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Simplificación de expresiones/ Descomposición en fracciones simples/ log(tan(x)^2+1)/8-log(tan(x)+1)/4+(tan(x)-1)/(4*tan(x)^2+4)

¿Cómo vas a descomponer esta log(tan(x)^2+1)/8-log(tan(x)+1)/4+(tan(x)-1)/(4*tan(x)^2+4) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /   2       \                                  
log\tan (x) + 1/   log(tan(x) + 1)     tan(x) - 1 
---------------- - --------------- + -------------
       8                  4               2       
                                     4*tan (x) + 4
(−log⁡(tan⁡(x)+1)4+log⁡(tan⁡2(x)+1)8)+tan⁡(x)−14tan⁡2(x)+4\left(- \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8}\right) + \frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} 
log(tan(x)^2 + 1)/8 - log(tan(x) + 1)/4 + (tan(x) - 1)/(4*tan(x)^2 + 4)
Simplificación general [src] 
                           /     2      \     ___    /      pi\
                        log|------------|   \/ 2 *cos|2*x + --|
  1   log(1 + tan(x))      \1 + cos(2*x)/            \      4 /
- - - --------------- + ----------------- - -------------------
  8          4                  8                    8
log⁡(2cos⁡(2x)+1)8−log⁡(tan⁡(x)+1)4−2cos⁡(2x+π4)8−18\frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8} 
-1/8 - log(1 + tan(x))/4 + log(2/(1 + cos(2*x)))/8 - sqrt(2)*cos(2*x + pi/4)/8
Denominador racional [src] 
                                       2       /       2   \        2                         /       2   \
-2 - 2*log(1 + tan(x)) + 2*tan(x) + tan (x)*log\1 + tan (x)/ - 2*tan (x)*log(1 + tan(x)) + log\1 + tan (x)/
-----------------------------------------------------------------------------------------------------------
                                                        2                                                  
                                               8 + 8*tan (x)
−2log⁡(tan⁡(x)+1)tan⁡2(x)−2log⁡(tan⁡(x)+1)+log⁡(tan⁡2(x)+1)tan⁡2(x)+log⁡(tan⁡2(x)+1)+2tan⁡(x)−28tan⁡2(x)+8\frac{- 2 \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - 2 \log{\left(\tan{\left(x \right)} + 1 \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} + 2 \tan{\left(x \right)} - 2}{8 \tan^{2}{\left(x \right)} + 8} 
(-2 - 2*log(1 + tan(x)) + 2*tan(x) + tan(x)^2*log(1 + tan(x)^2) - 2*tan(x)^2*log(1 + tan(x)) + log(1 + tan(x)^2))/(8 + 8*tan(x)^2)
Respuesta numérica [src] 
0.125*log(tan(x)^2 + 1) - 0.25*log(tan(x) + 1) + (-1.0 + tan(x))/(4.0 + 4.0*tan(x)^2)
0.125*log(tan(x)^2 + 1) - 0.25*log(tan(x) + 1) + (-1.0 + tan(x))/(4.0 + 4.0*tan(x)^2)
Denominador común [src] 
                       /       2   \                
  log(1 + tan(x))   log\1 + tan (x)/    -1 + tan(x) 
- --------------- + ---------------- + -------------
         4                 8                    2   
                                       4 + 4*tan (x)
tan⁡(x)−14tan⁡2(x)+4−log⁡(tan⁡(x)+1)4+log⁡(tan⁡2(x)+1)8\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8} 
-log(1 + tan(x))/4 + log(1 + tan(x)^2)/8 + (-1 + tan(x))/(4 + 4*tan(x)^2)
Unión de expresiones racionales [src] 
                /       2   \ /                        /       2   \\
-2 + 2*tan(x) + \1 + tan (x)/*\-2*log(1 + tan(x)) + log\1 + tan (x)//
---------------------------------------------------------------------
                             /       2   \                           
                           8*\1 + tan (x)/
(−2log⁡(tan⁡(x)+1)+log⁡(tan⁡2(x)+1))(tan⁡2(x)+1)+2tan⁡(x)−28(tan⁡2(x)+1)\frac{\left(- 2 \log{\left(\tan{\left(x \right)} + 1 \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)} - 2}{8 \left(\tan^{2}{\left(x \right)} + 1\right)} 
(-2 + 2*tan(x) + (1 + tan(x)^2)*(-2*log(1 + tan(x)) + log(1 + tan(x)^2)))/(8*(1 + tan(x)^2))
Potencias [src] 
                                   /                    2\                          
                                   |    /   I*x    -I*x\ |                          
     /      /   I*x    -I*x\\      |    \- e    + e    / |          /   I*x    -I*x\
     |    I*\- e    + e    /|   log|1 - -----------------|        I*\- e    + e    /
  log|1 + ------------------|      |                   2 |   -1 + ------------------
     |        I*x    -I*x   |      |     / I*x    -I*x\  |            I*x    -I*x   
     \       e    + e       /      \     \e    + e    /  /           e    + e       
- --------------------------- + -------------------------- + -----------------------
               4                            8                                      2
                                                                   /   I*x    -I*x\ 
                                                                 4*\- e    + e    / 
                                                             4 - -------------------
                                                                                 2  
                                                                   / I*x    -I*x\   
                                                                   \e    + e    /
log⁡(−(−eix+e−ix)2(eix+e−ix)2+1)8−log⁡(i(−eix+e−ix)eix+e−ix+1)4+i(−eix+e−ix)eix+e−ix−1−4(−eix+e−ix)2(eix+e−ix)2+4\frac{\log{\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1 \right)}}{8} - \frac{\log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1 \right)}}{4} + \frac{\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} - 1}{- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 4} 
                       /       2   \                
  log(1 + tan(x))   log\1 + tan (x)/    -1 + tan(x) 
- --------------- + ---------------- + -------------
         4                 8                    2   
                                       4 + 4*tan (x)
tan⁡(x)−14tan⁡2(x)+4−log⁡(tan⁡(x)+1)4+log⁡(tan⁡2(x)+1)8\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8} 
-log(1 + tan(x))/4 + log(1 + tan(x)^2)/8 + (-1 + tan(x))/(4 + 4*tan(x)^2)
Abrimos la expresión [src] 
                                       /   2       \                
        1         log(tan(x) + 1)   log\tan (x) + 1/       tan(x)   
- ------------- - --------------- + ---------------- + -------------
       2                 4                 8                2       
  4*tan (x) + 4                                        4*tan (x) + 4
−log⁡(tan⁡(x)+1)4+log⁡(tan⁡2(x)+1)8+tan⁡(x)4tan⁡2(x)+4−14tan⁡2(x)+4- \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8} + \frac{\tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 4} - \frac{1}{4 \tan^{2}{\left(x \right)} + 4} 
-1/(4*tan(x)^2 + 4) - log(tan(x) + 1)/4 + log(tan(x)^2 + 1)/8 + tan(x)/(4*tan(x)^2 + 4)
Compilar la expresión [src] 
                       /   2       \                
  log(tan(x) + 1)   log\tan (x) + 1/    -1 + tan(x) 
- --------------- + ---------------- + -------------
         4                 8                    2   
                                       4 + 4*tan (x)
tan⁡(x)−14tan⁡2(x)+4−log⁡(tan⁡(x)+1)4+log⁡(tan⁡2(x)+1)8\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8} 
-log(tan(x) + 1)/4 + log(tan(x)^2 + 1)/8 + (-1 + tan(x))/(4 + 4*tan(x)^2)
Combinatoria [src] 
 /       /       2   \                                     2       /       2   \        2                   \ 
-\2 - log\1 + tan (x)/ - 2*tan(x) + 2*log(1 + tan(x)) - tan (x)*log\1 + tan (x)/ + 2*tan (x)*log(1 + tan(x))/ 
--------------------------------------------------------------------------------------------------------------
                                                 /       2   \                                                
                                               8*\1 + tan (x)/
−2log⁡(tan⁡(x)+1)tan⁡2(x)+2log⁡(tan⁡(x)+1)−log⁡(tan⁡2(x)+1)tan⁡2(x)−log⁡(tan⁡2(x)+1)−2tan⁡(x)+28(tan⁡2(x)+1)- \frac{2 \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} + 2 \log{\left(\tan{\left(x \right)} + 1 \right)} - \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} - 2 \tan{\left(x \right)} + 2}{8 \left(\tan^{2}{\left(x \right)} + 1\right)} 
-(2 - log(1 + tan(x)^2) - 2*tan(x) + 2*log(1 + tan(x)) - tan(x)^2*log(1 + tan(x)^2) + 2*tan(x)^2*log(1 + tan(x)))/(8*(1 + tan(x)^2))
Parte trigonométrica [src] 
         /       sec(x)  \                                 
      log|1 + -----------|                                 
         |       /    pi\|                                 
         |    sec|x - --||      /   2   \          ___     
  1      \       \    2 //   log\sec (x)/        \/ 2      
- - - -------------------- + ------------ - ---------------
  8            4                  8              /      pi\
                                            8*sec|2*x + --|
                                                 \      4 /
−log⁡(sec⁡(x)sec⁡(x−π2)+1)4+log⁡(sec⁡2(x))8−18−28sec⁡(2x+π4)- \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1 \right)}}{4} + \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{8} - \frac{1}{8} - \frac{\sqrt{2}}{8 \sec{\left(2 x + \frac{\pi}{4} \right)}} 
                       /       2   \                
                       |    sec (x)|                
     /    sec(x)\   log|1 + -------|         sec(x) 
  log|1 + ------|      |       2   |    -1 + ------ 
     \    csc(x)/      \    csc (x)/         csc(x) 
- --------------- + ---------------- + -------------
         4                 8                    2   
                                           4*sec (x)
                                       4 + ---------
                                               2    
                                            csc (x)
−1+sec⁡(x)csc⁡(x)4+4sec⁡2(x)csc⁡2(x)+log⁡(1+sec⁡2(x)csc⁡2(x))8−log⁡(1+sec⁡(x)csc⁡(x))4\frac{-1 + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}}{4 + \frac{4 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}}{4} 
                       /       2   \                
  log(1 + tan(x))   log\1 + tan (x)/    -1 + tan(x) 
- --------------- + ---------------- + -------------
         4                 8                    2   
                                       4 + 4*tan (x)
tan⁡(x)−14tan⁡2(x)+4−log⁡(tan⁡(x)+1)4+log⁡(tan⁡2(x)+1)8\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8} 
                            /         2      \                   
     /       sec(x)  \      |      sec (x)   |           sec(x)  
  log|1 + -----------|   log|1 + ------------|   -1 + -----------
     |       /    pi\|      |       2/    pi\|           /    pi\
     |    sec|x - --||      |    sec |x - --||        sec|x - --|
     \       \    2 //      \        \    2 //           \    2 /
- -------------------- + --------------------- + ----------------
           4                       8                       2     
                                                      4*sec (x)  
                                                 4 + ------------
                                                        2/    pi\
                                                     sec |x - --|
                                                         \    2 /
sec⁡(x)sec⁡(x−π2)−14sec⁡2(x)sec⁡2(x−π2)+4−log⁡(sec⁡(x)sec⁡(x−π2)+1)4+log⁡(sec⁡2(x)sec⁡2(x−π2)+1)8\frac{\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - 1}{\frac{4 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 4} - \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1 \right)}}{8} 
                           /             2\                           
                           |/       2/x\\ |                           
                           ||1 + tan |-|| |                           
                           |\        \2// |                           
                        log|--------------|                           
                           |             2|                           
                           |/       2/x\\ |     ___ /       2/    pi\\
                           ||1 - tan |-|| |   \/ 2 *|1 - tan |x + --||
  1   log(1 + tan(x))      \\        \2// /         \        \    8 //
- - - --------------- + ------------------- - ------------------------
  8          4                   8                /       2/    pi\\  
                                                8*|1 + tan |x + --||  
                                                  \        \    8 //
−2(1−tan⁡2(x+π8))8(tan⁡2(x+π8)+1)+log⁡((tan⁡2(x2)+1)2(1−tan⁡2(x2))2)8−log⁡(tan⁡(x)+1)4−18- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(x + \frac{\pi}{8} \right)}\right)}{8 \left(\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1\right)} + \frac{\log{\left(\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{1}{8} 
                            /       2/    pi\\                     
     /       /    pi\\      |    cos |x - --||            /    pi\ 
     |    cos|x - --||      |        \    2 /|         cos|x - --| 
     |       \    2 /|   log|1 + ------------|            \    2 / 
  log|1 + -----------|      |         2      |    -1 + ----------- 
     \       cos(x)  /      \      cos (x)   /            cos(x)   
- -------------------- + --------------------- + ------------------
           4                       8                      2/    pi\
                                                     4*cos |x - --|
                                                           \    2 /
                                                 4 + --------------
                                                           2       
                                                        cos (x)
−1+cos⁡(x−π2)cos⁡(x)4+4cos⁡2(x−π2)cos⁡2(x)+log⁡(1+cos⁡2(x−π2)cos⁡2(x))8−log⁡(1+cos⁡(x−π2)cos⁡(x))4\frac{-1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}{4 + \frac{4 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{4} 
                                         ___    /      pi\
                           /   2   \   \/ 2 *cos|2*x + --|
  1   log(1 + tan(x))   log\sec (x)/            \      4 /
- - - --------------- + ------------ - -------------------
  8          4               8                  8
−log⁡(tan⁡(x)+1)4+log⁡(sec⁡2(x))8−2cos⁡(2x+π4)8−18- \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{8} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8} 
                                                    /   1   \
        ___    /      pi\   -2*log(1 + tan(x)) + log|-------|
  1 + \/ 2 *cos|2*x + --|                           |   2   |
               \      4 /                           \cos (x)/
- ----------------------- + ---------------------------------
             8                              8
−2cos⁡(2x+π4)+18+−2log⁡(tan⁡(x)+1)+log⁡(1cos⁡2(x))8- \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} + 1}{8} + \frac{- 2 \log{\left(\tan{\left(x \right)} + 1 \right)} + \log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{8} 
                                                 /   1   \
                          ___    /      pi\   log|-------|
                    1 + \/ 2 *cos|2*x + --|      |   2   |
  log(1 + tan(x))                \      4 /      \cos (x)/
- --------------- - ----------------------- + ------------
         4                     8                   8
−2cos⁡(2x+π4)+18−log⁡(tan⁡(x)+1)4+log⁡(1cos⁡2(x))8- \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} + 1}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{8} 
                       /       2   \                
                       |    sin (x)|                
     /    sin(x)\   log|1 + -------|         sin(x) 
  log|1 + ------|      |       2   |    -1 + ------ 
     \    cos(x)/      \    cos (x)/         cos(x) 
- --------------- + ---------------- + -------------
         4                 8                    2   
                                           4*sin (x)
                                       4 + ---------
                                               2    
                                            cos (x)
sin⁡(x)cos⁡(x)−14sin⁡2(x)cos⁡2(x)+4−log⁡(sin⁡(x)cos⁡(x)+1)4+log⁡(sin⁡2(x)cos⁡2(x)+1)8\frac{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1}{\frac{4 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 4} - \frac{\log{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 \right)}}{8} 
                           /              2\                            
                           | /       2/x\\ |                            
                           | |1 + cot |-|| |                            
                           | \        \2// |                            
                        log|---------------|                            
                           |              2|                            
         /      1   \      |/        2/x\\ |     ___ /        2/    pi\\
      log|1 + ------|      ||-1 + cot |-|| |   \/ 2 *|-1 + cot |x + --||
  1      \    cot(x)/      \\         \2// /         \         \    8 //
- - - --------------- + -------------------- - -------------------------
  8          4                   8                  /       2/    pi\\  
                                                  8*|1 + cot |x + --||  
                                                    \        \    8 //
−2(cot⁡2(x+π8)−1)8(cot⁡2(x+π8)+1)+log⁡((cot⁡2(x2)+1)2(cot⁡2(x2)−1)2)8−log⁡(1+1cot⁡(x))4−18- \frac{\sqrt{2} \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} - 1\right)}{8 \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} + 1\right)} + \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)}}{8} - \frac{\log{\left(1 + \frac{1}{\cot{\left(x \right)}} \right)}}{4} - \frac{1}{8} 
                       /       1   \              
     /      1   \   log|1 + -------|          1   
  log|1 + ------|      |       2   |   -1 + ------
     \    cot(x)/      \    cot (x)/        cot(x)
- --------------- + ---------------- + -----------
         4                 8                  4   
                                       4 + -------
                                              2   
                                           cot (x)
−1+1cot⁡(x)4+4cot⁡2(x)+log⁡(1+1cot⁡2(x))8−log⁡(1+1cot⁡(x))4\frac{-1 + \frac{1}{\cot{\left(x \right)}}}{4 + \frac{4}{\cot^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{1}{\cot{\left(x \right)}} \right)}}{4} 
                          /         4   \                 
     /         2   \      |    4*sin (x)|             2   
     |    2*sin (x)|   log|1 + ---------|        2*sin (x)
  log|1 + ---------|      |       2     |   -1 + ---------
     \     sin(2*x)/      \    sin (2*x)/         sin(2*x)
- ------------------ + ------------------ + --------------
          4                    8                      4   
                                                16*sin (x)
                                            4 + ----------
                                                   2      
                                                sin (2*x)
2sin⁡2(x)sin⁡(2x)−116sin⁡4(x)sin⁡2(2x)+4−log⁡(2sin⁡2(x)sin⁡(2x)+1)4+log⁡(4sin⁡4(x)sin⁡2(2x)+1)8\frac{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} - 1}{\frac{16 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 4} - \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1 \right)}}{8} 
                           /     2      \     ___ /       2/    pi\\
                        log|------------|   \/ 2 *|1 - tan |x + --||
  1   log(1 + tan(x))      \1 + cos(2*x)/         \        \    8 //
- - - --------------- + ----------------- - ------------------------
  8          4                  8               /       2/    pi\\  
                                              8*|1 + tan |x + --||  
                                                \        \    8 //
−2(1−tan⁡2(x+π8))8(tan⁡2(x+π8)+1)+log⁡(2cos⁡(2x)+1)8−log⁡(tan⁡(x)+1)4−18- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(x + \frac{\pi}{8} \right)}\right)}{8 \left(\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1\right)} + \frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{1}{8} 
                           /     2      \     ___    /      pi\
                        log|------------|   \/ 2 *cos|2*x + --|
  1   log(1 + tan(x))      \1 + cos(2*x)/            \      4 /
- - - --------------- + ----------------- - -------------------
  8          4                  8                    8
log⁡(2cos⁡(2x)+1)8−log⁡(tan⁡(x)+1)4−2cos⁡(2x+π4)8−18\frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8} 
         /       /    pi\\                                     
         |    cos|x - --||      /   1   \                      
         |       \    2 /|   log|-------|     ___    /      pi\
      log|1 + -----------|      |   2   |   \/ 2 *cos|2*x + --|
  1      \       cos(x)  /      \cos (x)/            \      4 /
- - - -------------------- + ------------ - -------------------
  8            4                  8                  8
−log⁡(1+cos⁡(x−π2)cos⁡(x))4+log⁡(1cos⁡2(x))8−2cos⁡(2x+π4)8−18- \frac{\log{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{4} + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{8} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8} 
                            /       2/pi    \\                     
     /       /pi    \\      |    csc |-- - x||            /pi    \ 
     |    csc|-- - x||      |        \2     /|         csc|-- - x| 
     |       \2     /|   log|1 + ------------|            \2     / 
  log|1 + -----------|      |         2      |    -1 + ----------- 
     \       csc(x)  /      \      csc (x)   /            csc(x)   
- -------------------- + --------------------- + ------------------
           4                       8                      2/pi    \
                                                     4*csc |-- - x|
                                                           \2     /
                                                 4 + --------------
                                                           2       
                                                        csc (x)
−1+csc⁡(−x+π2)csc⁡(x)4+4csc⁡2(−x+π2)csc⁡2(x)+log⁡(1+csc⁡2(−x+π2)csc⁡2(x))8−log⁡(1+csc⁡(−x+π2)csc⁡(x))4\frac{-1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}}{4 + \frac{4 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{4} 
                              /     1      \                        
         /         2   \   log|------------|                        
         |    2*sin (x)|      |   2/    pi\|     ___    /      3*pi\
      log|1 + ---------|      |sin |x + --||   \/ 2 *sin|2*x + ----|
  1      \     sin(2*x)/      \    \    2 //            \       4  /
- - - ------------------ + ----------------- - ---------------------
  8           4                    8                     8
−log⁡(2sin⁡2(x)sin⁡(2x)+1)4+log⁡(1sin⁡2(x+π2))8−2sin⁡(2x+3π4)8−18- \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{1}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} \right)}}{8} - \frac{\sqrt{2} \sin{\left(2 x + \frac{3 \pi}{4} \right)}}{8} - \frac{1}{8} 
         /       /pi    \\                                       
         |    csc|-- - x||                                       
         |       \2     /|      /   2/pi    \\                   
      log|1 + -----------|   log|csc |-- - x||          ___      
  1      \       csc(x)  /      \    \2     //        \/ 2       
- - - -------------------- + ----------------- - ----------------
  8            4                     8                /       pi\
                                                 8*csc|-2*x + --|
                                                      \       4 /
−log⁡(1+csc⁡(−x+π2)csc⁡(x))4+log⁡(csc⁡2(−x+π2))8−18−28csc⁡(−2x+π4)- \frac{\log{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{4} + \frac{\log{\left(\csc^{2}{\left(- x + \frac{\pi}{2} \right)} \right)}}{8} - \frac{1}{8} - \frac{\sqrt{2}}{8 \csc{\left(- 2 x + \frac{\pi}{4} \right)}} 
                           /     2      \     ___    2/    pi\ /        2/    pi\\
                        log|------------|   \/ 2 *sin |x + --|*|-1 + cot |x + --||
  1   log(1 + tan(x))      \1 + cos(2*x)/             \    8 / \         \    8 //
- - - --------------- + ----------------- - --------------------------------------
  8          4                  8                             8
−2(cot⁡2(x+π8)−1)sin⁡2(x+π8)8+log⁡(2cos⁡(2x)+1)8−log⁡(tan⁡(x)+1)4−18- \frac{\sqrt{2} \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(x + \frac{\pi}{8} \right)}}{8} + \frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{1}{8} 
-1/8 - log(1 + tan(x))/4 + log(2/(1 + cos(2*x)))/8 - sqrt(2)*sin(x + pi/8)^2*(-1 + cot(x + pi/8)^2)/8
    © Sr Examen – Калькулятор