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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(sin(2*x)/(sqrt(5)*(1+cos(2*x))))/sqrt(5)

¿Cómo vas a descomponer esta atan(sin(2*x)/(sqrt(5)*(1+cos(2*x))))/sqrt(5) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    /      sin(2*x)      \
atan|--------------------|
    |  ___               |
    \\/ 5 *(1 + cos(2*x))/
--------------------------
            ___           
          \/ 5
$$\frac{\operatorname{atan}{\left(\frac{\sin{\left(2 x \right)}}{\sqrt{5} \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{\sqrt{5}}$$ 
atan(sin(2*x)/((sqrt(5)*(1 + cos(2*x)))))/sqrt(5)
Descomposición de una fracción [src] 
sqrt(5)*atan(sin(2*x)/(sqrt(5) + sqrt(5)*cos(2*x)))/5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sin{\left(2 x \right)}}{\sqrt{5} \cos{\left(2 x \right)} + \sqrt{5}} \right)}}{5}$$ 
  ___     /       sin(2*x)       \
\/ 5 *atan|----------------------|
          |  ___     ___         |
          \\/ 5  + \/ 5 *cos(2*x)/
----------------------------------
                5
Simplificación general [src] 
          /  ___       \
  ___     |\/ 5 *tan(x)|
\/ 5 *atan|------------|
          \     5      /
------------------------
           5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \tan{\left(x \right)}}{5} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*tan(x)/5)/5
Respuesta numérica [src] 
0.447213595499958*atan(sin(2*x)/((sqrt(5)*(1 + cos(2*x)))))
0.447213595499958*atan(sin(2*x)/((sqrt(5)*(1 + cos(2*x)))))
Denominador racional [src] 
          /   ___          \
  ___     | \/ 5 *sin(2*x) |
\/ 5 *atan|----------------|
          \5*(1 + cos(2*x))/
----------------------------
             5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*sin(2*x)/(5*(1 + cos(2*x))))/5
Denominador común [src] 
          /   ___          \
  ___     | \/ 5 *sin(2*x) |
\/ 5 *atan|----------------|
          \5*(1 + cos(2*x))/
----------------------------
             5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*sin(2*x)/(5*(1 + cos(2*x))))/5
Potencias [src] 
           /    ___ /   -2*I*x    2*I*x\\ 
   ___     |I*\/ 5 *\- e       + e     /| 
-\/ 5 *atan|----------------------------| 
           |    /     -2*I*x    2*I*x\  | 
           |    |    e         e     |  | 
           | 10*|1 + ------- + ------|  | 
           \    \       2        2   /  / 
------------------------------------------
                    5
$$- \frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} i \left(e^{2 i x} - e^{- 2 i x}\right)}{10 \left(\frac{e^{2 i x}}{2} + 1 + \frac{e^{- 2 i x}}{2}\right)} \right)}}{5}$$ 
          /   ___          \
  ___     | \/ 5 *sin(2*x) |
\/ 5 *atan|----------------|
          \5*(1 + cos(2*x))/
----------------------------
             5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*sin(2*x)/(5*(1 + cos(2*x))))/5
Combinatoria [src] 
          /   ___          \
  ___     | \/ 5 *sin(2*x) |
\/ 5 *atan|----------------|
          \5*(1 + cos(2*x))/
----------------------------
             5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*sin(2*x)/(5*(1 + cos(2*x))))/5
Unión de expresiones racionales [src] 
          /   ___          \
  ___     | \/ 5 *sin(2*x) |
\/ 5 *atan|----------------|
          \5*(1 + cos(2*x))/
----------------------------
             5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*sin(2*x)/(5*(1 + cos(2*x))))/5
Abrimos la expresión [src] 
  ___     /   ___          \
\/ 5      | \/ 5 *sin(2*x) |
-----*atan|----------------|
  5       \5*(1 + cos(2*x))/
$$\frac{\sqrt{5}}{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}$$ 
          /  ___       \
  ___     |\/ 5 *sin(x)|
\/ 5 *atan|------------|
          \  5*cos(x)  /
------------------------
           5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(x \right)}}{5 \cos{\left(x \right)}} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*sin(x)/(5*cos(x)))/5
Compilar la expresión [src] 
  ___     /      sin(2*x)      \
\/ 5 *atan|--------------------|
          |  ___               |
          \\/ 5 *(1 + cos(2*x))/
--------------------------------
               5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sin{\left(2 x \right)}}{\sqrt{5} \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sin(2*x)/((sqrt(5)*(1 + cos(2*x)))))/5
Parte trigonométrica [src] 
          /      ___            \
  ___     |    \/ 5 *sin(2*x)   |
\/ 5 *atan|---------------------|
          |  /       /pi      \\|
          |5*|1 + sin|-- + 2*x|||
          \  \       \2       ///
---------------------------------
                5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\sin{\left(2 x + \frac{\pi}{2} \right)} + 1\right)} \right)}}{5}$$ 
          /              ___                 \
  ___     |          2*\/ 5 *cot(x)          |
\/ 5 *atan|----------------------------------|
          |                /            2   \|
          |  /       2   \ |    -1 + cot (x)||
          |5*\1 + cot (x)/*|1 + ------------||
          |                |           2    ||
          \                \    1 + cot (x) //
----------------------------------------------
                      5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{2 \sqrt{5} \cot{\left(x \right)}}{5 \left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)} \right)}}{5}$$ 
          /   ___  \
  ___     | \/ 5   |
\/ 5 *atan|--------|
          \5*cot(x)/
--------------------
         5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5 \cot{\left(x \right)}} \right)}}{5}$$ 
          /   ___       \
  ___     | \/ 5 *sec(x)|
\/ 5 *atan|-------------|
          |     /    pi\|
          |5*sec|x - --||
          \     \    2 //
-------------------------
            5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sec{\left(x \right)}}{5 \sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{5}$$ 
          /              ___             \
  ___     |            \/ 5              |
\/ 5 *atan|------------------------------|
          |  /       1    \    /      pi\|
          |5*|1 + --------|*sec|2*x - --||
          \  \    sec(2*x)/    \      2 //
------------------------------------------
                    5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5 \left(1 + \frac{1}{\sec{\left(2 x \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}} \right)}}{5}$$ 
          /            ___          \
  ___     |          \/ 5           |
\/ 5 *atan|-------------------------|
          |  /       1    \         |
          |5*|1 + --------|*csc(2*x)|
          \  \    sec(2*x)/         /
-------------------------------------
                  5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5 \left(1 + \frac{1}{\sec{\left(2 x \right)}}\right) \csc{\left(2 x \right)}} \right)}}{5}$$ 
          /              ___             \
  ___     |            \/ 5              |
\/ 5 *atan|------------------------------|
          |  /          1      \         |
          |5*|1 + -------------|*csc(2*x)|
          |  |       /pi      \|         |
          |  |    csc|-- - 2*x||         |
          \  \       \2       //         /
------------------------------------------
                    5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5 \left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right) \csc{\left(2 x \right)}} \right)}}{5}$$ 
          /  ___       \
  ___     |\/ 5 *tan(x)|
\/ 5 *atan|------------|
          \     5      /
------------------------
           5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \tan{\left(x \right)}}{5} \right)}}{5}$$ 
          /  ___    /      pi\\
          |\/ 5 *cos|2*x - --||
  ___     |         \      2 /|
\/ 5 *atan|-------------------|
          \  5*(1 + cos(2*x)) /
-------------------------------
               5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \cos{\left(2 x - \frac{\pi}{2} \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
          /    ___    2   \
  ___     |2*\/ 5 *sin (x)|
\/ 5 *atan|---------------|
          \   5*sin(2*x)  /
---------------------------
             5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{2 \sqrt{5} \sin^{2}{\left(x \right)}}{5 \sin{\left(2 x \right)}} \right)}}{5}$$ 
          /  ___    /pi    \\
          |\/ 5 *csc|-- - x||
  ___     |         \2     /|
\/ 5 *atan|-----------------|
          \     5*csc(x)    /
-----------------------------
              5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \csc{\left(- x + \frac{\pi}{2} \right)}}{5 \csc{\left(x \right)}} \right)}}{5}$$ 
          /              ___                \
  ___     |          2*\/ 5 *tan(x)         |
\/ 5 *atan|---------------------------------|
          |                /           2   \|
          |  /       2   \ |    1 - tan (x)||
          |5*\1 + tan (x)/*|1 + -----------||
          |                |           2   ||
          \                \    1 + tan (x)//
---------------------------------------------
                      5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{2 \sqrt{5} \tan{\left(x \right)}}{5 \left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)} \right)}}{5}$$ 
          /   ___          \
  ___     | \/ 5 *sin(2*x) |
\/ 5 *atan|----------------|
          \5*(1 + cos(2*x))/
----------------------------
             5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \sin{\left(2 x \right)}}{5 \left(\cos{\left(2 x \right)} + 1\right)} \right)}}{5}$$ 
          /  ___    /    pi\\
          |\/ 5 *cos|x - --||
  ___     |         \    2 /|
\/ 5 *atan|-----------------|
          \     5*cos(x)    /
-----------------------------
              5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \cos{\left(x - \frac{\pi}{2} \right)}}{5 \cos{\left(x \right)}} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*cos(x - pi/2)/(5*cos(x)))/5
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