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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(((6*sin(x))/(1+cos(x))+2)/(2*sqrt(5)))/sqrt(5)

¿Cómo vas a descomponer esta atan(((6*sin(x))/(1+cos(x))+2)/(2*sqrt(5)))/sqrt(5) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    / 6*sin(x)     \
    |---------- + 2|
    |1 + cos(x)    |
atan|--------------|
    |       ___    |
    \   2*\/ 5     /
--------------------
         ___        
       \/ 5
$$\frac{\operatorname{atan}{\left(\frac{2 + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{2 \sqrt{5}} \right)}}{\sqrt{5}}$$ 
atan(((6*sin(x))/(1 + cos(x)) + 2)/((2*sqrt(5))))/sqrt(5)
Descomposición de una fracción [src] 
sqrt(5)*atan(sqrt(5)/5 + 3*sqrt(5)*sin(x)/(5*(1 + cos(x))))/5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} + \frac{3 \sqrt{5} \sin{\left(x \right)}}{5 \left(\cos{\left(x \right)} + 1\right)} \right)}}{5}$$ 
          /  ___       ___       \
  ___     |\/ 5    3*\/ 5 *sin(x)|
\/ 5 *atan|----- + --------------|
          \  5     5*(1 + cos(x))/
----------------------------------
                5
Simplificación general [src] 
          /  ___                        \
  ___     |\/ 5 *(1 + 3*sin(x) + cos(x))|
\/ 5 *atan|-----------------------------|
          \        5*(1 + cos(x))       /
-----------------------------------------
                    5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(3 \sin{\left(x \right)} + \cos{\left(x \right)} + 1\right)}{5 \left(\cos{\left(x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*(1 + 3*sin(x) + cos(x))/(5*(1 + cos(x))))/5
Respuesta numérica [src] 
0.447213595499958*atan(((6*sin(x))/(1 + cos(x)) + 2)/((2*sqrt(5))))
0.447213595499958*atan(((6*sin(x))/(1 + cos(x)) + 2)/((2*sqrt(5))))
Combinatoria [src] 
          /  ___       ___       \
  ___     |\/ 5    3*\/ 5 *sin(x)|
\/ 5 *atan|----- + --------------|
          \  5     5*(1 + cos(x))/
----------------------------------
                5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} + \frac{3 \sqrt{5} \sin{\left(x \right)}}{5 \left(\cos{\left(x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)/5 + 3*sqrt(5)*sin(x)/(5*(1 + cos(x))))/5
Unión de expresiones racionales [src] 
          /  ___                        \
  ___     |\/ 5 *(1 + 3*sin(x) + cos(x))|
\/ 5 *atan|-----------------------------|
          \        5*(1 + cos(x))       /
-----------------------------------------
                    5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(3 \sin{\left(x \right)} + \cos{\left(x \right)} + 1\right)}{5 \left(\cos{\left(x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*(1 + 3*sin(x) + cos(x))/(5*(1 + cos(x))))/5
Denominador racional [src] 
          /  ___                        \
  ___     |\/ 5 *(1 + 3*sin(x) + cos(x))|
\/ 5 *atan|-----------------------------|
          \        5*(1 + cos(x))       /
-----------------------------------------
                    5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(3 \sin{\left(x \right)} + \cos{\left(x \right)} + 1\right)}{5 \left(\cos{\left(x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*(1 + 3*sin(x) + cos(x))/(5*(1 + cos(x))))/5
Denominador común [src] 
          /  ___       ___       \
  ___     |\/ 5    3*\/ 5 *sin(x)|
\/ 5 *atan|----- + --------------|
          \  5     5*(1 + cos(x))/
----------------------------------
                5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} + \frac{3 \sqrt{5} \sin{\left(x \right)}}{5 \left(\cos{\left(x \right)} + 1\right)} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)/5 + 3*sqrt(5)*sin(x)/(5*(1 + cos(x))))/5
Abrimos la expresión [src] 
          /  ___ / 6*sin(x)     \\
  ___     |\/ 5 *|---------- + 2||
\/ 5      |      \1 + cos(x)    /|
-----*atan|----------------------|
  5       \          10          /
$$\frac{\sqrt{5}}{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{10} \right)}$$ 
  ___     /  ___       ___       \
\/ 5      |\/ 5    3*\/ 5 *sin(x)|
-----*atan|----- + --------------|
  5       \  5     5*(1 + cos(x))/
$$\frac{\sqrt{5}}{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} + \frac{3 \sqrt{5} \sin{\left(x \right)}}{5 \left(\cos{\left(x \right)} + 1\right)} \right)}$$ 
(sqrt(5)/5)*atan(sqrt(5)/5 + 3*sqrt(5)*sin(x)/(5*(1 + cos(x))))
Compilar la expresión [src] 
          / 6*sin(x)     \
          |---------- + 2|
  ___     |1 + cos(x)    |
\/ 5 *atan|--------------|
          |       ___    |
          \   2*\/ 5     /
--------------------------
            5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{2 + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{2 \sqrt{5}} \right)}}{5}$$ 
sqrt(5)*atan(((6*sin(x))/(1 + cos(x)) + 2)/((2*sqrt(5))))/5
Parte trigonométrica [src] 
          /  ___ /        6*sin(x)   \\
          |\/ 5 *|2 + ---------------||
          |      |           /    pi\||
          |      |    1 + sin|x + --|||
  ___     |      \           \    2 //|
\/ 5 *atan|---------------------------|
          \             10            /
---------------------------------------
                   5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6 \sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}\right)}{10} \right)}}{5}$$ 
          /      /         /    pi\\\
          |      |    6*cos|x - --|||
          |  ___ |         \    2 /||
          |\/ 5 *|2 + -------------||
  ___     |      \      1 + cos(x) /|
\/ 5 *atan|-------------------------|
          \            10           /
-------------------------------------
                  5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}\right)}{10} \right)}}{5}$$ 
          /  ___ /               6            \\
          |\/ 5 *|2 + ------------------------||
          |      |    /         1     \       ||
          |      |    |1 + -----------|*csc(x)||
          |      |    |       /pi    \|       ||
          |      |    |    csc|-- - x||       ||
  ___     |      \    \       \2     //       /|
\/ 5 *atan|------------------------------------|
          \                 10                 /
------------------------------------------------
                       5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}\right)}{10} \right)}}{5}$$ 
          /  ___ /             6         \\
          |\/ 5 *|2 + -------------------||
          |      |    /      1   \       ||
          |      |    |1 + ------|*csc(x)||
  ___     |      \    \    sec(x)/       /|
\/ 5 *atan|-------------------------------|
          \               10              /
-------------------------------------------
                     5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}\right)}{10} \right)}}{5}$$ 
          /  ___ /               6            \\
          |\/ 5 *|2 + ------------------------||
          |      |    /      1   \    /    pi\||
          |      |    |1 + ------|*sec|x - --|||
  ___     |      \    \    sec(x)/    \    2 //|
\/ 5 *atan|------------------------------------|
          \                 10                 /
------------------------------------------------
                       5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}\right)}{10} \right)}}{5}$$ 
          /      /                     /x\           \\
          |      |               12*tan|-|           ||
          |  ___ |                     \2/           ||
          |\/ 5 *|2 + -------------------------------||
          |      |                  /           2/x\\||
          |      |                  |    1 - tan |-||||
          |      |    /       2/x\\ |            \2/|||
          |      |    |1 + tan |-||*|1 + -----------|||
          |      |    \        \2// |           2/x\|||
          |      |                  |    1 + tan |-||||
  ___     |      \                  \            \2///|
\/ 5 *atan|-------------------------------------------|
          \                     10                    /
-------------------------------------------------------
                           5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{12 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}\right)}{10} \right)}}{5}$$ 
          /      /                     /x\            \\
          |      |               12*cot|-|            ||
          |  ___ |                     \2/            ||
          |\/ 5 *|2 + --------------------------------||
          |      |                  /            2/x\\||
          |      |                  |    -1 + cot |-||||
          |      |    /       2/x\\ |             \2/|||
          |      |    |1 + cot |-||*|1 + ------------|||
          |      |    \        \2// |           2/x\ |||
          |      |                  |    1 + cot |-| |||
  ___     |      \                  \            \2/ //|
\/ 5 *atan|--------------------------------------------|
          \                     10                     /
--------------------------------------------------------
                           5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{12 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}\right)}{10} \right)}}{5}$$ 
          /  ___ /     6*sin(x) \\
          |\/ 5 *|2 + ----------||
  ___     |      \    1 + cos(x)/|
\/ 5 *atan|----------------------|
          \          10          /
----------------------------------
                5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{10} \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*(2 + 6*sin(x)/(1 + cos(x)))/10)/5
Potencias [src] 
          /      /        /   -I*x    I*x\\\
          |  ___ |    3*I*\- e     + e   /||
          |\/ 5 *|2 - --------------------||
          |      |           I*x    -I*x  ||
          |      |          e      e      ||
          |      |      1 + ---- + -----  ||
  ___     |      \           2       2    /|
\/ 5 *atan|--------------------------------|
          \               10               /
--------------------------------------------
                     5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(- \frac{3 i \left(e^{i x} - e^{- i x}\right)}{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} + 2\right)}{10} \right)}}{5}$$ 
          /  ___ /     6*sin(x) \\
          |\/ 5 *|2 + ----------||
  ___     |      \    1 + cos(x)/|
\/ 5 *atan|----------------------|
          \          10          /
----------------------------------
                5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(2 + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{10} \right)}}{5}$$ 
  ___     /  ___ /1      3*sin(x)   \\
\/ 5 *atan|\/ 5 *|- + --------------||
          \      \5   5*(1 + cos(x))//
--------------------------------------
                  5
$$\frac{\sqrt{5} \operatorname{atan}{\left(\sqrt{5} \left(\frac{1}{5} + \frac{3 \sin{\left(x \right)}}{5 \left(\cos{\left(x \right)} + 1\right)}\right) \right)}}{5}$$ 
sqrt(5)*atan(sqrt(5)*(1/5 + 3*sin(x)/(5*(1 + cos(x)))))/5
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