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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
   atan(x)    /log(cos(x))   atan(x)*sin(x)\
cos       (x)*|----------- - --------------|
              |        2         cos(x)    |
              \   1 + x                    /
(−sin⁡(x)atan⁡(x)cos⁡(x)+log⁡(cos⁡(x))x2+1)cos⁡atan⁡(x)(x)\left(- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} 
cos(x)^atan(x)*(log(cos(x))/(1 + x^2) - atan(x)*sin(x)/cos(x))
Simplificación general [src] 
   -1 + atan(x)    /                     /     2\               \
cos            (x)*\cos(x)*log(cos(x)) - \1 + x /*atan(x)*sin(x)/
-----------------------------------------------------------------
                                   2                             
                              1 + x
(−(x2+1)sin⁡(x)atan⁡(x)+log⁡(cos⁡(x))cos⁡(x))cos⁡atan⁡(x)−1(x)x2+1\frac{\left(- \left(x^{2} + 1\right) \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)} - 1}{\left(x \right)}}{x^{2} + 1} 
cos(x)^(-1 + atan(x))*(cos(x)*log(cos(x)) - (1 + x^2)*atan(x)*sin(x))/(1 + x^2)
Respuesta numérica [src] 
cos(x)^atan(x)*(log(cos(x))/(1.0 + x^2) - atan(x)*sin(x)/cos(x))
cos(x)^atan(x)*(log(cos(x))/(1.0 + x^2) - atan(x)*sin(x)/cos(x))
Unión de expresiones racionales [src] 
   atan(x)    /                     /     2\               \
cos       (x)*\cos(x)*log(cos(x)) - \1 + x /*atan(x)*sin(x)/
------------------------------------------------------------
                      /     2\                              
                      \1 + x /*cos(x)
(−(x2+1)sin⁡(x)atan⁡(x)+log⁡(cos⁡(x))cos⁡(x))cos⁡atan⁡(x)(x)(x2+1)cos⁡(x)\frac{\left(- \left(x^{2} + 1\right) \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cos{\left(x \right)}} 
cos(x)^atan(x)*(cos(x)*log(cos(x)) - (1 + x^2)*atan(x)*sin(x))/((1 + x^2)*cos(x))
Denominador común [src] 
 /   atan(x)                        atan(x)                          2    atan(x)                  \ 
-\cos       (x)*atan(x)*sin(x) - cos       (x)*cos(x)*log(cos(x)) + x *cos       (x)*atan(x)*sin(x)/ 
-----------------------------------------------------------------------------------------------------
                                           2                                                         
                                          x *cos(x) + cos(x)
−x2sin⁡(x)cos⁡atan⁡(x)(x)atan⁡(x)−log⁡(cos⁡(x))cos⁡(x)cos⁡atan⁡(x)(x)+sin⁡(x)cos⁡atan⁡(x)(x)atan⁡(x)x2cos⁡(x)+cos⁡(x)- \frac{x^{2} \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)} - \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} + \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x^{2} \cos{\left(x \right)} + \cos{\left(x \right)}} 
-(cos(x)^atan(x)*atan(x)*sin(x) - cos(x)^atan(x)*cos(x)*log(cos(x)) + x^2*cos(x)^atan(x)*atan(x)*sin(x))/(x^2*cos(x) + cos(x))
Denominador racional [src] 
   atan(x)    /                     /     2\               \
cos       (x)*\cos(x)*log(cos(x)) - \1 + x /*atan(x)*sin(x)/
------------------------------------------------------------
                      /     2\                              
                      \1 + x /*cos(x)
(−(x2+1)sin⁡(x)atan⁡(x)+log⁡(cos⁡(x))cos⁡(x))cos⁡atan⁡(x)(x)(x2+1)cos⁡(x)\frac{\left(- \left(x^{2} + 1\right) \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cos{\left(x \right)}} 
cos(x)^atan(x)*(cos(x)*log(cos(x)) - (1 + x^2)*atan(x)*sin(x))/((1 + x^2)*cos(x))
Potencias [src] 
                      /   / I*x    -I*x\                             \
              atan(x) |   |e      e    |                             |
/ I*x    -I*x\        |log|---- + -----|     /   -I*x    I*x\        |
|e      e    |        |   \ 2       2  /   I*\- e     + e   /*atan(x)|
|---- + -----|       *|----------------- + --------------------------|
\ 2       2  /        |           2               / I*x    -I*x\     |
                      |      1 + x                |e      e    |     |
                      |                         2*|---- + -----|     |
                      \                           \ 2       2  /     /
(i(eix−e−ix)atan⁡(x)2(eix2+e−ix2)+log⁡(eix2+e−ix2)x2+1)(eix2+e−ix2)atan⁡(x)\left(\frac{i \left(e^{i x} - e^{- i x}\right) \operatorname{atan}{\left(x \right)}}{2 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)} + \frac{\log{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}}{x^{2} + 1}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\operatorname{atan}{\left(x \right)}} 
(exp(i*x)/2 + exp(-i*x)/2)^atan(x)*(log(exp(i*x)/2 + exp(-i*x)/2)/(1 + x^2) + i*(-exp(-i*x) + exp(i*x))*atan(x)/(2*(exp(i*x)/2 + exp(-i*x)/2)))
Combinatoria [src] 
    atan(x)    /                                       2               \ 
-cos       (x)*\atan(x)*sin(x) - cos(x)*log(cos(x)) + x *atan(x)*sin(x)/ 
-------------------------------------------------------------------------
                             /     2\                                    
                             \1 + x /*cos(x)
−(x2sin⁡(x)atan⁡(x)−log⁡(cos⁡(x))cos⁡(x)+sin⁡(x)atan⁡(x))cos⁡atan⁡(x)(x)(x2+1)cos⁡(x)- \frac{\left(x^{2} \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} - \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} + 1\right) \cos{\left(x \right)}} 
-cos(x)^atan(x)*(atan(x)*sin(x) - cos(x)*log(cos(x)) + x^2*atan(x)*sin(x))/((1 + x^2)*cos(x))
Abrimos la expresión [src] 
   atan(x)                     atan(x)                  
cos       (x)*log(cos(x))   cos       (x)*atan(x)*sin(x)
------------------------- - ----------------------------
               2                       cos(x)           
          1 + x
−sin⁡(x)cos⁡atan⁡(x)(x)atan⁡(x)cos⁡(x)+log⁡(cos⁡(x))cos⁡atan⁡(x)(x)x2+1- \frac{\sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)} \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)}}{x^{2} + 1} 
cos(x)^atan(x)*log(cos(x))/(1 + x^2) - cos(x)^atan(x)*atan(x)*sin(x)/cos(x)
Parte trigonométrica [src] 
                   /   /   /    pi\\                 \
                   |log|sin|x + --||                 |
   atan(x)/    pi\ |   \   \    2 //   atan(x)*sin(x)|
sin       |x + --|*|---------------- - --------------|
          \    2 / |          2            /    pi\  |
                   |     1 + x          sin|x + --|  |
                   \                       \    2 /  /
(−sin⁡(x)atan⁡(x)sin⁡(x+π2)+log⁡(sin⁡(x+π2))x2+1)sin⁡atan⁡(x)(x+π2)\left(- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x^{2} + 1}\right) \sin^{\operatorname{atan}{\left(x \right)}}{\left(x + \frac{\pi}{2} \right)} 
                      /   /        2/x\\                   \
                      |   |-1 + cot |-||                   |
                      |   |         \2/|                   |
              atan(x) |log|------------|                   |
/        2/x\\        |   |       2/x\ |                /x\|
|-1 + cot |-||        |   |1 + cot |-| |   2*atan(x)*cot|-||
|         \2/|        |   \        \2/ /                \2/|
|------------|       *|----------------- - ----------------|
|       2/x\ |        |           2                  2/x\  |
|1 + cot |-| |        |      1 + x           -1 + cot |-|  |
\        \2/ /        \                               \2/  /
(cot⁡2(x2)−1cot⁡2(x2)+1)atan⁡(x)(−2cot⁡(x2)atan⁡(x)cot⁡2(x2)−1+log⁡(cot⁡2(x2)−1cot⁡2(x2)+1)x2+1)\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \frac{2 \cot{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(x \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1}\right) 
                /   /  1   \                 \
        atan(x) |log|------|                 |
/  1   \        |   \sec(x)/   atan(x)*sec(x)|
|------|       *|----------- - --------------|
\sec(x)/        |        2         /    pi\  |
                |   1 + x       sec|x - --|  |
                \                  \    2 /  /
(−atan⁡(x)sec⁡(x)sec⁡(x−π2)+log⁡(1sec⁡(x))x2+1)(1sec⁡(x))atan⁡(x)\left(- \frac{\operatorname{atan}{\left(x \right)} \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} 
                /   /  1   \                 \
        atan(x) |log|------|                 |
/  1   \        |   \sec(x)/   atan(x)*sec(x)|
|------|       *|----------- - --------------|
\sec(x)/        |        2         csc(x)    |
                \   1 + x                    /
(−atan⁡(x)sec⁡(x)csc⁡(x)+log⁡(1sec⁡(x))x2+1)(1sec⁡(x))atan⁡(x)\left(- \frac{\operatorname{atan}{\left(x \right)} \sec{\left(x \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x \right)}} 
                     /   /       2/x\\                   \
                     |   |1 - tan |-||                   |
                     |   |        \2/|                   |
             atan(x) |log|-----------|                   |
/       2/x\\        |   |       2/x\|                /x\|
|1 - tan |-||        |   |1 + tan |-||   2*atan(x)*tan|-||
|        \2/|        |   \        \2//                \2/|
|-----------|       *|---------------- - ----------------|
|       2/x\|        |          2                 2/x\   |
|1 + tan |-||        |     1 + x           1 - tan |-|   |
\        \2//        \                             \2/   /
(1−tan⁡2(x2)tan⁡2(x2)+1)atan⁡(x)(log⁡(1−tan⁡2(x2)tan⁡2(x2)+1)x2+1−2tan⁡(x2)atan⁡(x)1−tan⁡2(x2))\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(\frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1} - \frac{2 \tan{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(x \right)}}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\right) 
   atan(x)    /log(cos(x))                 \
cos       (x)*|----------- - atan(x)*tan(x)|
              |        2                   |
              \   1 + x                    /
(−tan⁡(x)atan⁡(x)+log⁡(cos⁡(x))x2+1)cos⁡atan⁡(x)(x)\left(- \tan{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} 
                   /   /   /    pi\\                    \
                   |log|sin|x + --||        2           |
   atan(x)/    pi\ |   \   \    2 //   2*sin (x)*atan(x)|
sin       |x + --|*|---------------- - -----------------|
          \    2 / |          2             sin(2*x)    |
                   \     1 + x                          /
(−2sin⁡2(x)atan⁡(x)sin⁡(2x)+log⁡(sin⁡(x+π2))x2+1)sin⁡atan⁡(x)(x+π2)\left(- \frac{2 \sin^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{x^{2} + 1}\right) \sin^{\operatorname{atan}{\left(x \right)}}{\left(x + \frac{\pi}{2} \right)} 
              /                         /    pi\\
              |              atan(x)*cos|x - --||
   atan(x)    |log(cos(x))              \    2 /|
cos       (x)*|----------- - -------------------|
              |        2            cos(x)      |
              \   1 + x                         /
(−cos⁡(x−π2)atan⁡(x)cos⁡(x)+log⁡(cos⁡(x))x2+1)cos⁡atan⁡(x)(x)\left(- \frac{\cos{\left(x - \frac{\pi}{2} \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x \right)}}{\left(x \right)} 
                      /   /        2/x\\          \
                      |   |-1 + cot |-||          |
                      |   |         \2/|          |
              atan(x) |log|------------|          |
/        2/x\\        |   |       2/x\ |          |
|-1 + cot |-||        |   |1 + cot |-| |          |
|         \2/|        |   \        \2/ /   atan(x)|
|------------|       *|----------------- - -------|
|       2/x\ |        |           2         cot(x)|
|1 + cot |-| |        \      1 + x                /
\        \2/ /
(cot⁡2(x2)−1cot⁡2(x2)+1)atan⁡(x)(−atan⁡(x)cot⁡(x)+log⁡(cot⁡2(x2)−1cot⁡2(x2)+1)x2+1)\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \frac{\operatorname{atan}{\left(x \right)}}{\cot{\left(x \right)}} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1}\right) 
                     /   /       2/x\\                 \
                     |   |1 - tan |-||                 |
                     |   |        \2/|                 |
             atan(x) |log|-----------|                 |
/       2/x\\        |   |       2/x\|                 |
|1 - tan |-||        |   |1 + tan |-||                 |
|        \2/|        |   \        \2//                 |
|-----------|       *|---------------- - atan(x)*tan(x)|
|       2/x\|        |          2                      |
|1 + tan |-||        \     1 + x                       /
\        \2//
(1−tan⁡2(x2)tan⁡2(x2)+1)atan⁡(x)(−tan⁡(x)atan⁡(x)+log⁡(1−tan⁡2(x2)tan⁡2(x2)+1)x2+1)\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x \right)}} \left(- \tan{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{2} + 1}\right) 
                     /   /     1     \                      \
                     |log|-----------|                      |
                     |   |   /pi    \|              /pi    \|
             atan(x) |   |csc|-- - x||   atan(x)*csc|-- - x||
/     1     \        |   \   \2     //              \2     /|
|-----------|       *|---------------- - -------------------|
|   /pi    \|        |          2               csc(x)      |
|csc|-- - x||        \     1 + x                            /
\   \2     //
(−atan⁡(x)csc⁡(−x+π2)csc⁡(x)+log⁡(1csc⁡(−x+π2))x2+1)(1csc⁡(−x+π2))atan⁡(x)\left(- \frac{\operatorname{atan}{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{x^{2} + 1}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x \right)}} 
(1/csc(pi/2 - x))^atan(x)*(log(1/csc(pi/2 - x))/(1 + x^2) - atan(x)*csc(pi/2 - x)/csc(x))
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