosx/32expx/3/3-2cosx/3expx/3sinx/3/3
cosx/3^2expx/3/3-2cosx/3expx/3sinx/3/3
cos(x divide by 3)^2*exp(x divide by 3) divide by 3-2*cos(x divide by 3)*exp(x divide by 3)*sin(x divide by 3) divide by 3
Similar expressions
cos(x/3)^2*exp(x/3)/3+2*cos(x/3)*exp(x/3)*sin(x/3)/3

Expression simplification/ Least common denominator/ cos(x/3)^2*exp(x/3)/3-2*cos(x/3)*exp(x/3)*sin(x/3)/3

Least common denominator cos(x/3)^2exp(x/3)/3-2cos(x/3)exp(x/3)sin(x/3)/3

The solution

You have entered [src]

         x             x       
         -             -       
   2/x\  3        /x\  3    /x\
cos |-|*e    2*cos|-|*e *sin|-|
    \3/           \3/       \3/
---------- - ------------------
    3                3

$$\frac{e^{\frac{x}{3}} \cos^{2}{\left(\frac{x}{3} \right)}}{3} - \frac{e^{\frac{x}{3}} \cdot 2 \cos{\left(\frac{x}{3} \right)} \sin{\left(\frac{x}{3} \right)}}{3}$$

(cos(x/3)^2*exp(x/3))/3 - ((2*cos(x/3))*exp(x/3))*sin(x/3)/3

General simplification [src]

/       /2*x\           \  x
|    cos|---|           |  -
|1      \ 3 /      /2*x\|  3
|- + -------- - sin|---||*e 
\2      2          \ 3 //   
----------------------------
             3

$$\frac{\left(- \sin{\left(\frac{2 x}{3} \right)} + \frac{\cos{\left(\frac{2 x}{3} \right)}}{2} + \frac{1}{2}\right) e^{\frac{x}{3}}}{3}$$

(1/2 + cos(2*x/3)/2 - sin(2*x/3))*exp(x/3)/3

Common denominator [src]

         x             x       
         -             -       
   2/x\  3        /x\  3    /x\
cos |-|*e    2*cos|-|*e *sin|-|
    \3/           \3/       \3/
---------- - ------------------
    3                3

$$- \frac{2 e^{\frac{x}{3}} \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} + \frac{e^{\frac{x}{3}} \cos^{2}{\left(\frac{x}{3} \right)}}{3}$$

cos(x/3)^2*exp(x/3)/3 - 2*cos(x/3)*exp(x/3)*sin(x/3)/3

Numerical answer [src]

0.333333333333333*cos(x/3)^2*exp(x/3) - 0.666666666666667*cos(x/3)*exp(x/3)*sin(x/3)

0.333333333333333*cos(x/3)^2*exp(x/3) - 0.666666666666667*cos(x/3)*exp(x/3)*sin(x/3)

Combining rational expressions [src]

                              x
                              -
/       /x\      /x\\    /x\  3
|- 2*sin|-| + cos|-||*cos|-|*e 
\       \3/      \3//    \3/   
-------------------------------
               3

$$\frac{\left(- 2 \sin{\left(\frac{x}{3} \right)} + \cos{\left(\frac{x}{3} \right)}\right) e^{\frac{x}{3}} \cos{\left(\frac{x}{3} \right)}}{3}$$

(-2*sin(x/3) + cos(x/3))*cos(x/3)*exp(x/3)/3

Combinatorics [src]

                              x
                              -
/       /x\      /x\\    /x\  3
|- 2*sin|-| + cos|-||*cos|-|*e 
\       \3/      \3//    \3/   
-------------------------------
               3

$$\frac{\left(- 2 \sin{\left(\frac{x}{3} \right)} + \cos{\left(\frac{x}{3} \right)}\right) e^{\frac{x}{3}} \cos{\left(\frac{x}{3} \right)}}{3}$$

(-2*sin(x/3) + cos(x/3))*cos(x/3)*exp(x/3)/3

Rational denominator [src]

         x             x       
         -             -       
   2/x\  3        /x\  3    /x\
cos |-|*e  - 2*cos|-|*e *sin|-|
    \3/           \3/       \3/
-------------------------------
               3

$$\frac{- 2 e^{\frac{x}{3}} \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)} + e^{\frac{x}{3}} \cos^{2}{\left(\frac{x}{3} \right)}}{3}$$

(cos(x/3)^2*exp(x/3) - 2*cos(x/3)*exp(x/3)*sin(x/3))/3

Powers [src]

               2                                            
/ -I*x     I*x\                                             
| -----    ---|   x                                         
|   3       3 |   -     /   -I*x     I*x\ / -I*x     I*x\  x
|e        e   |   3     |   -----    ---| | -----    ---|  -
|------ + ----| *e      |     3       3 | |   3       3 |  3
\  2       2  /       I*\- e      + e   /*\e      + e   /*e 
------------------- + --------------------------------------
         3                              6

$$\frac{\left(\frac{e^{\frac{i x}{3}}}{2} + \frac{e^{- \frac{i x}{3}}}{2}\right)^{2} e^{\frac{x}{3}}}{3} + \frac{i \left(e^{\frac{i x}{3}} - e^{- \frac{i x}{3}}\right) \left(e^{\frac{i x}{3}} + e^{- \frac{i x}{3}}\right) e^{\frac{x}{3}}}{6}$$

         x             x       
         -             -       
   2/x\  3        /x\  3    /x\
cos |-|*e    2*cos|-|*e *sin|-|
    \3/           \3/       \3/
---------- - ------------------
    3                3

$$- \frac{2 e^{\frac{x}{3}} \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} + \frac{e^{\frac{x}{3}} \cos^{2}{\left(\frac{x}{3} \right)}}{3}$$

cos(x/3)^2*exp(x/3)/3 - 2*cos(x/3)*exp(x/3)*sin(x/3)/3

Expand expression [src]

         x             x       
         -             -       
   2/x\  3        /x\  3    /x\
cos |-|*e    2*cos|-|*e *sin|-|
    \3/           \3/       \3/
---------- - ------------------
    3                3

$$- \frac{2 e^{\frac{x}{3}} \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} + \frac{e^{\frac{x}{3}} \cos^{2}{\left(\frac{x}{3} \right)}}{3}$$

cos(x/3)^2*exp(x/3)/3 - 2*cos(x/3)*exp(x/3)*sin(x/3)/3

Assemble expression [src]

         x             x       
         -             -       
   2/x\  3        /x\  3    /x\
cos |-|*e    2*cos|-|*e *sin|-|
    \3/           \3/       \3/
---------- - ------------------
    3                3

$$- \frac{2 e^{\frac{x}{3}} \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} + \frac{e^{\frac{x}{3}} \cos^{2}{\left(\frac{x}{3} \right)}}{3}$$

cos(x/3)^2*exp(x/3)/3 - 2*cos(x/3)*exp(x/3)*sin(x/3)/3

Trigonometric part [src]

                                  x
                                  -
/       1              2       \  3
|1 + -------- - ---------------|*e 
|       /2*x\      /  pi   2*x\|   
|    sec|---|   sec|- -- + ---||   
\       \ 3 /      \  2     3 //   
-----------------------------------
                 6

$$\frac{\left(1 - \frac{2}{\sec{\left(\frac{2 x}{3} - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(\frac{2 x}{3} \right)}}\right) e^{\frac{x}{3}}}{6}$$

      /x\    /2*x\      /2*x\     /x\   /       /2*x\\     /x\   /       /2*x\\     /x\
  cosh|-|*sin|---|   sin|---|*sinh|-|   |1 + cos|---||*cosh|-|   |1 + cos|---||*sinh|-|
      \3/    \ 3 /      \ 3 /     \3/   \       \ 3 //     \3/   \       \ 3 //     \3/
- ---------------- - ---------------- + ---------------------- + ----------------------
         3                  3                     6                        6

$$\frac{\left(\cos{\left(\frac{2 x}{3} \right)} + 1\right) \sinh{\left(\frac{x}{3} \right)}}{6} + \frac{\left(\cos{\left(\frac{2 x}{3} \right)} + 1\right) \cosh{\left(\frac{x}{3} \right)}}{6} - \frac{\sin{\left(\frac{2 x}{3} \right)} \sinh{\left(\frac{x}{3} \right)}}{3} - \frac{\sin{\left(\frac{2 x}{3} \right)} \cosh{\left(\frac{x}{3} \right)}}{3}$$

     x                  x         
     -                  -         
     3                  3         
    e                2*e          
--------- - ----------------------
     2/x\        /x\    /  pi   x\
3*sec |-|   3*sec|-|*sec|- -- + -|
      \3/        \3/    \  2    3/

$$- \frac{2 e^{\frac{x}{3}}}{3 \sec{\left(\frac{x}{3} \right)} \sec{\left(\frac{x}{3} - \frac{\pi}{2} \right)}} + \frac{e^{\frac{x}{3}}}{3 \sec^{2}{\left(\frac{x}{3} \right)}}$$

/            2/x\          /x\ \  x
|    -1 + cot |-|     4*cot|-| |  -
|             \3/          \3/ |  3
|1 + ------------ - -----------|*e 
|           2/x\           2/x\|   
|    1 + cot |-|    1 + cot |-||   
\            \3/            \3//   
-----------------------------------
                 6

$$\frac{\left(\frac{\cot^{2}{\left(\frac{x}{3} \right)} - 1}{\cot^{2}{\left(\frac{x}{3} \right)} + 1} + 1 - \frac{4 \cot{\left(\frac{x}{3} \right)}}{\cot^{2}{\left(\frac{x}{3} \right)} + 1}\right) e^{\frac{x}{3}}}{6}$$

      /x\    /2*x\      /2*x\     /x\      2/x\     /x\      2/x\     /x\
  cosh|-|*sin|---|   sin|---|*sinh|-|   cos |-|*cosh|-|   cos |-|*sinh|-|
      \3/    \ 3 /      \ 3 /     \3/       \3/     \3/       \3/     \3/
- ---------------- - ---------------- + --------------- + ---------------
         3                  3                  3                 3

$$- \frac{\sin{\left(\frac{2 x}{3} \right)} \sinh{\left(\frac{x}{3} \right)}}{3} - \frac{\sin{\left(\frac{2 x}{3} \right)} \cosh{\left(\frac{x}{3} \right)}}{3} + \frac{\cos^{2}{\left(\frac{x}{3} \right)} \sinh{\left(\frac{x}{3} \right)}}{3} + \frac{\cos^{2}{\left(\frac{x}{3} \right)} \cosh{\left(\frac{x}{3} \right)}}{3}$$

           /x\                 /x\             /x\         /x\ 
       cosh|-|             sinh|-|         cosh|-|     sinh|-| 
           \3/                 \3/             \3/         \3/ 
- ----------------- - ----------------- + --------- + ---------
       /  pi   2*x\        /  pi   2*x\        2/x\        2/x\
  3*sec|- -- + ---|   3*sec|- -- + ---|   3*sec |-|   3*sec |-|
       \  2     3 /        \  2     3 /         \3/         \3/

$$- \frac{\sinh{\left(\frac{x}{3} \right)}}{3 \sec{\left(\frac{2 x}{3} - \frac{\pi}{2} \right)}} + \frac{\sinh{\left(\frac{x}{3} \right)}}{3 \sec^{2}{\left(\frac{x}{3} \right)}} - \frac{\cosh{\left(\frac{x}{3} \right)}}{3 \sec{\left(\frac{2 x}{3} - \frac{\pi}{2} \right)}} + \frac{\cosh{\left(\frac{x}{3} \right)}}{3 \sec^{2}{\left(\frac{x}{3} \right)}}$$

     /  pi   2*x\     /x\      /  pi   2*x\     /x\      2/x\     /x\      2/x\     /x\
  cos|- -- + ---|*cosh|-|   cos|- -- + ---|*sinh|-|   cos |-|*cosh|-|   cos |-|*sinh|-|
     \  2     3 /     \3/      \  2     3 /     \3/       \3/     \3/       \3/     \3/
- ----------------------- - ----------------------- + --------------- + ---------------
             3                         3                     3                 3

$$\frac{\cos^{2}{\left(\frac{x}{3} \right)} \sinh{\left(\frac{x}{3} \right)}}{3} + \frac{\cos^{2}{\left(\frac{x}{3} \right)} \cosh{\left(\frac{x}{3} \right)}}{3} - \frac{\cos{\left(\frac{2 x}{3} - \frac{\pi}{2} \right)} \sinh{\left(\frac{x}{3} \right)}}{3} - \frac{\cos{\left(\frac{2 x}{3} - \frac{\pi}{2} \right)} \cosh{\left(\frac{x}{3} \right)}}{3}$$

   2/x\ /    /x\       /x\\     /    /x\       /x\\    /x\    /x\
cos |-|*|cosh|-| + sinh|-||   2*|cosh|-| + sinh|-||*cos|-|*sin|-|
    \3/ \    \3/       \3//     \    \3/       \3//    \3/    \3/
--------------------------- - -----------------------------------
             3                                 3

$$- \frac{2 \left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} + \frac{\left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \cos^{2}{\left(\frac{x}{3} \right)}}{3}$$

/           2/x\          /x\ \  x
|    1 - tan |-|     4*tan|-| |  -
|            \3/          \3/ |  3
|1 + ----------- - -----------|*e 
|           2/x\          2/x\|   
|    1 + tan |-|   1 + tan |-||   
\            \3/           \3//   
----------------------------------
                6

$$\frac{\left(\frac{1 - \tan^{2}{\left(\frac{x}{3} \right)}}{\tan^{2}{\left(\frac{x}{3} \right)} + 1} + 1 - \frac{4 \tan{\left(\frac{x}{3} \right)}}{\tan^{2}{\left(\frac{x}{3} \right)} + 1}\right) e^{\frac{x}{3}}}{6}$$

       x                    x        
       -                    -        
       3                    3        
      e                  2*e         
-------------- - --------------------
     2/pi   x\        /x\    /pi   x\
3*csc |-- - -|   3*csc|-|*csc|-- - -|
      \2    3/        \3/    \2    3/

$$\frac{e^{\frac{x}{3}}}{3 \csc^{2}{\left(- \frac{x}{3} + \frac{\pi}{2} \right)}} - \frac{2 e^{\frac{x}{3}}}{3 \csc{\left(\frac{x}{3} \right)} \csc{\left(- \frac{x}{3} + \frac{\pi}{2} \right)}}$$

                             x
                             -
/         /2*x\      /2*x\\  3
|1 - 2*sin|---| + cos|---||*e 
\         \ 3 /      \ 3 //   
------------------------------
              6

$$\frac{\left(- 2 \sin{\left(\frac{2 x}{3} \right)} + \cos{\left(\frac{2 x}{3} \right)} + 1\right) e^{\frac{x}{3}}}{6}$$

                x                    x       
             2  -                    -       
/       2/x\\   3     /       2/x\\  3    /x\
|1 - tan |-|| *e    4*|1 - tan |-||*e *tan|-|
\        \6//         \        \6//       \6/
----------------- - -------------------------
                2                       2    
   /       2/x\\           /       2/x\\     
 3*|1 + tan |-||         3*|1 + tan |-||     
   \        \6//           \        \6//

$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{6} \right)}\right)^{2} e^{\frac{x}{3}}}{3 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}} - \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{6} \right)}\right) e^{\frac{x}{3}} \tan{\left(\frac{x}{6} \right)}}{3 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}}$$

/    /x\       /x\\ /         /2*x\      /2*x\\
|cosh|-| + sinh|-||*|1 - 2*sin|---| + cos|---||
\    \3/       \3// \         \ 3 /      \ 3 //
-----------------------------------------------
                       6

$$\frac{\left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \left(- 2 \sin{\left(\frac{2 x}{3} \right)} + \cos{\left(\frac{2 x}{3} \right)} + 1\right)}{6}$$

   2/x\ /    /x\       /x\\     /    /x\       /x\\    /x\    /  pi   x\
cos |-|*|cosh|-| + sinh|-||   2*|cosh|-| + sinh|-||*cos|-|*cos|- -- + -|
    \3/ \    \3/       \3//     \    \3/       \3//    \3/    \  2    3/
--------------------------- - ------------------------------------------
             3                                    3

$$\frac{\left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \cos^{2}{\left(\frac{x}{3} \right)}}{3} - \frac{2 \left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \cos{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} - \frac{\pi}{2} \right)}}{3}$$

       /x\          /x\            /x\              /x\    
   cosh|-|      sinh|-|        cosh|-|          sinh|-|    
       \3/          \3/            \3/              \3/    
- ---------- - ---------- + -------------- + --------------
       /2*x\        /2*x\        2/pi   x\        2/pi   x\
  3*csc|---|   3*csc|---|   3*csc |-- - -|   3*csc |-- - -|
       \ 3 /        \ 3 /         \2    3/         \2    3/

$$\frac{\sinh{\left(\frac{x}{3} \right)}}{3 \csc^{2}{\left(- \frac{x}{3} + \frac{\pi}{2} \right)}} - \frac{\sinh{\left(\frac{x}{3} \right)}}{3 \csc{\left(\frac{2 x}{3} \right)}} + \frac{\cosh{\left(\frac{x}{3} \right)}}{3 \csc^{2}{\left(- \frac{x}{3} + \frac{\pi}{2} \right)}} - \frac{\cosh{\left(\frac{x}{3} \right)}}{3 \csc{\left(\frac{2 x}{3} \right)}}$$

/    /x\       /x\\ /         /  pi   2*x\      /2*x\\
|cosh|-| + sinh|-||*|1 - 2*cos|- -- + ---| + cos|---||
\    \3/       \3// \         \  2     3 /      \ 3 //
------------------------------------------------------
                          6

$$\frac{\left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \left(\cos{\left(\frac{2 x}{3} \right)} - 2 \cos{\left(\frac{2 x}{3} - \frac{\pi}{2} \right)} + 1\right)}{6}$$

                                x
                                -
/          1            2    \  3
|1 + ------------- - --------|*e 
|       /pi   2*x\      /2*x\|   
|    csc|-- - ---|   csc|---||   
\       \2     3 /      \ 3 //   
---------------------------------
                6

$$\frac{\left(1 + \frac{1}{\csc{\left(- \frac{2 x}{3} + \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(\frac{2 x}{3} \right)}}\right) e^{\frac{x}{3}}}{6}$$

     x               x     
     -               -     
     3               3     
    e             2*e      
--------- - ---------------
     2/x\        /x\    /x\
3*sec |-|   3*csc|-|*sec|-|
      \3/        \3/    \3/

$$\frac{e^{\frac{x}{3}}}{3 \sec^{2}{\left(\frac{x}{3} \right)}} - \frac{2 e^{\frac{x}{3}}}{3 \csc{\left(\frac{x}{3} \right)} \sec{\left(\frac{x}{3} \right)}}$$

                 x                            x
              2  -                            -
/        2/x\\   3     /        2/x\\    /x\  3
|-1 + cot |-|| *e    4*|-1 + cot |-||*cot|-|*e 
\         \6//         \         \6//    \6/   
------------------ - --------------------------
                2                        2     
   /       2/x\\            /       2/x\\      
 3*|1 + cot |-||          3*|1 + cot |-||      
   \        \6//            \        \6//

$$\frac{\left(\cot^{2}{\left(\frac{x}{6} \right)} - 1\right)^{2} e^{\frac{x}{3}}}{3 \left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}} - \frac{4 \left(\cot^{2}{\left(\frac{x}{6} \right)} - 1\right) e^{\frac{x}{3}} \cot{\left(\frac{x}{6} \right)}}{3 \left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}}$$

                                                      2                         2        
        /x\    /x\        /x\     /x\   /        2/x\\      /x\   /        2/x\\      /x\
  2*cosh|-|*cot|-|   2*cot|-|*sinh|-|   |-1 + cot |-|| *cosh|-|   |-1 + cot |-|| *sinh|-|
        \3/    \3/        \3/     \3/   \         \6//      \3/   \         \6//      \3/
- ---------------- - ---------------- + ----------------------- + -----------------------
    /       2/x\\      /       2/x\\                       2                         2   
  3*|1 + cot |-||    3*|1 + cot |-||          /       2/x\\             /       2/x\\    
    \        \3//      \        \3//        3*|1 + cot |-||           3*|1 + cot |-||    
                                              \        \6//             \        \6//

$$\frac{\left(\cot^{2}{\left(\frac{x}{6} \right)} - 1\right)^{2} \sinh{\left(\frac{x}{3} \right)}}{3 \left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}} + \frac{\left(\cot^{2}{\left(\frac{x}{6} \right)} - 1\right)^{2} \cosh{\left(\frac{x}{3} \right)}}{3 \left(\cot^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}} - \frac{2 \cot{\left(\frac{x}{3} \right)} \sinh{\left(\frac{x}{3} \right)}}{3 \left(\cot^{2}{\left(\frac{x}{3} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{x}{3} \right)} \cosh{\left(\frac{x}{3} \right)}}{3 \left(\cot^{2}{\left(\frac{x}{3} \right)} + 1\right)}$$

   2/pi   x\ /    /x\       /x\\     /    /x\       /x\\    /x\    /pi   x\
sin |-- + -|*|cosh|-| + sinh|-||   2*|cosh|-| + sinh|-||*sin|-|*sin|-- + -|
    \2    3/ \    \3/       \3//     \    \3/       \3//    \3/    \2    3/
-------------------------------- - ----------------------------------------
               3                                      3

$$- \frac{2 \left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \sin{\left(\frac{x}{3} \right)} \sin{\left(\frac{x}{3} + \frac{\pi}{2} \right)}}{3} + \frac{\left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \sin^{2}{\left(\frac{x}{3} + \frac{\pi}{2} \right)}}{3}$$

                                                     2                        2        
        /x\    /x\         /x\    /x\   /       2/x\\      /x\   /       2/x\\      /x\
  2*cosh|-|*tan|-|   2*sinh|-|*tan|-|   |1 - tan |-|| *cosh|-|   |1 - tan |-|| *sinh|-|
        \3/    \3/         \3/    \3/   \        \6//      \3/   \        \6//      \3/
- ---------------- - ---------------- + ---------------------- + ----------------------
    /       2/x\\      /       2/x\\                      2                        2   
  3*|1 + tan |-||    3*|1 + tan |-||         /       2/x\\            /       2/x\\    
    \        \3//      \        \3//       3*|1 + tan |-||          3*|1 + tan |-||    
                                             \        \6//            \        \6//

$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{6} \right)}\right)^{2} \sinh{\left(\frac{x}{3} \right)}}{3 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}} + \frac{\left(1 - \tan^{2}{\left(\frac{x}{6} \right)}\right)^{2} \cosh{\left(\frac{x}{3} \right)}}{3 \left(\tan^{2}{\left(\frac{x}{6} \right)} + 1\right)^{2}} - \frac{2 \tan{\left(\frac{x}{3} \right)} \sinh{\left(\frac{x}{3} \right)}}{3 \left(\tan^{2}{\left(\frac{x}{3} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{x}{3} \right)} \cosh{\left(\frac{x}{3} \right)}}{3 \left(\tan^{2}{\left(\frac{x}{3} \right)} + 1\right)}$$

      /x\    /2*x\      /2*x\     /x\      2/pi   x\     /x\      2/pi   x\     /x\
  cosh|-|*sin|---|   sin|---|*sinh|-|   sin |-- + -|*cosh|-|   sin |-- + -|*sinh|-|
      \3/    \ 3 /      \ 3 /     \3/       \2    3/     \3/       \2    3/     \3/
- ---------------- - ---------------- + -------------------- + --------------------
         3                  3                    3                      3

$$- \frac{\sin{\left(\frac{2 x}{3} \right)} \sinh{\left(\frac{x}{3} \right)}}{3} - \frac{\sin{\left(\frac{2 x}{3} \right)} \cosh{\left(\frac{x}{3} \right)}}{3} + \frac{\sin^{2}{\left(\frac{x}{3} + \frac{\pi}{2} \right)} \sinh{\left(\frac{x}{3} \right)}}{3} + \frac{\sin^{2}{\left(\frac{x}{3} + \frac{\pi}{2} \right)} \cosh{\left(\frac{x}{3} \right)}}{3}$$

/    /x\       /x\\ /         /2*x\      /pi   2*x\\
|cosh|-| + sinh|-||*|1 - 2*sin|---| + sin|-- + ---||
\    \3/       \3// \         \ 3 /      \2     3 //
----------------------------------------------------
                         6

$$\frac{\left(\sinh{\left(\frac{x}{3} \right)} + \cosh{\left(\frac{x}{3} \right)}\right) \left(- 2 \sin{\left(\frac{2 x}{3} \right)} + \sin{\left(\frac{2 x}{3} + \frac{\pi}{2} \right)} + 1\right)}{6}$$

(cosh(x/3) + sinh(x/3))*(1 - 2*sin(2*x/3) + sin(pi/2 + 2*x/3))/6

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Least common denominator cos(x/3)^2*exp(x/3)/3-2*cos(x/3)*exp(x/3)*sin(x/3)/3

The solution

Least common denominator cos(x/3)^2exp(x/3)/3-2cos(x/3)exp(x/3)sin(x/3)/3