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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(((4*sin(2*x))/(1+cos(2*x))+1)/sqrt(31))/sqrt(31)

¿Cómo vas a descomponer esta atan(((4*sin(2*x))/(1+cos(2*x))+1)/sqrt(31))/sqrt(31) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    / 4*sin(2*x)     \
    |------------ + 1|
    |1 + cos(2*x)    |
atan|----------------|
    |       ____     |
    \     \/ 31      /
----------------------
          ____        
        \/ 31
$$\frac{\operatorname{atan}{\left(\frac{1 + \frac{4 \sin{\left(2 x \right)}}{\cos{\left(2 x \right)} + 1}}{\sqrt{31}} \right)}}{\sqrt{31}}$$ 
atan(((4*sin(2*x))/(1 + cos(2*x)) + 1)/sqrt(31))/sqrt(31)
Descomposición de una fracción [src] 
sqrt(31)*atan(sqrt(31)/31 + 4*(sqrt(31)/31)*sin(2*x)/(1 + cos(2*x)))/31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31}}{31} + \frac{4 \frac{\sqrt{31}}{31} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)} + 1} \right)}}{31}$$ 
           /             ____         \
           |           \/ 31          |
           |  ____   4*------*sin(2*x)|
  ____     |\/ 31        31           |
\/ 31 *atan|------ + -----------------|
           \  31        1 + cos(2*x)  /
---------------------------------------
                   31
Simplificación general [src] 
           /  ____               \
  ____     |\/ 31 *(1 + 4*tan(x))|
\/ 31 *atan|---------------------|
           \          31         /
----------------------------------
                31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(4 \tan{\left(x \right)} + 1\right)}{31} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)*(1 + 4*tan(x))/31)/31
Respuesta numérica [src] 
0.179605302026775*atan(((4*sin(2*x))/(1 + cos(2*x)) + 1)/sqrt(31))
0.179605302026775*atan(((4*sin(2*x))/(1 + cos(2*x)) + 1)/sqrt(31))
Unión de expresiones racionales [src] 
           /  ____                            \
  ____     |\/ 31 *(1 + 4*sin(2*x) + cos(2*x))|
\/ 31 *atan|----------------------------------|
           \        31*(1 + cos(2*x))         /
-----------------------------------------------
                       31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(4 \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + 1\right)}{31 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)*(1 + 4*sin(2*x) + cos(2*x))/(31*(1 + cos(2*x))))/31
Potencias [src] 
  ____     /  ____ /1        4*sin(2*x)   \\
\/ 31 *atan|\/ 31 *|-- + -----------------||
           \       \31   31*(1 + cos(2*x))//
--------------------------------------------
                     31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\sqrt{31} \left(\frac{1}{31} + \frac{4 \sin{\left(2 x \right)}}{31 \left(\cos{\left(2 x \right)} + 1\right)}\right) \right)}}{31}$$ 
           /       /        /   -2*I*x    2*I*x\\\
           |  ____ |    2*I*\- e       + e     /||
           |\/ 31 *|1 - ------------------------||
           |       |           -2*I*x    2*I*x  ||
           |       |          e         e       ||
           |       |      1 + ------- + ------  ||
  ____     |       \             2        2     /|
\/ 31 *atan|-------------------------------------|
           \                  31                 /
--------------------------------------------------
                        31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(- \frac{2 i \left(e^{2 i x} - e^{- 2 i x}\right)}{\frac{e^{2 i x}}{2} + 1 + \frac{e^{- 2 i x}}{2}} + 1\right)}{31} \right)}}{31}$$ 
           /  ____ /     4*sin(2*x) \\
           |\/ 31 *|1 + ------------||
  ____     |       \    1 + cos(2*x)/|
\/ 31 *atan|-------------------------|
           \            31           /
--------------------------------------
                  31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4 \sin{\left(2 x \right)}}{\cos{\left(2 x \right)} + 1}\right)}{31} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)*(1 + 4*sin(2*x)/(1 + cos(2*x)))/31)/31
Denominador común [src] 
           /  ____       ____         \
  ____     |\/ 31    4*\/ 31 *sin(2*x)|
\/ 31 *atan|------ + -----------------|
           \  31     31*(1 + cos(2*x))/
---------------------------------------
                   31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31}}{31} + \frac{4 \sqrt{31} \sin{\left(2 x \right)}}{31 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)/31 + 4*sqrt(31)*sin(2*x)/(31*(1 + cos(2*x))))/31
Denominador racional [src] 
           /  ____                            \
  ____     |\/ 31 *(1 + 4*sin(2*x) + cos(2*x))|
\/ 31 *atan|----------------------------------|
           \        31*(1 + cos(2*x))         /
-----------------------------------------------
                       31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(4 \sin{\left(2 x \right)} + \cos{\left(2 x \right)} + 1\right)}{31 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)*(1 + 4*sin(2*x) + cos(2*x))/(31*(1 + cos(2*x))))/31
Combinatoria [src] 
           /  ____       ____         \
  ____     |\/ 31    4*\/ 31 *sin(2*x)|
\/ 31 *atan|------ + -----------------|
           \  31     31*(1 + cos(2*x))/
---------------------------------------
                   31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31}}{31} + \frac{4 \sqrt{31} \sin{\left(2 x \right)}}{31 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)/31 + 4*sqrt(31)*sin(2*x)/(31*(1 + cos(2*x))))/31
Compilar la expresión [src] 
           / 4*sin(2*x)     \
           |------------ + 1|
  ____     |1 + cos(2*x)    |
\/ 31 *atan|----------------|
           |       ____     |
           \     \/ 31      /
-----------------------------
              31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{1 + \frac{4 \sin{\left(2 x \right)}}{\cos{\left(2 x \right)} + 1}}{\sqrt{31}} \right)}}{31}$$ 
sqrt(31)*atan(((4*sin(2*x))/(1 + cos(2*x)) + 1)/sqrt(31))/31
Abrimos la expresión [src] 
           /  ____       ____       \
  ____     |\/ 31    4*\/ 31 *sin(x)|
\/ 31 *atan|------ + ---------------|
           \  31        31*cos(x)   /
-------------------------------------
                  31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{4 \sqrt{31} \sin{\left(x \right)}}{31 \cos{\left(x \right)}} + \frac{\sqrt{31}}{31} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)/31 + 4*sqrt(31)*sin(x)/(31*cos(x)))/31
Parte trigonométrica [src] 
           /       /         /    pi\\\
           |       |    4*cos|x - --|||
           |  ____ |         \    2 /||
           |\/ 31 *|1 + -------------||
  ____     |       \        cos(x)   /|
\/ 31 *atan|--------------------------|
           \            31            /
---------------------------------------
                   31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)}{31} \right)}}{31}$$ 
           /  ____ /      4   \\
           |\/ 31 *|1 + ------||
  ____     |       \    cot(x)/|
\/ 31 *atan|-------------------|
           \         31        /
--------------------------------
               31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4}{\cot{\left(x \right)}}\right)}{31} \right)}}{31}$$ 
           /  ____ /                8*tan(x)           \\
           |\/ 31 *|1 + -------------------------------||
           |       |                  /           2   \||
           |       |    /       2   \ |    1 - tan (x)|||
           |       |    \1 + tan (x)/*|1 + -----------|||
           |       |                  |           2   |||
  ____     |       \                  \    1 + tan (x)//|
\/ 31 *atan|--------------------------------------------|
           \                     31                     /
---------------------------------------------------------
                            31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{8 \tan{\left(x \right)}}{\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}\right)}{31} \right)}}{31}$$ 
           /       /         /pi    \\\
           |       |    4*csc|-- - x|||
           |  ____ |         \2     /||
           |\/ 31 *|1 + -------------||
  ____     |       \        csc(x)   /|
\/ 31 *atan|--------------------------|
           \            31            /
---------------------------------------
                   31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4 \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)}{31} \right)}}{31}$$ 
           /  ____ /     4*sin(2*x) \\
           |\/ 31 *|1 + ------------||
  ____     |       \    1 + cos(2*x)/|
\/ 31 *atan|-------------------------|
           \            31           /
--------------------------------------
                  31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4 \sin{\left(2 x \right)}}{\cos{\left(2 x \right)} + 1}\right)}{31} \right)}}{31}$$ 
           /  ____               \
  ____     |\/ 31 *(1 + 4*tan(x))|
\/ 31 *atan|---------------------|
           \          31         /
----------------------------------
                31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(4 \tan{\left(x \right)} + 1\right)}{31} \right)}}{31}$$ 
           /  ____ /                 4              \\
           |\/ 31 *|1 + ----------------------------||
           |       |    /       1    \    /      pi\||
           |       |    |1 + --------|*sec|2*x - --|||
  ____     |       \    \    sec(2*x)/    \      2 //|
\/ 31 *atan|-----------------------------------------|
           \                    31                   /
------------------------------------------------------
                          31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4}{\left(1 + \frac{1}{\sec{\left(2 x \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}}\right)}{31} \right)}}{31}$$ 
           /       /         2   \\
           |  ____ |    8*sin (x)||
           |\/ 31 *|1 + ---------||
  ____     |       \     sin(2*x)/|
\/ 31 *atan|----------------------|
           \          31          /
-----------------------------------
                 31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(\frac{8 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right)}{31} \right)}}{31}$$ 
           /  ____ /                 4              \\
           |\/ 31 *|1 + ----------------------------||
           |       |    /          1      \         ||
           |       |    |1 + -------------|*csc(2*x)||
           |       |    |       /pi      \|         ||
           |       |    |    csc|-- - 2*x||         ||
  ____     |       \    \       \2       //         /|
\/ 31 *atan|-----------------------------------------|
           \                    31                   /
------------------------------------------------------
                          31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4}{\left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right) \csc{\left(2 x \right)}}\right)}{31} \right)}}{31}$$ 
           /  ____ /               4           \\
           |\/ 31 *|1 + -----------------------||
           |       |    /       1    \         ||
           |       |    |1 + --------|*csc(2*x)||
  ____     |       \    \    sec(2*x)/         /|
\/ 31 *atan|------------------------------------|
           \                 31                 /
-------------------------------------------------
                        31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4}{\left(1 + \frac{1}{\sec{\left(2 x \right)}}\right) \csc{\left(2 x \right)}}\right)}{31} \right)}}{31}$$ 
           /  ____ /      4*sec(x) \\
           |\/ 31 *|1 + -----------||
           |       |       /    pi\||
           |       |    sec|x - --|||
  ____     |       \       \    2 //|
\/ 31 *atan|------------------------|
           \           31           /
-------------------------------------
                  31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(\frac{4 \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)}{31} \right)}}{31}$$ 
           /  ____ /        4*sin(2*x)   \\
           |\/ 31 *|1 + -----------------||
           |       |           /pi      \||
           |       |    1 + sin|-- + 2*x|||
  ____     |       \           \2       //|
\/ 31 *atan|------------------------------|
           \              31              /
-------------------------------------------
                     31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4 \sin{\left(2 x \right)}}{\sin{\left(2 x + \frac{\pi}{2} \right)} + 1}\right)}{31} \right)}}{31}$$ 
           /       /         /      pi\\\
           |       |    4*cos|2*x - --|||
           |  ____ |         \      2 /||
           |\/ 31 *|1 + ---------------||
  ____     |       \      1 + cos(2*x) /|
\/ 31 *atan|----------------------------|
           \             31             /
-----------------------------------------
                    31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{4 \cos{\left(2 x - \frac{\pi}{2} \right)}}{\cos{\left(2 x \right)} + 1}\right)}{31} \right)}}{31}$$ 
           /  ____ /                8*cot(x)            \\
           |\/ 31 *|1 + --------------------------------||
           |       |                  /            2   \||
           |       |    /       2   \ |    -1 + cot (x)|||
           |       |    \1 + cot (x)/*|1 + ------------|||
           |       |                  |           2    |||
  ____     |       \                  \    1 + cot (x) //|
\/ 31 *atan|---------------------------------------------|
           \                      31                     /
----------------------------------------------------------
                            31
$$\frac{\sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31} \left(1 + \frac{8 \cot{\left(x \right)}}{\left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}\right)}{31} \right)}}{31}$$ 
sqrt(31)*atan(sqrt(31)*(1 + 8*cot(x)/((1 + cot(x)^2)*(1 + (-1 + cot(x)^2)/(1 + cot(x)^2))))/31)/31
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