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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(x)/x^2+x^(4/5)*(-1-cot(x)^2)-(1+tan(x)^2)/x+4*cot(x)/(5*x^(1/5))

¿Cómo vas a descomponer esta tan(x)/x^2+x^(4/5)*(-1-cot(x)^2)-(1+tan(x)^2)/x+4*cot(x)/(5*x^(1/5)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                                      2              
tan(x)    4/5 /        2   \   1 + tan (x)   4*cot(x)
------ + x   *\-1 - cot (x)/ - ----------- + --------
   2                                x          5 ___ 
  x                                          5*\/ x
$$\left(\left(x^{\frac{4}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) + \frac{\tan{\left(x \right)}}{x^{2}}\right) - \frac{\tan^{2}{\left(x \right)} + 1}{x}\right) + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
tan(x)/x^2 + x^(4/5)*(-1 - cot(x)^2) - (1 + tan(x)^2)/x + (4*cot(x))/((5*x^(1/5)))
Simplificación general [src] 
                        4/5            
tan(x)       1         x       4*cot(x)
------ - --------- - ------- + --------
   2          2         2        5 ___ 
  x      x*cos (x)   sin (x)   5*\/ x
$$- \frac{x^{\frac{4}{5}}}{\sin^{2}{\left(x \right)}} - \frac{1}{x \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
tan(x)/x^2 - 1/(x*cos(x)^2) - x^(4/5)/sin(x)^2 + 4*cot(x)/(5*x^(1/5))
Respuesta numérica [src] 
tan(x)/x^2 + x^0.8*(-1.0 - cot(x)^2) - (1.0 + tan(x)^2)/x + 0.8*x^(-0.2)*cot(x)
tan(x)/x^2 + x^0.8*(-1.0 - cot(x)^2) - (1.0 + tan(x)^2)/x + 0.8*x^(-0.2)*cot(x)
Denominador común [src] 
 /   4      11/5      6/5             3             4    2         11/5    2   \ 
-\5*x  + 5*x     - 5*x   *tan(x) - 4*x *cot(x) + 5*x *cot (x) + 5*x    *tan (x)/ 
---------------------------------------------------------------------------------
                                        16/5                                     
                                     5*x
$$- \frac{5 x^{\frac{11}{5}} \tan^{2}{\left(x \right)} + 5 x^{\frac{11}{5}} - 5 x^{\frac{6}{5}} \tan{\left(x \right)} + 5 x^{4} \cot^{2}{\left(x \right)} + 5 x^{4} - 4 x^{3} \cot{\left(x \right)}}{5 x^{\frac{16}{5}}}$$ 
-(5*x^4 + 5*x^(11/5) - 5*x^(6/5)*tan(x) - 4*x^3*cot(x) + 5*x^4*cot(x)^2 + 5*x^(11/5)*tan(x)^2)/(5*x^(16/5))
Denominador racional [src] 
   3            5 ___ /  / 14/5 /        2   \         \    2 /        2   \\
4*x *cot(x) + 5*\/ x *\x*\x    *\-1 - cot (x)/ + tan(x)/ + x *\-1 - tan (x)//
-----------------------------------------------------------------------------
                                      16/5                                   
                                   5*x
$$\frac{5 \sqrt[5]{x} \left(x^{2} \left(- \tan^{2}{\left(x \right)} - 1\right) + x \left(x^{\frac{14}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) + \tan{\left(x \right)}\right)\right) + 4 x^{3} \cot{\left(x \right)}}{5 x^{\frac{16}{5}}}$$ 
(4*x^3*cot(x) + 5*x^(1/5)*(x*(x^(14/5)*(-1 - cot(x)^2) + tan(x)) + x^2*(-1 - tan(x)^2)))/(5*x^(16/5))
Potencias [src] 
                                          2                                
                          /   I*x    -I*x\                                 
                          \- e    + e    /                                 
                      1 - -----------------                                
                                         2                                 
                           / I*x    -I*x\                  /   I*x    -I*x\
 4/5 /        2   \        \e    + e    /     4*cot(x)   I*\- e    + e    /
x   *\-1 - cot (x)/ - --------------------- + -------- + ------------------
                                x               5 ___     2 / I*x    -I*x\ 
                                              5*\/ x     x *\e    + e    /
$$x^{\frac{4}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) - \frac{- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1}{x} + \frac{i \left(- e^{i x} + e^{- i x}\right)}{x^{2} \left(e^{i x} + e^{- i x}\right)} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
        2                                             
-1 - tan (x)   tan(x)    4/5 /        2   \   4*cot(x)
------------ + ------ + x   *\-1 - cot (x)/ + --------
     x            2                             5 ___ 
                 x                            5*\/ x
$$x^{\frac{4}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) + \frac{- \tan^{2}{\left(x \right)} - 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
                                      2              
tan(x)    4/5 /        2   \   1 + tan (x)   4*cot(x)
------ + x   *\-1 - cot (x)/ - ----------- + --------
   2                                x          5 ___ 
  x                                          5*\/ x
$$x^{\frac{4}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
tan(x)/x^2 + x^(4/5)*(-1 - cot(x)^2) - (1 + tan(x)^2)/x + 4*cot(x)/(5*x^(1/5))
Unión de expresiones racionales [src] 
   2            5 ___ / 14/5 /        2   \     /       2   \         \
4*x *cot(x) + 5*\/ x *\x    *\-1 - cot (x)/ - x*\1 + tan (x)/ + tan(x)/
-----------------------------------------------------------------------
                                   11/5                                
                                5*x
$$\frac{5 \sqrt[5]{x} \left(x^{\frac{14}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) - x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) + 4 x^{2} \cot{\left(x \right)}}{5 x^{\frac{11}{5}}}$$ 
(4*x^2*cot(x) + 5*x^(1/5)*(x^(14/5)*(-1 - cot(x)^2) - x*(1 + tan(x)^2) + tan(x)))/(5*x^(11/5))
Abrimos la expresión [src] 
                                      2              
 4/5 /        2   \   tan(x)   1 + tan (x)   4*cot(x)
x   *\-1 - cot (x)/ + ------ - ----------- + --------
                         2          x          5 ___ 
                        x                    5*\/ x
$$x^{\frac{4}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) + \frac{\tan{\left(x \right)}}{x^{2}} - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
                         2                             
  1    4/5   tan(x)   tan (x)    4/5    2      4*cot(x)
- - - x    + ------ - ------- - x   *cot (x) + --------
  x             2        x                       5 ___ 
               x                               5*\/ x
$$- x^{\frac{4}{5}} \cot^{2}{\left(x \right)} - x^{\frac{4}{5}} - \frac{\tan^{2}{\left(x \right)}}{x} - \frac{1}{x} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
-1/x - x^(4/5) + tan(x)/x^2 - tan(x)^2/x - x^(4/5)*cot(x)^2 + 4*cot(x)/(5*x^(1/5))
Compilar la expresión [src] 
                                      2              
tan(x)    4/5 /        2   \   1 + tan (x)   4*cot(x)
------ + x   *\-1 - cot (x)/ - ----------- + --------
   2                                x          5 ___ 
  x                                          5*\/ x
$$x^{\frac{4}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
tan(x)/x^2 + x^(4/5)*(-1 - cot(x)^2) - (1 + tan(x)^2)/x + 4*cot(x)/(5*x^(1/5))
Combinatoria [src] 
 /   3      6/5     5 ___             2             3    2         6/5    2   \ 
-\5*x  + 5*x    - 5*\/ x *tan(x) - 4*x *cot(x) + 5*x *cot (x) + 5*x   *tan (x)/ 
--------------------------------------------------------------------------------
                                       11/5                                     
                                    5*x
$$- \frac{5 x^{\frac{6}{5}} \tan^{2}{\left(x \right)} + 5 x^{\frac{6}{5}} - 5 \sqrt[5]{x} \tan{\left(x \right)} + 5 x^{3} \cot^{2}{\left(x \right)} + 5 x^{3} - 4 x^{2} \cot{\left(x \right)}}{5 x^{\frac{11}{5}}}$$ 
-(5*x^3 + 5*x^(6/5) - 5*x^(1/5)*tan(x) - 4*x^2*cot(x) + 5*x^3*cot(x)^2 + 5*x^(6/5)*tan(x)^2)/(5*x^(11/5))
Parte trigonométrica [src] 
tan(x)       1        4/5    2            4       
------ - --------- - x   *csc (x) + --------------
   2          2                       5 ___       
  x      x*cos (x)                  5*\/ x *tan(x)
$$- x^{\frac{4}{5}} \csc^{2}{\left(x \right)} - \frac{1}{x \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4}{5 \sqrt[5]{x} \tan{\left(x \right)}}$$ 
            2         4/5                  
tan(x)   sec (x)     x             4       
------ - ------- - ------- + --------------
   2        x         2        5 ___       
  x                sin (x)   5*\/ x *tan(x)
$$- \frac{x^{\frac{4}{5}}}{\sin^{2}{\left(x \right)}} - \frac{\sec^{2}{\left(x \right)}}{x} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4}{5 \sqrt[5]{x} \tan{\left(x \right)}}$$ 
                                                      /    pi\ 
     2                                           4*sec|x - --| 
  sec (x)    4/5    2/    pi\       sec(x)            \    2 / 
- ------- - x   *sec |x - --| + -------------- + --------------
     x               \    2 /    2    /    pi\     5 ___       
                                x *sec|x - --|   5*\/ x *sec(x)
                                      \    2 /
$$- x^{\frac{4}{5}} \sec^{2}{\left(x - \frac{\pi}{2} \right)} - \frac{\sec^{2}{\left(x \right)}}{x} + \frac{\sec{\left(x \right)}}{x^{2} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{4 \sec{\left(x - \frac{\pi}{2} \right)}}{5 \sqrt[5]{x} \sec{\left(x \right)}}$$ 
                                  2/pi    \                                    
                               csc |-- - x|                                    
                                   \2     /                                    
                           1 + ------------      /pi    \                      
     /          2      \            2         csc|-- - x|                      
 4/5 |       csc (x)   |         csc (x)         \2     /         4*csc(x)     
x   *|-1 - ------------| - ---------------- + ----------- + -------------------
     |        2/pi    \|          x             2             5 ___    /pi    \
     |     csc |-- - x||                       x *csc(x)    5*\/ x *csc|-- - x|
     \         \2     //                                               \2     /
$$x^{\frac{4}{5}} \left(- \frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 1\right) - \frac{1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{x} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{x^{2} \csc{\left(x \right)}} + \frac{4 \csc{\left(x \right)}}{5 \sqrt[5]{x} \csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
                                    2                                        
                                 sec (x)                                     
                           1 + ------------                                  
     /        2/    pi\\          2/    pi\                         /    pi\ 
     |     sec |x - --||       sec |x - --|                    4*sec|x - --| 
 4/5 |         \    2 /|           \    2 /       sec(x)            \    2 / 
x   *|-1 - ------------| - ---------------- + -------------- + --------------
     |          2      |          x            2    /    pi\     5 ___       
     \       sec (x)   /                      x *sec|x - --|   5*\/ x *sec(x)
                                                    \    2 /
$$x^{\frac{4}{5}} \left(-1 - \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) - \frac{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1}{x} + \frac{\sec{\left(x \right)}}{x^{2} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{4 \sec{\left(x - \frac{\pi}{2} \right)}}{5 \sqrt[5]{x} \sec{\left(x \right)}}$$ 
                                  4/5              
tan(x)          2              2*x         4*cot(x)
------ - ---------------- - ------------ + --------
   2     x*(1 + cos(2*x))   1 - cos(2*x)     5 ___ 
  x                                        5*\/ x
$$- \frac{2 x^{\frac{4}{5}}}{1 - \cos{\left(2 x \right)}} - \frac{2}{x \left(\cos{\left(2 x \right)} + 1\right)} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
     2/pi    \                     /pi    \                      
  csc |-- - x|                  csc|-- - x|                      
      \2     /    4/5    2         \2     /         4*csc(x)     
- ------------ - x   *csc (x) + ----------- + -------------------
       x                          2             5 ___    /pi    \
                                 x *csc(x)    5*\/ x *csc|-- - x|
                                                         \2     /
$$- x^{\frac{4}{5}} \csc^{2}{\left(x \right)} - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{x} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{x^{2} \csc{\left(x \right)}} + \frac{4 \csc{\left(x \right)}}{5 \sqrt[5]{x} \csc{\left(- x + \frac{\pi}{2} \right)}}$$ 
                                      2                      2
                         /       2/x\\      4/5 /       2/x\\ 
                         |1 + cot |-||     x   *|1 + cot |-|| 
    1       4*cot(x)     \        \2//          \        \2// 
--------- + -------- - ----------------- - -------------------
 2            5 ___                    2             2/x\     
x *cot(x)   5*\/ x       /        2/x\\         4*cot |-|     
                       x*|-1 + cot |-||               \2/     
                         \         \2//
$$- \frac{x^{\frac{4}{5}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \cot^{2}{\left(\frac{x}{2} \right)}} - \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{x \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}} + \frac{1}{x^{2} \cot{\left(x \right)}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
                        4/5                    
tan(x)       1         x            1          
------ - --------- - ------- + 4*-------*cot(x)
   2          2         2          5 ___       
  x      x*cos (x)   sin (x)     5*\/ x
$$4 \frac{1}{5 \sqrt[5]{x}} \cot{\left(x \right)} - \frac{x^{\frac{4}{5}}}{\sin^{2}{\left(x \right)}} - \frac{1}{x \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x^{2}}$$ 
                             2                                
                          sin (x)                             
                      1 + -------                             
     /        2   \          2                                
 4/5 |     cos (x)|       cos (x)     sin(x)       4*cos(x)   
x   *|-1 - -------| - ----------- + --------- + --------------
     |        2   |        x         2            5 ___       
     \     sin (x)/                 x *cos(x)   5*\/ x *sin(x)
$$x^{\frac{4}{5}} \left(-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1}{x} + \frac{\sin{\left(x \right)}}{x^{2} \cos{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{5 \sqrt[5]{x} \sin{\left(x \right)}}$$ 
                                /    pi\                      
                   4/5       cos|x - --|                      
      1           x             \    2 /         4*cos(x)     
- --------- - ------------ + ----------- + -------------------
       2         2/    pi\     2             5 ___    /    pi\
  x*cos (x)   cos |x - --|    x *cos(x)    5*\/ x *cos|x - --|
                  \    2 /                            \    2 /
$$- \frac{x^{\frac{4}{5}}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{x \cos^{2}{\left(x \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x^{2} \cos{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{5 \sqrt[5]{x} \cos{\left(x - \frac{\pi}{2} \right)}}$$ 
                                  4/5                    
tan(x)          2              2*x               4       
------ - ---------------- - ------------ + --------------
   2     x*(1 + cos(2*x))   1 - cos(2*x)     5 ___       
  x                                        5*\/ x *tan(x)
$$- \frac{2 x^{\frac{4}{5}}}{1 - \cos{\left(2 x \right)}} - \frac{2}{x \left(\cos{\left(2 x \right)} + 1\right)} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4}{5 \sqrt[5]{x} \tan{\left(x \right)}}$$ 
                                  2/    pi\                                    
                               cos |x - --|                                    
                                   \    2 /                                    
                           1 + ------------      /    pi\                      
     /          2      \            2         cos|x - --|                      
 4/5 |       cos (x)   |         cos (x)         \    2 /         4*cos(x)     
x   *|-1 - ------------| - ---------------- + ----------- + -------------------
     |        2/    pi\|          x             2             5 ___    /    pi\
     |     cos |x - --||                       x *cos(x)    5*\/ x *cos|x - --|
     \         \    2 //                                               \    2 /
$$x^{\frac{4}{5}} \left(- \frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 1\right) - \frac{1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}}{x} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{x^{2} \cos{\left(x \right)}} + \frac{4 \cos{\left(x \right)}}{5 \sqrt[5]{x} \cos{\left(x - \frac{\pi}{2} \right)}}$$ 
                      4/5          2                      
        1            x        2*sin (x)       2*sin(2*x)  
- -------------- - ------- + ----------- + ---------------
       2/    pi\      2       2              5 ___    2   
  x*sin |x + --|   sin (x)   x *sin(2*x)   5*\/ x *sin (x)
        \    2 /
$$- \frac{x^{\frac{4}{5}}}{\sin^{2}{\left(x \right)}} - \frac{1}{x \sin^{2}{\left(x + \frac{\pi}{2} \right)}} + \frac{2 \sin^{2}{\left(x \right)}}{x^{2} \sin{\left(2 x \right)}} + \frac{2 \sin{\left(2 x \right)}}{5 \sqrt[5]{x} \sin^{2}{\left(x \right)}}$$ 
                             2                                
                          sec (x)                             
                      1 + -------                             
     /        2   \          2                                
 4/5 |     csc (x)|       csc (x)     sec(x)       4*csc(x)   
x   *|-1 - -------| - ----------- + --------- + --------------
     |        2   |        x         2            5 ___       
     \     sec (x)/                 x *csc(x)   5*\/ x *sec(x)
$$x^{\frac{4}{5}} \left(- \frac{\csc^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)}} - 1\right) - \frac{1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}}{x} + \frac{\sec{\left(x \right)}}{x^{2} \csc{\left(x \right)}} + \frac{4 \csc{\left(x \right)}}{5 \sqrt[5]{x} \sec{\left(x \right)}}$$ 
                        4/5                  
tan(x)       1         x             4       
------ - --------- - ------- + --------------
   2          2         2        5 ___       
  x      x*cos (x)   sin (x)   5*\/ x *tan(x)
$$- \frac{x^{\frac{4}{5}}}{\sin^{2}{\left(x \right)}} - \frac{1}{x \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4}{5 \sqrt[5]{x} \tan{\left(x \right)}}$$ 
                        4/5            
tan(x)       1         x       4*cot(x)
------ - --------- - ------- + --------
   2          2         2        5 ___ 
  x      x*cos (x)   sin (x)   5*\/ x
$$- \frac{x^{\frac{4}{5}}}{\sin^{2}{\left(x \right)}} - \frac{1}{x \cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
                                 4                                   
                            4*sin (x)                                
                        1 + ---------                                
     /        2     \          2              2                      
 4/5 |     sin (2*x)|       sin (2*x)    2*sin (x)       2*sin(2*x)  
x   *|-1 - ---------| - ------------- + ----------- + ---------------
     |          4   |         x          2              5 ___    2   
     \     4*sin (x)/                   x *sin(2*x)   5*\/ x *sin (x)
$$x^{\frac{4}{5}} \left(-1 - \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) - \frac{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1}{x} + \frac{2 \sin^{2}{\left(x \right)}}{x^{2} \sin{\left(2 x \right)}} + \frac{2 \sin{\left(2 x \right)}}{5 \sqrt[5]{x} \sin^{2}{\left(x \right)}}$$ 
                                        2                      2
                           /       2/x\\      4/5 /       2/x\\ 
                           |1 + tan |-||     x   *|1 + tan |-|| 
tan(x)         4           \        \2//          \        \2// 
------ + -------------- - ---------------- - -------------------
   2       5 ___                         2             2/x\     
  x      5*\/ x *tan(x)     /       2/x\\         4*tan |-|     
                          x*|1 - tan |-||               \2/     
                            \        \2//
$$- \frac{x^{\frac{4}{5}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \tan^{2}{\left(\frac{x}{2} \right)}} - \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{x \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4}{5 \sqrt[5]{x} \tan{\left(x \right)}}$$ 
                                         1              
                                  1 + -------           
                                         2              
    1        4/5 /        2   \       cot (x)   4*cot(x)
--------- + x   *\-1 - cot (x)/ - ----------- + --------
 2                                     x          5 ___ 
x *cot(x)                                       5*\/ x
$$x^{\frac{4}{5}} \left(- \cot^{2}{\left(x \right)} - 1\right) - \frac{1 + \frac{1}{\cot^{2}{\left(x \right)}}}{x} + \frac{1}{x^{2} \cot{\left(x \right)}} + \frac{4 \cot{\left(x \right)}}{5 \sqrt[5]{x}}$$ 
                                      2                    
tan(x)    4/5 /        1   \   1 + tan (x)         4       
------ + x   *|-1 - -------| - ----------- + --------------
   2          |        2   |        x          5 ___       
  x           \     tan (x)/                 5*\/ x *tan(x)
$$x^{\frac{4}{5}} \left(-1 - \frac{1}{\tan^{2}{\left(x \right)}}\right) - \frac{\tan^{2}{\left(x \right)} + 1}{x} + \frac{\tan{\left(x \right)}}{x^{2}} + \frac{4}{5 \sqrt[5]{x} \tan{\left(x \right)}}$$ 
tan(x)/x^2 + x^(4/5)*(-1 - 1/tan(x)^2) - (1 + tan(x)^2)/x + 4/(5*x^(1/5)*tan(x))
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