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Simplificación de expresiones/ Descomposición en fracciones simples/ cot(x)^atan(4*x)*(4*log(cot(x))/(1+16*x^2)+(-1-cot(x)^2)*atan(4*x)/cot(x))

¿Cómo vas a descomponer esta cot(x)^atan(4*x)*(4*log(cot(x))/(1+16*x^2)+(-1-cot(x)^2)*atan(4*x)/cot(x)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                /                /        2   \          \
   atan(4*x)    |4*log(cot(x))   \-1 - cot (x)/*atan(4*x)|
cot         (x)*|------------- + ------------------------|
                |          2              cot(x)         |
                \  1 + 16*x                              /
$$\left(\frac{\left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{atan}{\left(4 x \right)}}{\cot{\left(x \right)}} + \frac{4 \log{\left(\cot{\left(x \right)} \right)}}{16 x^{2} + 1}\right) \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)}$$ 
cot(x)^atan(4*x)*((4*log(cot(x)))/(1 + 16*x^2) + ((-1 - cot(x)^2)*atan(4*x))/cot(x))
Simplificación general [src] 
                     /                       /        2\          \
   -1 + atan(4*x)    |                       \1 + 16*x /*atan(4*x)|
cot              (x)*|4*cot(x)*log(cot(x)) - ---------------------|
                     |                                 2          |
                     \                              sin (x)       /
-------------------------------------------------------------------
                                     2                             
                             1 + 16*x
$$\frac{\left(- \frac{\left(16 x^{2} + 1\right) \operatorname{atan}{\left(4 x \right)}}{\sin^{2}{\left(x \right)}} + 4 \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)}\right) \cot^{\operatorname{atan}{\left(4 x \right)} - 1}{\left(x \right)}}{16 x^{2} + 1}$$ 
cot(x)^(-1 + atan(4*x))*(4*cot(x)*log(cot(x)) - (1 + 16*x^2)*atan(4*x)/sin(x)^2)/(1 + 16*x^2)
Respuesta numérica [src] 
cot(x)^atan(4*x)*(4.0*log(cot(x))/(1.0 + 16.0*x^2) + (-1.0 - cot(x)^2)*atan(4*x)/cot(x))
cot(x)^atan(4*x)*(4.0*log(cot(x))/(1.0 + 16.0*x^2) + (-1.0 - cot(x)^2)*atan(4*x)/cot(x))
Denominador común [src] 
     atan(4*x)                     atan(4*x)                             2    atan(4*x)                                                
  cot         (x)*atan(4*x) - 4*cot         (x)*cot(x)*log(cot(x)) + 16*x *cot         (x)*atan(4*x)      atan(4*x)                    
- -------------------------------------------------------------------------------------------------- - cot         (x)*atan(4*x)*cot(x)
                                            2                                                                                          
                                        16*x *cot(x) + cot(x)
$$- \cot{\left(x \right)} \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)} \operatorname{atan}{\left(4 x \right)} - \frac{16 x^{2} \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)} \operatorname{atan}{\left(4 x \right)} - 4 \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)} \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)} + \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)} \operatorname{atan}{\left(4 x \right)}}{16 x^{2} \cot{\left(x \right)} + \cot{\left(x \right)}}$$ 
-(cot(x)^atan(4*x)*atan(4*x) - 4*cot(x)^atan(4*x)*cot(x)*log(cot(x)) + 16*x^2*cot(x)^atan(4*x)*atan(4*x))/(16*x^2*cot(x) + cot(x)) - cot(x)^atan(4*x)*atan(4*x)*cot(x)
Denominador racional [src] 
   atan(4*x)    /                       /        2\ /        2   \          \
cot         (x)*\4*cot(x)*log(cot(x)) + \1 + 16*x /*\-1 - cot (x)/*atan(4*x)/
-----------------------------------------------------------------------------
                              /        2\                                    
                              \1 + 16*x /*cot(x)
$$\frac{\left(\left(16 x^{2} + 1\right) \left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{atan}{\left(4 x \right)} + 4 \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)}\right) \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)}}{\left(16 x^{2} + 1\right) \cot{\left(x \right)}}$$ 
cot(x)^atan(4*x)*(4*cot(x)*log(cot(x)) + (1 + 16*x^2)*(-1 - cot(x)^2)*atan(4*x))/((1 + 16*x^2)*cot(x))
Unión de expresiones racionales [src] 
   atan(4*x)    /                       /        2\ /        2   \          \
cot         (x)*\4*cot(x)*log(cot(x)) + \1 + 16*x /*\-1 - cot (x)/*atan(4*x)/
-----------------------------------------------------------------------------
                              /        2\                                    
                              \1 + 16*x /*cot(x)
$$\frac{\left(\left(16 x^{2} + 1\right) \left(- \cot^{2}{\left(x \right)} - 1\right) \operatorname{atan}{\left(4 x \right)} + 4 \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)}\right) \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)}}{\left(16 x^{2} + 1\right) \cot{\left(x \right)}}$$ 
cot(x)^atan(4*x)*(4*cot(x)*log(cot(x)) + (1 + 16*x^2)*(-1 - cot(x)^2)*atan(4*x))/((1 + 16*x^2)*cot(x))
Combinatoria [src] 
    atan(4*x)    /   2                                           2                 2    2                         \ 
-cot         (x)*\cot (x)*atan(4*x) - 4*cot(x)*log(cot(x)) + 16*x *atan(4*x) + 16*x *cot (x)*atan(4*x) + atan(4*x)/ 
--------------------------------------------------------------------------------------------------------------------
                                                 /        2\                                                        
                                                 \1 + 16*x /*cot(x)
$$- \frac{\left(16 x^{2} \cot^{2}{\left(x \right)} \operatorname{atan}{\left(4 x \right)} + 16 x^{2} \operatorname{atan}{\left(4 x \right)} - 4 \log{\left(\cot{\left(x \right)} \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)} \operatorname{atan}{\left(4 x \right)} + \operatorname{atan}{\left(4 x \right)}\right) \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)}}{\left(16 x^{2} + 1\right) \cot{\left(x \right)}}$$ 
-cot(x)^atan(4*x)*(cot(x)^2*atan(4*x) - 4*cot(x)*log(cot(x)) + 16*x^2*atan(4*x) + 16*x^2*cot(x)^2*atan(4*x) + atan(4*x))/((1 + 16*x^2)*cot(x))
Abrimos la expresión [src] 
     atan(4*x)                                                        atan(4*x)               
  cot         (x)*atan(4*x)      atan(4*x)                       4*cot         (x)*log(cot(x))
- ------------------------- - cot         (x)*atan(4*x)*cot(x) + -----------------------------
            cot(x)                                                                 2          
                                                                           1 + 16*x
$$- \cot{\left(x \right)} \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)} \operatorname{atan}{\left(4 x \right)} - \frac{\cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)} \operatorname{atan}{\left(4 x \right)}}{\cot{\left(x \right)}} + \frac{4 \log{\left(\cot{\left(x \right)} \right)} \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)}}{16 x^{2} + 1}$$ 
-cot(x)^atan(4*x)*atan(4*x)/cot(x) - cot(x)^atan(4*x)*atan(4*x)*cot(x) + 4*cot(x)^atan(4*x)*log(cot(x))/(1 + 16*x^2)
Parte trigonométrica [src] 
                       /                     /        2/    pi\\                 \
                       |     /   /    pi\\   |     sec |x - --||                 |
             atan(4*x) |     |sec|x - --||   |         \    2 /|                 |
/   /    pi\\          |     |   \    2 /|   |-1 - ------------|*atan(4*x)*sec(x)|
|sec|x - --||          |4*log|-----------|   |          2      |                 |
|   \    2 /|          |     \   sec(x)  /   \       sec (x)   /                 |
|-----------|         *|------------------ + ------------------------------------|
\   sec(x)  /          |            2                       /    pi\             |
                       |    1 + 16*x                     sec|x - --|             |
                       \                                    \    2 /             /
$$\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(\frac{\left(-1 - \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(4 x \right)} \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{4 \log{\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                /                /       2   \          \
   atan(4*x)    |4*log(cot(x))   \1 + cot (x)/*atan(4*x)|
cot         (x)*|------------- - -----------------------|
                |          2              cot(x)        |
                \  1 + 16*x                             /
$$\left(- \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(4 x \right)}}{\cot{\left(x \right)}} + \frac{4 \log{\left(\cot{\left(x \right)} \right)}}{16 x^{2} + 1}\right) \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)}$$ 
                  /     /  1   \                                  \
        atan(4*x) |4*log|------|                                  |
/  1   \          |     \tan(x)/   /        1   \                 |
|------|         *|------------- + |-1 - -------|*atan(4*x)*tan(x)|
\tan(x)/          |          2     |        2   |                 |
                  \  1 + 16*x      \     tan (x)/                 /
$$\left(\left(-1 - \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)} \operatorname{atan}{\left(4 x \right)} + \frac{4 \log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{16 x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}}$$ 
                  /                             /  1   \\
        atan(4*x) |                        4*log|------||
/  1   \          |                             \tan(x)/|
|------|         *|-2*atan(4*x)*csc(2*x) + -------------|
\tan(x)/          |                                  2  |
                  \                          1 + 16*x   /
$$\left(- 2 \operatorname{atan}{\left(4 x \right)} \csc{\left(2 x \right)} + \frac{4 \log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{16 x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}}$$ 
                       /                                   /   /    pi\\\
             atan(4*x) |                                   |sec|x - --|||
/   /    pi\\          |                                   |   \    2 /||
|sec|x - --||          |                              4*log|-----------||
|   \    2 /|          |                 /      pi\        \   sec(x)  /|
|-----------|         *|- 2*atan(4*x)*sec|2*x - --| + ------------------|
\   sec(x)  /          |                 \      2 /               2     |
                       \                                  1 + 16*x      /
$$\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(- 2 \operatorname{atan}{\left(4 x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)} + \frac{4 \log{\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                  /     /  1   \                          \
        atan(4*x) |4*log|------|   /       2   \          |
/  1   \          |     \tan(x)/   \1 + tan (x)/*atan(4*x)|
|------|         *|------------- - -----------------------|
\tan(x)/          |          2              tan(x)        |
                  \  1 + 16*x                             /
$$\left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(4 x \right)}}{\tan{\left(x \right)}} + \frac{4 \log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{16 x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}}$$ 
                     /                             /        2     \          \
                     |     / sin(2*x)\        2    |     sin (2*x)|          |
                     |4*log|---------|   2*sin (x)*|-1 - ---------|*atan(4*x)|
           atan(4*x) |     |     2   |             |          4   |          |
/ sin(2*x)\          |     \2*sin (x)/             \     4*sin (x)/          |
|---------|         *|---------------- + ------------------------------------|
|     2   |          |           2                     sin(2*x)              |
\2*sin (x)/          \   1 + 16*x                                            /
$$\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(\frac{2 \left(-1 - \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)} \operatorname{atan}{\left(4 x \right)}}{\sin{\left(2 x \right)}} + \frac{4 \log{\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                       /                       /   cos(x)  \\
                       |                  4*log|-----------||
                       |                       |   /    pi\||
             atan(4*x) |                       |cos|x - --|||
/   cos(x)  \          |   2*atan(4*x)         \   \    2 //|
|-----------|         *|- ------------- + ------------------|
|   /    pi\|          |     /      pi\               2     |
|cos|x - --||          |  cos|2*x - --|       1 + 16*x      |
\   \    2 //          \     \      2 /                     /
$$\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(- \frac{2 \operatorname{atan}{\left(4 x \right)}}{\cos{\left(2 x - \frac{\pi}{2} \right)}} + \frac{4 \log{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                       /                     /          2      \                      \
                       |     /   cos(x)  \   |       cos (x)   |              /    pi\|
                       |4*log|-----------|   |-1 - ------------|*atan(4*x)*cos|x - --||
                       |     |   /    pi\|   |        2/    pi\|              \    2 /|
             atan(4*x) |     |cos|x - --||   |     cos |x - --||                      |
/   cos(x)  \          |     \   \    2 //   \         \    2 //                      |
|-----------|         *|------------------ + -----------------------------------------|
|   /    pi\|          |            2                          cos(x)                 |
|cos|x - --||          \    1 + 16*x                                                  /
\   \    2 //
$$\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(\frac{\left(- \frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(x - \frac{\pi}{2} \right)} \operatorname{atan}{\left(4 x \right)}}{\cos{\left(x \right)}} + \frac{4 \log{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                       /                             /   csc(x)  \\
                       |                        4*log|-----------||
                       |                             |   /pi    \||
             atan(4*x) |                             |csc|-- - x|||
/   csc(x)  \          |                             \   \2     //|
|-----------|         *|-2*atan(4*x)*csc(2*x) + ------------------|
|   /pi    \|          |                                    2     |
|csc|-- - x||          \                            1 + 16*x      /
\   \2     //
$$\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(- 2 \operatorname{atan}{\left(4 x \right)} \csc{\left(2 x \right)} + \frac{4 \log{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                  /                /        2   \                 \
                  |                |     csc (x)|                 |
                  |     /csc(x)\   |-1 - -------|*atan(4*x)*sec(x)|
        atan(4*x) |4*log|------|   |        2   |                 |
/csc(x)\          |     \sec(x)/   \     sec (x)/                 |
|------|         *|------------- + -------------------------------|
\sec(x)/          |          2                  csc(x)            |
                  \  1 + 16*x                                     /
$$\left(\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(\frac{\left(- \frac{\csc^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)}} - 1\right) \operatorname{atan}{\left(4 x \right)} \sec{\left(x \right)}}{\csc{\left(x \right)}} + \frac{4 \log{\left(\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                     /                     / sin(2*x)\\
                     |                4*log|---------||
           atan(4*x) |                     |     2   ||
/ sin(2*x)\          |  2*atan(4*x)        \2*sin (x)/|
|---------|         *|- ----------- + ----------------|
|     2   |          |    sin(2*x)               2    |
\2*sin (x)/          \                   1 + 16*x     /
$$\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(- \frac{2 \operatorname{atan}{\left(4 x \right)}}{\sin{\left(2 x \right)}} + \frac{4 \log{\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                  /                /        2   \                 \
                  |                |     cos (x)|                 |
                  |     /cos(x)\   |-1 - -------|*atan(4*x)*sin(x)|
        atan(4*x) |4*log|------|   |        2   |                 |
/cos(x)\          |     \sin(x)/   \     sin (x)/                 |
|------|         *|------------- + -------------------------------|
\sin(x)/          |          2                  cos(x)            |
                  \  1 + 16*x                                     /
$$\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(\frac{\left(-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)} \operatorname{atan}{\left(4 x \right)}}{\cos{\left(x \right)}} + \frac{4 \log{\left(\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
                       /                     /          2      \                      \
                       |     /   csc(x)  \   |       csc (x)   |              /pi    \|
                       |4*log|-----------|   |-1 - ------------|*atan(4*x)*csc|-- - x||
                       |     |   /pi    \|   |        2/pi    \|              \2     /|
             atan(4*x) |     |csc|-- - x||   |     csc |-- - x||                      |
/   csc(x)  \          |     \   \2     //   \         \2     //                      |
|-----------|         *|------------------ + -----------------------------------------|
|   /pi    \|          |            2                          csc(x)                 |
|csc|-- - x||          \    1 + 16*x                                                  /
\   \2     //
$$\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}} \left(\frac{\left(- \frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 1\right) \operatorname{atan}{\left(4 x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{4 \log{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{16 x^{2} + 1}\right)$$ 
   atan(4*x)    /  2*atan(4*x)   4*log(cot(x))\
cot         (x)*|- ----------- + -------------|
                |    sin(2*x)              2  |
                \                  1 + 16*x   /
$$\left(- \frac{2 \operatorname{atan}{\left(4 x \right)}}{\sin{\left(2 x \right)}} + \frac{4 \log{\left(\cot{\left(x \right)} \right)}}{16 x^{2} + 1}\right) \cot^{\operatorname{atan}{\left(4 x \right)}}{\left(x \right)}$$ 
                  /                     /  1   \\
        atan(4*x) |                4*log|------||
/  1   \          |  2*atan(4*x)        \tan(x)/|
|------|         *|- ----------- + -------------|
\tan(x)/          |    sin(2*x)              2  |
                  \                  1 + 16*x   /
$$\left(- \frac{2 \operatorname{atan}{\left(4 x \right)}}{\sin{\left(2 x \right)}} + \frac{4 \log{\left(\frac{1}{\tan{\left(x \right)}} \right)}}{16 x^{2} + 1}\right) \left(\frac{1}{\tan{\left(x \right)}}\right)^{\operatorname{atan}{\left(4 x \right)}}$$ 
(1/tan(x))^atan(4*x)*(-2*atan(4*x)/sin(2*x) + 4*log(1/tan(x))/(1 + 16*x^2))
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