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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
((-log(cos(two *x)+ one)/ two)+log(cos(two *x)- one)/ two +cos(two *x))/ two
(( minus logarithm of ( co sinus of e of (2 multiply by x) plus 1) divide by 2) plus logarithm of ( co sinus of e of (2 multiply by x) minus 1) divide by 2 plus co sinus of e of (2 multiply by x)) divide by 2
(( minus logarithm of ( co sinus of e of (two multiply by x) plus one) divide by two) plus logarithm of ( co sinus of e of (two multiply by x) minus one) divide by two plus co sinus of e of (two multiply by x)) divide by two
((-log(cos(2x)+1)/2)+log(cos(2x)-1)/2+cos(2x))/2
-logcos2x+1/2+logcos2x-1/2+cos2x/2
((-log(cos(2*x)+1) divide by 2)+log(cos(2*x)-1) divide by 2+cos(2*x)) divide by 2
Similar expressions
((-log(cos(2*x)-1)/2)+log(cos(2*x)-1)/2+cos(2*x))/2
((-log(cos(2*x)+1)/2)-log(cos(2*x)-1)/2+cos(2*x))/2
((-log(cos(2*x)+1)/2)+log(cos(2*x)+1)/2+cos(2*x))/2
((log(cos(2*x)+1)/2)+log(cos(2*x)-1)/2+cos(2*x))/2
((-log(cos(2*x)+1)/2)+log(cos(2*x)-1)/2-cos(2*x))/2

Expression simplification/ Least common denominator/ ((-log(cos(2*x)+1)/2)+log(cos(2*x)-1)/2+cos(2*x))/2

Least common denominator ((-log(cos(2x)+1)/2)+log(cos(2x)-1)/2+cos(2*x))/2

The solution

You have entered [src]

-log(cos(2*x) + 1)    log(cos(2*x) - 1)           
------------------- + ----------------- + cos(2*x)
         2                    2                   
--------------------------------------------------
                        2

$$\frac{\left(\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)}}{2} + \frac{\left(-1\right) \log{\left(\cos{\left(2 x \right)} + 1 \right)}}{2}\right) + \cos{\left(2 x \right)}}{2}$$

((-log(cos(2*x) + 1))/2 + log(cos(2*x) - 1)/2 + cos(2*x))/2

General simplification [src]

                   /   2   \      /        2   \
  1      2      log\cos (x)/   log\-1 + cos (x)/
- - + cos (x) - ------------ + -----------------
  2                  4                 4

$$\frac{\log{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{4} + \cos^{2}{\left(x \right)} - \frac{1}{2}$$

-1/2 + cos(x)^2 - log(cos(x)^2)/4 + log(-1 + cos(x)^2)/4

Rational denominator [src]

-log(1 + cos(2*x)) + 2*cos(2*x) + log(-1 + cos(2*x))
----------------------------------------------------
                         4

$$\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)} - \log{\left(\cos{\left(2 x \right)} + 1 \right)} + 2 \cos{\left(2 x \right)}}{4}$$

(-log(1 + cos(2*x)) + 2*cos(2*x) + log(-1 + cos(2*x)))/4

Numerical answer [src]

0.25*log(cos(2*x) - 1) + 0.5*cos(2*x) - 0.25*log(cos(2*x) + 1)

0.25*log(cos(2*x) - 1) + 0.5*cos(2*x) - 0.25*log(cos(2*x) + 1)

Powers [src]

     /     -2*I*x    2*I*x\                         /      -2*I*x    2*I*x\
     |    e         e     |                         |     e         e     |
  log|1 + ------- + ------|    -2*I*x    2*I*x   log|-1 + ------- + ------|
     \       2        2   /   e         e           \        2        2   /
- ------------------------- + ------- + ------ + --------------------------
              4                  4        4                  4

$$\frac{e^{2 i x}}{4} + \frac{\log{\left(\frac{e^{2 i x}}{2} - 1 + \frac{e^{- 2 i x}}{2} \right)}}{4} - \frac{\log{\left(\frac{e^{2 i x}}{2} + 1 + \frac{e^{- 2 i x}}{2} \right)}}{4} + \frac{e^{- 2 i x}}{4}$$

cos(2*x)   log(1 + cos(2*x))   log(-1 + cos(2*x))
-------- - ----------------- + ------------------
   2               4                   4

$$\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)}}{4} - \frac{\log{\left(\cos{\left(2 x \right)} + 1 \right)}}{4} + \frac{\cos{\left(2 x \right)}}{2}$$

cos(2*x)/2 - log(1 + cos(2*x))/4 + log(-1 + cos(2*x))/4

Combining rational expressions [src]

-log(1 + cos(2*x)) + 2*cos(2*x) + log(-1 + cos(2*x))
----------------------------------------------------
                         4

$$\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)} - \log{\left(\cos{\left(2 x \right)} + 1 \right)} + 2 \cos{\left(2 x \right)}}{4}$$

(-log(1 + cos(2*x)) + 2*cos(2*x) + log(-1 + cos(2*x)))/4

Combinatorics [src]

cos(2*x)   log(1 + cos(2*x))   log(-1 + cos(2*x))
-------- - ----------------- + ------------------
   2               4                   4

$$\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)}}{4} - \frac{\log{\left(\cos{\left(2 x \right)} + 1 \right)}}{4} + \frac{\cos{\left(2 x \right)}}{2}$$

cos(2*x)/2 - log(1 + cos(2*x))/4 + log(-1 + cos(2*x))/4

Common denominator [src]

cos(2*x)   log(1 + cos(2*x))   log(-1 + cos(2*x))
-------- - ----------------- + ------------------
   2               4                   4

$$\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)}}{4} - \frac{\log{\left(\cos{\left(2 x \right)} + 1 \right)}}{4} + \frac{\cos{\left(2 x \right)}}{2}$$

cos(2*x)/2 - log(1 + cos(2*x))/4 + log(-1 + cos(2*x))/4

Expand expression [src]

                            /   2   \      /          2   \
  1      2      log(2)   log\cos (x)/   log\-2 + 2*cos (x)/
- - + cos (x) - ------ - ------------ + -------------------
  2               4           4                  4

$$\frac{\log{\left(2 \cos^{2}{\left(x \right)} - 2 \right)}}{4} - \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{4} + \cos^{2}{\left(x \right)} - \frac{1}{2} - \frac{\log{\left(2 \right)}}{4}$$

-1/2 + cos(x)^2 - log(2)/4 - log(cos(x)^2)/4 + log(-2 + 2*cos(x)^2)/4

Assemble expression [src]

cos(2*x)   log(cos(2*x) + 1)   log(cos(2*x) - 1)
-------- - ----------------- + -----------------
   2               4                   4

$$\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)}}{4} - \frac{\log{\left(\cos{\left(2 x \right)} + 1 \right)}}{4} + \frac{\cos{\left(2 x \right)}}{2}$$

cos(2*x)/2 - log(cos(2*x) + 1)/4 + log(cos(2*x) - 1)/4

Trigonometric part [src]

                /       1    \      /        1    \
             log|1 + --------|   log|-1 + --------|
    1           \    sec(2*x)/      \     sec(2*x)/
---------- - ----------------- + ------------------
2*sec(2*x)           4                   4

$$\frac{\log{\left(-1 + \frac{1}{\sec{\left(2 x \right)}} \right)}}{4} - \frac{\log{\left(1 + \frac{1}{\sec{\left(2 x \right)}} \right)}}{4} + \frac{1}{2 \sec{\left(2 x \right)}}$$

         /             2\      /                  2\                 
         |/       2/x\\ |      |     /       2/x\\ |                 
         ||1 - tan |-|| |      |     |1 - tan |-|| |                 
         |\        \2// |      |     \        \2// |                 
      log|--------------|   log|-1 + --------------|                 
         |             2|      |                  2|                2
         |/       2/x\\ |      |     /       2/x\\ |   /       2/x\\ 
         ||1 + tan |-|| |      |     |1 + tan |-|| |   |1 - tan |-|| 
  1      \\        \2// /      \     \        \2// /   \        \2// 
- - - ------------------- + ------------------------ + --------------
  2            4                       4                            2
                                                       /       2/x\\ 
                                                       |1 + tan |-|| 
                                                       \        \2//

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

     /            2   \      /             2   \                  
     |    -1 + cot (x)|      |     -1 + cot (x)|                  
  log|1 + ------------|   log|-1 + ------------|                  
     |           2    |      |            2    |             2    
     \    1 + cot (x) /      \     1 + cot (x) /     -1 + cot (x) 
- --------------------- + ---------------------- + ---------------
            4                       4                /       2   \
                                                   2*\1 + cot (x)/

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

                   /   2   \      /        2   \
  1      2      log\cos (x)/   log\-1 + cos (x)/
- - + cos (x) - ------------ + -----------------
  2                  4                 4

$$\frac{\log{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{4} + \cos^{2}{\left(x \right)} - \frac{1}{2}$$

                   /   2   \      /    2   \
  1      2      log\cos (x)/   log\-sin (x)/
- - + cos (x) - ------------ + -------------
  2                  4               4

$$\frac{\log{\left(- \sin^{2}{\left(x \right)} \right)}}{4} - \frac{\log{\left(\cos^{2}{\left(x \right)} \right)}}{4} + \cos^{2}{\left(x \right)} - \frac{1}{2}$$

                                           /-1 + cos(2*x)\
                                        log|-------------|
cos(2*x)   log(1 + cos(2*x))   log(2)      \      2      /
-------- - ----------------- + ------ + ------------------
   2               4             4              4

$$\frac{\log{\left(\frac{\cos{\left(2 x \right)} - 1}{2} \right)}}{4} - \frac{\log{\left(\cos{\left(2 x \right)} + 1 \right)}}{4} + \frac{\cos{\left(2 x \right)}}{2} + \frac{\log{\left(2 \right)}}{4}$$

     /           2   \      /            2   \                  
     |    1 - tan (x)|      |     1 - tan (x)|                  
  log|1 + -----------|   log|-1 + -----------|                  
     |           2   |      |            2   |            2     
     \    1 + tan (x)/      \     1 + tan (x)/     1 - tan (x)  
- -------------------- + --------------------- + ---------------
           4                       4               /       2   \
                                                 2*\1 + tan (x)/

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

                   /   1   \      /        1   \
                log|-------|   log|-1 + -------|
                   |   2   |      |        2   |
  1      1         \sec (x)/      \     sec (x)/
- - + ------- - ------------ + -----------------
  2      2           4                 4        
      sec (x)

$$\frac{\log{\left(-1 + \frac{1}{\sec^{2}{\left(x \right)}} \right)}}{4} - \frac{\log{\left(\frac{1}{\sec^{2}{\left(x \right)}} \right)}}{4} - \frac{1}{2} + \frac{1}{\sec^{2}{\left(x \right)}}$$

                        /     1      \      /          1      \
                     log|------------|   log|-1 + ------------|
                        |   2/pi    \|      |        2/pi    \|
                        |csc |-- - x||      |     csc |-- - x||
  1        1            \    \2     //      \         \2     //
- - + ------------ - ----------------- + ----------------------
  2      2/pi    \           4                     4           
      csc |-- - x|                                             
          \2     /

$$\frac{\log{\left(-1 + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4} - \frac{\log{\left(\frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} \right)}}{4} - \frac{1}{2} + \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$

cos(2*x)   log(1 + cos(2*x))   log(-1 + cos(2*x))
-------- - ----------------- + ------------------
   2               4                   4

$$\frac{\log{\left(\cos{\left(2 x \right)} - 1 \right)}}{4} - \frac{\log{\left(\cos{\left(2 x \right)} + 1 \right)}}{4} + \frac{\cos{\left(2 x \right)}}{2}$$

                        /   2/    pi\\      /        2/    pi\\
                     log|sin |x + --||   log|-1 + sin |x + --||
  1      2/    pi\      \    \    2 //      \         \    2 //
- - + sin |x + --| - ----------------- + ----------------------
  2       \    2 /           4                     4

$$\frac{\log{\left(\sin^{2}{\left(x + \frac{\pi}{2} \right)} - 1 \right)}}{4} - \frac{\log{\left(\sin^{2}{\left(x + \frac{\pi}{2} \right)} \right)}}{4} + \sin^{2}{\left(x + \frac{\pi}{2} \right)} - \frac{1}{2}$$

         /              2\      /                   2\                  
         |/        2/x\\ |      |     /        2/x\\ |                  
         ||-1 + cot |-|| |      |     |-1 + cot |-|| |                  
         |\         \2// |      |     \         \2// |                  
      log|---------------|   log|-1 + ---------------|                  
         |              2|      |                   2|                 2
         | /       2/x\\ |      |      /       2/x\\ |   /        2/x\\ 
         | |1 + cot |-|| |      |      |1 + cot |-|| |   |-1 + cot |-|| 
  1      \ \        \2// /      \      \        \2// /   \         \2// 
- - - -------------------- + ------------------------- + ---------------
  2            4                         4                             2
                                                          /       2/x\\ 
                                                          |1 + cot |-|| 
                                                          \        \2//

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

   /pi      \      /       /pi      \\      /        /pi      \\
sin|-- + 2*x|   log|1 + sin|-- + 2*x||   log|-1 + sin|-- + 2*x||
   \2       /      \       \2       //      \        \2       //
------------- - ---------------------- + -----------------------
      2                   4                         4

$$\frac{\log{\left(\sin{\left(2 x + \frac{\pi}{2} \right)} - 1 \right)}}{4} - \frac{\log{\left(\sin{\left(2 x + \frac{\pi}{2} \right)} + 1 \right)}}{4} + \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{2}$$

                /   2   \      /        2   \
cos(2*x)   - log\cos (x)/ + log\-1 + cos (x)/
-------- + ----------------------------------
   2                       4

$$\frac{\log{\left(\cos^{2}{\left(x \right)} - 1 \right)} - \log{\left(\cos^{2}{\left(x \right)} \right)}}{4} + \frac{\cos{\left(2 x \right)}}{2}$$

                     /          1      \      /           1      \
                  log|1 + -------------|   log|-1 + -------------|
                     |       /pi      \|      |        /pi      \|
                     |    csc|-- - 2*x||      |     csc|-- - 2*x||
       1             \       \2       //      \        \2       //
--------------- - ---------------------- + -----------------------
     /pi      \             4                         4           
2*csc|-- - 2*x|                                                   
     \2       /

$$\frac{\log{\left(-1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{4} - \frac{\log{\left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} \right)}}{4} + \frac{1}{2 \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$

1/(2*csc(pi/2 - 2*x)) - log(1 + 1/csc(pi/2 - 2*x))/4 + log(-1 + 1/csc(pi/2 - 2*x))/4

Other calculators

How to use it?

Identical expressions

Similar expressions

Least common denominator ((-log(cos(2*x)+1)/2)+log(cos(2*x)-1)/2+cos(2*x))/2

The solution

Least common denominator ((-log(cos(2x)+1)/2)+log(cos(2x)-1)/2+cos(2*x))/2