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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
           /                       2   \
x ________ |-log(tan(x))    1 + tan (x)|
\/ tan(x) *|------------- + -----------|
           |       2          x*tan(x) |
           \      x                    /
(tan⁡2(x)+1xtan⁡(x)+(−1)log⁡(tan⁡(x))x2)tan⁡1x(x)\left(\frac{\tan^{2}{\left(x \right)} + 1}{x \tan{\left(x \right)}} + \frac{\left(-1\right) \log{\left(\tan{\left(x \right)} \right)}}{x^{2}}\right) \tan^{\frac{1}{x}}{\left(x \right)} 
tan(x)^(1/x)*((-log(tan(x)))/x^2 + (1 + tan(x)^2)/((x*tan(x))))
Simplificación general [src] 
             1                               
        -1 + -                               
             x /   x                        \
(tan(x))      *|------- - log(tan(x))*tan(x)|
               |   2                        |
               \cos (x)                     /
---------------------------------------------
                       2                     
                      x
(xcos⁡2(x)−log⁡(tan⁡(x))tan⁡(x))tan⁡−1+1x(x)x2\frac{\left(\frac{x}{\cos^{2}{\left(x \right)}} - \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \tan^{-1 + \frac{1}{x}}{\left(x \right)}}{x^{2}} 
tan(x)^(-1 + 1/x)*(x/cos(x)^2 - log(tan(x))*tan(x))/x^2
Denominador racional [src] 
x ________ / 2 /       2   \                       \
\/ tan(x) *\x *\1 + tan (x)/ - x*log(tan(x))*tan(x)/
----------------------------------------------------
                      3                             
                     x *tan(x)
(x2(tan⁡2(x)+1)−xlog⁡(tan⁡(x))tan⁡(x))tan⁡1x(x)x3tan⁡(x)\frac{\left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) - x \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \tan^{\frac{1}{x}}{\left(x \right)}}{x^{3} \tan{\left(x \right)}} 
tan(x)^(1/x)*(x^2*(1 + tan(x)^2) - x*log(tan(x))*tan(x))/(x^3*tan(x))
Combinatoria [src] 
x ________ /         2                        \
\/ tan(x) *\x + x*tan (x) - log(tan(x))*tan(x)/
-----------------------------------------------
                    2                          
                   x *tan(x)
(xtan⁡2(x)+x−log⁡(tan⁡(x))tan⁡(x))tan⁡1x(x)x2tan⁡(x)\frac{\left(x \tan^{2}{\left(x \right)} + x - \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \tan^{\frac{1}{x}}{\left(x \right)}}{x^{2} \tan{\left(x \right)}} 
tan(x)^(1/x)*(x + x*tan(x)^2 - log(tan(x))*tan(x))/(x^2*tan(x))
Denominador común [src] 
  x ________        2    x ________   x ________                   
x*\/ tan(x)  + x*tan (x)*\/ tan(x)  - \/ tan(x) *log(tan(x))*tan(x)
-------------------------------------------------------------------
                              2                                    
                             x *tan(x)
xtan⁡2(x)tan⁡1x(x)+xtan⁡1x(x)−log⁡(tan⁡(x))tan⁡(x)tan⁡1x(x)x2tan⁡(x)\frac{x \tan^{2}{\left(x \right)} \tan^{\frac{1}{x}}{\left(x \right)} + x \tan^{\frac{1}{x}}{\left(x \right)} - \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)} \tan^{\frac{1}{x}}{\left(x \right)}}{x^{2} \tan{\left(x \right)}} 
(x*tan(x)^(1/x) + x*tan(x)^2*tan(x)^(1/x) - tan(x)^(1/x)*log(tan(x))*tan(x))/(x^2*tan(x))
Unión de expresiones racionales [src] 
x ________ /  /       2   \                     \
\/ tan(x) *\x*\1 + tan (x)/ - log(tan(x))*tan(x)/
-------------------------------------------------
                     2                           
                    x *tan(x)
(x(tan⁡2(x)+1)−log⁡(tan⁡(x))tan⁡(x))tan⁡1x(x)x2tan⁡(x)\frac{\left(x \left(\tan^{2}{\left(x \right)} + 1\right) - \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}\right) \tan^{\frac{1}{x}}{\left(x \right)}}{x^{2} \tan{\left(x \right)}} 
tan(x)^(1/x)*(x*(1 + tan(x)^2) - log(tan(x))*tan(x))/(x^2*tan(x))
Abrimos la expresión [src] 
           /                       2   \
x ________ |-log(tan(x))    1 + tan (x)|
\/ tan(x) *|------------- + -----------|
           |       2          x*tan(x) |
           \      x                    /
((−1)log⁡(tan⁡(x))x2+tan⁡2(x)+1xtan⁡(x))tan⁡1x(x)\left(\frac{\left(-1\right) \log{\left(\tan{\left(x \right)} \right)}}{x^{2}} + \frac{\tan^{2}{\left(x \right)} + 1}{x \tan{\left(x \right)}}\right) \tan^{\frac{1}{x}}{\left(x \right)} 
x ________   x ________          x ________            
\/ tan(x)    \/ tan(x) *tan(x)   \/ tan(x) *log(tan(x))
---------- + ----------------- - ----------------------
 x*tan(x)            x                      2          
                                           x
tan⁡(x)tan⁡1x(x)x+tan⁡1x(x)xtan⁡(x)−log⁡(tan⁡(x))tan⁡1x(x)x2\frac{\tan{\left(x \right)} \tan^{\frac{1}{x}}{\left(x \right)}}{x} + \frac{\tan^{\frac{1}{x}}{\left(x \right)}}{x \tan{\left(x \right)}} - \frac{\log{\left(\tan{\left(x \right)} \right)} \tan^{\frac{1}{x}}{\left(x \right)}}{x^{2}} 
tan(x)^(1/x)/(x*tan(x)) + tan(x)^(1/x)*tan(x)/x - tan(x)^(1/x)*log(tan(x))/x^2
Respuesta numérica [src] 
tan(x)^(1/x)*(-log(tan(x))/x^2 + (1.0 + tan(x)^2)/(x*tan(x)))
tan(x)^(1/x)*(-log(tan(x))/x^2 + (1.0 + tan(x)^2)/(x*tan(x)))
Parte trigonométrica [src] 
             /                /       2   \       \
             |                |    sin (x)|       |
             |     /sin(x)\   |1 + -------|*cos(x)|
    ________ |  log|------|   |       2   |       |
   / sin(x)  |     \cos(x)/   \    cos (x)/       |
x /  ------ *|- ----------- + --------------------|
\/   cos(x)  |        2             x*sin(x)      |
             \       x                            /
(sin⁡(x)cos⁡(x))1x((sin⁡2(x)cos⁡2(x)+1)cos⁡(x)xsin⁡(x)−log⁡(sin⁡(x)cos⁡(x))x2)\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}\right)^{\frac{1}{x}} \left(\frac{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(x \right)}}{x \sin{\left(x \right)}} - \frac{\log{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} \right)}}{x^{2}}\right) 
           /       2                 \
x ________ |1 + tan (x)   log(tan(x))|
\/ tan(x) *|----------- - -----------|
           \   tan(x)          x     /
--------------------------------------
                  x
(tan⁡2(x)+1tan⁡(x)−log⁡(tan⁡(x))x)tan⁡1x(x)x\frac{\left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)}} - \frac{\log{\left(\tan{\left(x \right)} \right)}}{x}\right) \tan^{\frac{1}{x}}{\left(x \right)}}{x} 
                    /                     /         2      \            \
                    |     /   sec(x)  \   |      sec (x)   |    /    pi\|
                    |  log|-----------|   |1 + ------------|*sec|x - --||
                    |     |   /    pi\|   |       2/    pi\|    \    2 /|
      _____________ |     |sec|x - --||   |    sec |x - --||            |
     /    sec(x)    |     \   \    2 //   \        \    2 //            |
    /  ----------- *|- ---------------- + ------------------------------|
   /      /    pi\  |          2                     x*sec(x)           |
x /    sec|x - --|  \         x                                         /
\/        \    2 /
(sec⁡(x)sec⁡(x−π2))1x((sec⁡2(x)sec⁡2(x−π2)+1)sec⁡(x−π2)xsec⁡(x)−log⁡(sec⁡(x)sec⁡(x−π2))x2)\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{x}} \left(\frac{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}{x \sec{\left(x \right)}} - \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{x^{2}}\right) 
             /                /       2   \       \
             |                |    sec (x)|       |
             |     /sec(x)\   |1 + -------|*csc(x)|
    ________ |  log|------|   |       2   |       |
   / sec(x)  |     \csc(x)/   \    csc (x)/       |
x /  ------ *|- ----------- + --------------------|
\/   csc(x)  |        2             x*sec(x)      |
             \       x                            /
(sec⁡(x)csc⁡(x))1x((1+sec⁡2(x)csc⁡2(x))csc⁡(x)xsec⁡(x)−log⁡(sec⁡(x)csc⁡(x))x2)\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right)^{\frac{1}{x}} \left(\frac{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}{x \sec{\left(x \right)}} - \frac{\log{\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}}{x^{2}}\right) 
                 /              /     2   \\
     ___________ |              |2*sin (x)||
    /      2     |           log|---------||
   /  2*sin (x)  |   2          \ sin(2*x)/|
x /   --------- *|-------- - --------------|
\/     sin(2*x)  \sin(2*x)         x       /
--------------------------------------------
                     x
(2sin⁡2(x)sin⁡(2x))1x(2sin⁡(2x)−log⁡(2sin⁡2(x)sin⁡(2x))x)x\frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}}\right)^{\frac{1}{x}} \left(\frac{2}{\sin{\left(2 x \right)}} - \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x}\right)}{x} 
                    /                     /   sec(x)  \\
                    |                  log|-----------||
                    |                     |   /    pi\||
      _____________ |                     |sec|x - --|||
     /    sec(x)    |     /      pi\      \   \    2 //|
    /  ----------- *|2*sec|2*x - --| - ----------------|
   /      /    pi\  \     \      2 /          x        /
x /    sec|x - --|                                      
\/        \    2 /                                      
--------------------------------------------------------
                           x
(sec⁡(x)sec⁡(x−π2))1x(2sec⁡(2x−π2)−log⁡(sec⁡(x)sec⁡(x−π2))x)x\frac{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{x}} \left(2 \sec{\left(2 x - \frac{\pi}{2} \right)} - \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{x}\right)}{x} 
                 /                   /         4   \         \
                 |     /     2   \   |    4*sin (x)|         |
     ___________ |     |2*sin (x)|   |1 + ---------|*sin(2*x)|
    /      2     |  log|---------|   |       2     |         |
   /  2*sin (x)  |     \ sin(2*x)/   \    sin (2*x)/         |
x /   --------- *|- -------------- + ------------------------|
\/     sin(2*x)  |         2                      2          |
                 \        x                2*x*sin (x)       /
(2sin⁡2(x)sin⁡(2x))1x((4sin⁡4(x)sin⁡2(2x)+1)sin⁡(2x)2xsin⁡2(x)−log⁡(2sin⁡2(x)sin⁡(2x))x2)\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}}\right)^{\frac{1}{x}} \left(\frac{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}{2 x \sin^{2}{\left(x \right)}} - \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2}}\right) 
                    /                     /       2/    pi\\       \
                    |     /   /    pi\\   |    cos |x - --||       |
      _____________ |     |cos|x - --||   |        \    2 /|       |
     /    /    pi\  |     |   \    2 /|   |1 + ------------|*cos(x)|
    /  cos|x - --|  |  log|-----------|   |         2      |       |
   /      \    2 /  |     \   cos(x)  /   \      cos (x)   /       |
x /    ----------- *|- ---------------- + -------------------------|
\/        cos(x)    |          2                     /    pi\      |
                    |         x                 x*cos|x - --|      |
                    \                                \    2 /      /
(cos⁡(x−π2)cos⁡(x))1x((1+cos⁡2(x−π2)cos⁡2(x))cos⁡(x)xcos⁡(x−π2)−log⁡(cos⁡(x−π2)cos⁡(x))x2)\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{\frac{1}{x}} \left(\frac{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{x \cos{\left(x - \frac{\pi}{2} \right)}} - \frac{\log{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2}}\right) 
           /                       2   \
x ________ |  log(tan(x))   1 + tan (x)|
\/ tan(x) *|- ----------- + -----------|
           |        2         x*tan(x) |
           \       x                   /
(tan⁡2(x)+1xtan⁡(x)−log⁡(tan⁡(x))x2)tan⁡1x(x)\left(\frac{\tan^{2}{\left(x \right)} + 1}{x \tan{\left(x \right)}} - \frac{\log{\left(\tan{\left(x \right)} \right)}}{x^{2}}\right) \tan^{\frac{1}{x}}{\left(x \right)} 
             /                 /  1   \\
    ________ |       2      log|------||
   /   1     |1 + cot (x)      \cot(x)/|
x /  ------ *|----------- - -----------|
\/   cot(x)  \   cot(x)          x     /
----------------------------------------
                   x
(cot⁡2(x)+1cot⁡(x)−log⁡(1cot⁡(x))x)(1cot⁡(x))1xx\frac{\left(\frac{\cot^{2}{\left(x \right)} + 1}{\cot{\left(x \right)}} - \frac{\log{\left(\frac{1}{\cot{\left(x \right)}} \right)}}{x}\right) \left(\frac{1}{\cot{\left(x \right)}}\right)^{\frac{1}{x}}}{x} 
x ________ /   2       log(tan(x))\
\/ tan(x) *|-------- - -----------|
           \sin(2*x)        x     /
-----------------------------------
                 x
(2sin⁡(2x)−log⁡(tan⁡(x))x)tan⁡1x(x)x\frac{\left(\frac{2}{\sin{\left(2 x \right)}} - \frac{\log{\left(\tan{\left(x \right)} \right)}}{x}\right) \tan^{\frac{1}{x}}{\left(x \right)}}{x} 
                    /                /   /pi    \\\
      _____________ |                |csc|-- - x|||
     /    /pi    \  |                |   \2     /||
    /  csc|-- - x|  |             log|-----------||
   /      \2     /  |                \   csc(x)  /|
x /    ----------- *|2*csc(2*x) - ----------------|
\/        csc(x)    \                    x        /
---------------------------------------------------
                         x
(csc⁡(−x+π2)csc⁡(x))1x(2csc⁡(2x)−log⁡(csc⁡(−x+π2)csc⁡(x))x)x\frac{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{\frac{1}{x}} \left(2 \csc{\left(2 x \right)} - \frac{\log{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{x}\right)}{x} 
x ________ /             log(tan(x))\
\/ tan(x) *|2*csc(2*x) - -----------|
           \                  x     /
-------------------------------------
                  x
(2csc⁡(2x)−log⁡(tan⁡(x))x)tan⁡1x(x)x\frac{\left(2 \csc{\left(2 x \right)} - \frac{\log{\left(\tan{\left(x \right)} \right)}}{x}\right) \tan^{\frac{1}{x}}{\left(x \right)}}{x} 
                    /                     /       2/pi    \\       \
                    |     /   /pi    \\   |    csc |-- - x||       |
      _____________ |     |csc|-- - x||   |        \2     /|       |
     /    /pi    \  |     |   \2     /|   |1 + ------------|*csc(x)|
    /  csc|-- - x|  |  log|-----------|   |         2      |       |
   /      \2     /  |     \   csc(x)  /   \      csc (x)   /       |
x /    ----------- *|- ---------------- + -------------------------|
\/        csc(x)    |          2                     /pi    \      |
                    |         x                 x*csc|-- - x|      |
                    \                                \2     /      /
(csc⁡(−x+π2)csc⁡(x))1x((1+csc⁡2(−x+π2)csc⁡2(x))csc⁡(x)xcsc⁡(−x+π2)−log⁡(csc⁡(−x+π2)csc⁡(x))x2)\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{\frac{1}{x}} \left(\frac{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}{x \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{\log{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{x^{2}}\right) 
                    /                   /   /    pi\\\
      _____________ |                   |cos|x - --|||
     /    /    pi\  |                   |   \    2 /||
    /  cos|x - --|  |                log|-----------||
   /      \    2 /  |      2            \   cos(x)  /|
x /    ----------- *|------------- - ----------------|
\/        cos(x)    |   /      pi\          x        |
                    |cos|2*x - --|                   |
                    \   \      2 /                   /
------------------------------------------------------
                          x
(cos⁡(x−π2)cos⁡(x))1x(2cos⁡(2x−π2)−log⁡(cos⁡(x−π2)cos⁡(x))x)x\frac{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{\frac{1}{x}} \left(\frac{2}{\cos{\left(2 x - \frac{\pi}{2} \right)}} - \frac{\log{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x}\right)}{x} 
             /                /       1   \       \
             |     /  1   \   |1 + -------|*cot(x)|
    ________ |  log|------|   |       2   |       |
   /   1     |     \cot(x)/   \    cot (x)/       |
x /  ------ *|- ----------- + --------------------|
\/   cot(x)  |        2                x          |
             \       x                            /
((1+1cot⁡2(x))cot⁡(x)x−log⁡(1cot⁡(x))x2)(1cot⁡(x))1x\left(\frac{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}}{x} - \frac{\log{\left(\frac{1}{\cot{\left(x \right)}} \right)}}{x^{2}}\right) \left(\frac{1}{\cot{\left(x \right)}}\right)^{\frac{1}{x}} 
(1/cot(x))^(1/x)*(-log(1/cot(x))/x^2 + (1 + cot(x)^(-2))*cot(x)/x)
Potencias [src] 
                           /                              /                    2\               \
                           |                              |    /   I*x    -I*x\ |               |
                           |     /  /   I*x    -I*x\\     |    \- e    + e    / | / I*x    -I*x\|
                           |     |I*\- e    + e    /|   I*|1 - -----------------|*\e    + e    /|
      ____________________ |  log|------------------|     |                   2 |               |
     /   /   I*x    -I*x\  |     |    I*x    -I*x   |     |     / I*x    -I*x\  |               |
    /  I*\- e    + e    /  |     \   e    + e       /     \     \e    + e    /  /               |
   /   ------------------ *|- ----------------------- - ----------------------------------------|
x /        I*x    -I*x     |              2                          /   I*x    -I*x\           |
\/        e    + e         \             x                         x*\- e    + e    /           /
(i(−eix+e−ix)eix+e−ix)1x(−i(−(−eix+e−ix)2(eix+e−ix)2+1)(eix+e−ix)x(−eix+e−ix)−log⁡(i(−eix+e−ix)eix+e−ix)x2)\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}}\right)^{\frac{1}{x}} \left(- \frac{i \left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(e^{i x} + e^{- i x}\right)}{x \left(- e^{i x} + e^{- i x}\right)} - \frac{\log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} \right)}}{x^{2}}\right) 
           /                       2   \
x ________ |  log(tan(x))   1 + tan (x)|
\/ tan(x) *|- ----------- + -----------|
           |        2         x*tan(x) |
           \       x                   /
(tan⁡2(x)+1xtan⁡(x)−log⁡(tan⁡(x))x2)tan⁡1x(x)\left(\frac{\tan^{2}{\left(x \right)} + 1}{x \tan{\left(x \right)}} - \frac{\log{\left(\tan{\left(x \right)} \right)}}{x^{2}}\right) \tan^{\frac{1}{x}}{\left(x \right)} 
tan(x)^(1/x)*(-log(tan(x))/x^2 + (1 + tan(x)^2)/(x*tan(x)))
Compilar la expresión [src] 
           /                       2   \
x ________ |  log(tan(x))   1 + tan (x)|
\/ tan(x) *|- ----------- + -----------|
           |        2         x*tan(x) |
           \       x                   /
(tan⁡2(x)+1xtan⁡(x)−log⁡(tan⁡(x))x2)tan⁡1x(x)\left(\frac{\tan^{2}{\left(x \right)} + 1}{x \tan{\left(x \right)}} - \frac{\log{\left(\tan{\left(x \right)} \right)}}{x^{2}}\right) \tan^{\frac{1}{x}}{\left(x \right)} 
tan(x)^(1/x)*(-log(tan(x))/x^2 + (1 + tan(x)^2)/(x*tan(x)))
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