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Simplificación de expresiones/ Descomposición en fracciones simples/ sqrt(2)*atan(((2*sin(x))/(1+cos(x))-2)/2^(3/2))

¿Cómo vas a descomponer esta sqrt(2)*atan(((2*sin(x))/(1+cos(x))-2)/2^(3/2)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
          / 2*sin(x)     \
          |---------- - 2|
  ___     |1 + cos(x)    |
\/ 2 *atan|--------------|
          |      3/2     |
          \     2        /
2atan⁡(−2+2sin⁡(x)cos⁡(x)+1232)\sqrt{2} \operatorname{atan}{\left(\frac{-2 + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{2^{\frac{3}{2}}} \right)} 
sqrt(2)*atan(((2*sin(x))/(1 + cos(x)) - 2)/2^(3/2))
Descomposición de una fracción [src] 
sqrt(2)*atan(-sqrt(2)/2 + 2*(sqrt(2)/4)*sin(x)/(1 + cos(x)))
2atan⁡(−22+224sin⁡(x)cos⁡(x)+1)\sqrt{2} \operatorname{atan}{\left(- \frac{\sqrt{2}}{2} + \frac{2 \frac{\sqrt{2}}{4} \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} \right)} 
          /                ___       \
          |              \/ 2        |
          |      ___   2*-----*sin(x)|
  ___     |  2*\/ 2        4         |
\/ 2 *atan|- ------- + --------------|
          \     4        1 + cos(x)  /
Simplificación general [src] 
           /  ___              \
           |\/ 2       /    pi\|
           |----- + cos|x + --||
   ___     |  2        \    4 /|
-\/ 2 *atan|-------------------|
           \     1 + cos(x)    /
−2atan⁡(cos⁡(x+π4)+22cos⁡(x)+1)- \sqrt{2} \operatorname{atan}{\left(\frac{\cos{\left(x + \frac{\pi}{4} \right)} + \frac{\sqrt{2}}{2}}{\cos{\left(x \right)} + 1} \right)} 
-sqrt(2)*atan((sqrt(2)/2 + cos(x + pi/4))/(1 + cos(x)))
Respuesta numérica [src] 
1.4142135623731*atan(((2*sin(x))/(1 + cos(x)) - 2)/2^(3/2))
1.4142135623731*atan(((2*sin(x))/(1 + cos(x)) - 2)/2^(3/2))
Denominador racional [src] 
           /  ___                      \
   ___     |\/ 2 *(1 - sin(x) + cos(x))|
-\/ 2 *atan|---------------------------|
           \       2*(1 + cos(x))      /
−2atan⁡(2(−sin⁡(x)+cos⁡(x)+1)2(cos⁡(x)+1))- \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(- \sin{\left(x \right)} + \cos{\left(x \right)} + 1\right)}{2 \left(\cos{\left(x \right)} + 1\right)} \right)} 
-sqrt(2)*atan(sqrt(2)*(1 - sin(x) + cos(x))/(2*(1 + cos(x))))
Potencias [src] 
  ___     /  ___ /  1       sin(x)    \\
\/ 2 *atan|\/ 2 *|- - + --------------||
          \      \  2   2*(1 + cos(x))//
2atan⁡(2(−12+sin⁡(x)2(cos⁡(x)+1)))\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \left(- \frac{1}{2} + \frac{\sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)}\right) \right)} 
           /       /       /   -I*x    I*x\\ \
           |   ___ |     I*\- e     + e   /| |
           |-\/ 2 *|-2 - ------------------| |
           |       |           I*x    -I*x | |
           |       |          e      e     | |
           |       |      1 + ---- + ----- | |
   ___     |       \           2       2   / |
-\/ 2 *atan|---------------------------------|
           \                4                /
−2atan⁡(−2(−i(eix−e−ix)eix2+1+e−ix2−2)4)- \sqrt{2} \operatorname{atan}{\left(- \frac{\sqrt{2} \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} - 2\right)}{4} \right)} 
          /  ___ /      2*sin(x) \\
          |\/ 2 *|-2 + ----------||
  ___     |      \     1 + cos(x)/|
\/ 2 *atan|-----------------------|
          \           4           /
2atan⁡(2(−2+2sin⁡(x)cos⁡(x)+1)4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{4} \right)} 
sqrt(2)*atan(sqrt(2)*(-2 + 2*sin(x)/(1 + cos(x)))/4)
Combinatoria [src] 
           /  ___      ___        \
   ___     |\/ 2     \/ 2 *sin(x) |
-\/ 2 *atan|----- - --------------|
           \  2     2*(1 + cos(x))/
−2atan⁡(22−2sin⁡(x)2(cos⁡(x)+1))- \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} \sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} \right)} 
-sqrt(2)*atan(sqrt(2)/2 - sqrt(2)*sin(x)/(2*(1 + cos(x))))
Unión de expresiones racionales [src] 
          /  ___                       \
  ___     |\/ 2 *(-1 - cos(x) + sin(x))|
\/ 2 *atan|----------------------------|
          \       2*(1 + cos(x))       /
2atan⁡(2(sin⁡(x)−cos⁡(x)−1)2(cos⁡(x)+1))\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(\sin{\left(x \right)} - \cos{\left(x \right)} - 1\right)}{2 \left(\cos{\left(x \right)} + 1\right)} \right)} 
sqrt(2)*atan(sqrt(2)*(-1 - cos(x) + sin(x))/(2*(1 + cos(x))))
Denominador común [src] 
           /  ___      ___        \
   ___     |\/ 2     \/ 2 *sin(x) |
-\/ 2 *atan|----- - --------------|
           \  2     2*(1 + cos(x))/
−2atan⁡(22−2sin⁡(x)2(cos⁡(x)+1))- \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} \sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} \right)} 
-sqrt(2)*atan(sqrt(2)/2 - sqrt(2)*sin(x)/(2*(1 + cos(x))))
Abrimos la expresión [src] 
          /                ___       \
          |              \/ 2        |
          |      ___   2*-----*sin(x)|
  ___     |  2*\/ 2        4         |
\/ 2 *atan|- ------- + --------------|
          \     4        1 + cos(x)  /
2atan⁡(−22+224sin⁡(x)cos⁡(x)+1)\sqrt{2} \operatorname{atan}{\left(- \frac{\sqrt{2}}{2} + \frac{2 \frac{\sqrt{2}}{4} \sin{\left(x \right)}}{\cos{\left(x \right)} + 1} \right)} 
sqrt(2)*atan(-sqrt(2)/2 + 2*(sqrt(2)/4)*sin(x)/(1 + cos(x)))
Parte trigonométrica [src] 
          /  ___ /                2            \\
          |\/ 2 *|-2 + ------------------------||
          |      |     /         1     \       ||
          |      |     |1 + -----------|*csc(x)||
          |      |     |       /pi    \|       ||
          |      |     |    csc|-- - x||       ||
  ___     |      \     \       \2     //       /|
\/ 2 *atan|-------------------------------------|
          \                  4                  /
2atan⁡(2(−2+2(1+1csc⁡(−x+π2))csc⁡(x))4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{2}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}\right)}{4} \right)} 
          /      /                      /x\            \\
          |      |                 4*cot|-|            ||
          |  ___ |                      \2/            ||
          |\/ 2 *|-2 + --------------------------------||
          |      |                   /            2/x\\||
          |      |                   |    -1 + cot |-||||
          |      |     /       2/x\\ |             \2/|||
          |      |     |1 + cot |-||*|1 + ------------|||
          |      |     \        \2// |           2/x\ |||
          |      |                   |    1 + cot |-| |||
  ___     |      \                   \            \2/ //|
\/ 2 *atan|---------------------------------------------|
          \                      4                      /
2atan⁡(2(−2+4cot⁡(x2)(cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)+1))4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{4 \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)}{4} \right)} 
          /  ___ /                2            \\
          |\/ 2 *|-2 + ------------------------||
          |      |     /      1   \    /    pi\||
          |      |     |1 + ------|*sec|x - --|||
  ___     |      \     \    sec(x)/    \    2 //|
\/ 2 *atan|-------------------------------------|
          \                  4                  /
2atan⁡(2(−2+2(1+1sec⁡(x))sec⁡(x−π2))4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{2}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}\right)}{4} \right)} 
          /      /                      /x\           \\
          |      |                 4*tan|-|           ||
          |  ___ |                      \2/           ||
          |\/ 2 *|-2 + -------------------------------||
          |      |                   /           2/x\\||
          |      |                   |    1 - tan |-||||
          |      |     /       2/x\\ |            \2/|||
          |      |     |1 + tan |-||*|1 + -----------|||
          |      |     \        \2// |           2/x\|||
          |      |                   |    1 + tan |-||||
  ___     |      \                   \            \2///|
\/ 2 *atan|--------------------------------------------|
          \                     4                      /
2atan⁡(2(−2+4tan⁡(x2)(1−tan⁡2(x2)tan⁡2(x2)+1+1)(tan⁡2(x2)+1))4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{4 \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)}{4} \right)} 
          /  ___ /         2*sin(x)   \\
          |\/ 2 *|-2 + ---------------||
          |      |            /    pi\||
          |      |     1 + sin|x + --|||
  ___     |      \            \    2 //|
\/ 2 *atan|----------------------------|
          \             4              /
2atan⁡(2(−2+2sin⁡(x)sin⁡(x+π2)+1)4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{2 \sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}\right)}{4} \right)} 
          /  ___ /              2         \\
          |\/ 2 *|-2 + -------------------||
          |      |     /      1   \       ||
          |      |     |1 + ------|*csc(x)||
  ___     |      \     \    sec(x)/       /|
\/ 2 *atan|--------------------------------|
          \               4                /
2atan⁡(2(−2+2(1+1sec⁡(x))csc⁡(x))4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{2}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}\right)}{4} \right)} 
          /  ___ /      2*sin(x) \\
          |\/ 2 *|-2 + ----------||
  ___     |      \     1 + cos(x)/|
\/ 2 *atan|-----------------------|
          \           4           /
2atan⁡(2(−2+2sin⁡(x)cos⁡(x)+1)4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{4} \right)} 
          /      /          /    pi\\\
          |      |     2*cos|x - --|||
          |  ___ |          \    2 /||
          |\/ 2 *|-2 + -------------||
  ___     |      \       1 + cos(x) /|
\/ 2 *atan|--------------------------|
          \            4             /
2atan⁡(2(−2+2cos⁡(x−π2)cos⁡(x)+1)4)\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(-2 + \frac{2 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}\right)}{4} \right)} 
sqrt(2)*atan(sqrt(2)*(-2 + 2*cos(x - pi/2)/(1 + cos(x)))/4)
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