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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
3 _________                  sin(x)       
\/ atan(x) *cos(x) + ---------------------
                       /     2\     2/3   
                     3*\1 + x /*atan   (x)
cos⁡(x)atan⁡(x)3+sin⁡(x)3(x2+1)atan⁡23(x)\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
atan(x)^(1/3)*cos(x) + sin(x)/(((3*(1 + x^2))*atan(x)^(2/3)))
Simplificación general [src] 
sin(x)   /     2\               
------ + \1 + x /*atan(x)*cos(x)
  3                             
--------------------------------
      /     2\     2/3          
      \1 + x /*atan   (x)
(x2+1)cos⁡(x)atan⁡(x)+sin⁡(x)3(x2+1)atan⁡23(x)\frac{\left(x^{2} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(x \right)}}{3}}{\left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
(sin(x)/3 + (1 + x^2)*atan(x)*cos(x))/((1 + x^2)*atan(x)^(2/3))
Respuesta numérica [src] 
atan(x)^0.333333333333333*cos(x) + atan(x)^(-0.666666666666667)*sin(x)/(3.0 + 3.0*x^2)
atan(x)^0.333333333333333*cos(x) + atan(x)^(-0.666666666666667)*sin(x)/(3.0 + 3.0*x^2)
Denominador común [src] 
                      2                        
3*atan(x)*cos(x) + 3*x *atan(x)*cos(x) + sin(x)
-----------------------------------------------
               2/3         2     2/3           
         3*atan   (x) + 3*x *atan   (x)
3x2cos⁡(x)atan⁡(x)+sin⁡(x)+3cos⁡(x)atan⁡(x)3x2atan⁡23(x)+3atan⁡23(x)\frac{3 x^{2} \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)} + 3 \cos{\left(x \right)} \operatorname{atan}{\left(x \right)}}{3 x^{2} \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} + 3 \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
(3*atan(x)*cos(x) + 3*x^2*atan(x)*cos(x) + sin(x))/(3*atan(x)^(2/3) + 3*x^2*atan(x)^(2/3))
Unión de expresiones racionales [src] 
  /     2\                        
3*\1 + x /*atan(x)*cos(x) + sin(x)
----------------------------------
        /     2\     2/3          
      3*\1 + x /*atan   (x)
3(x2+1)cos⁡(x)atan⁡(x)+sin⁡(x)3(x2+1)atan⁡23(x)\frac{3 \left(x^{2} + 1\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)}}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
(3*(1 + x^2)*atan(x)*cos(x) + sin(x))/(3*(1 + x^2)*atan(x)^(2/3))
Potencias [src] 
3 _________                  sin(x)       
\/ atan(x) *cos(x) + ---------------------
                     /       2\     2/3   
                     \3 + 3*x /*atan   (x)
cos⁡(x)atan⁡(x)3+sin⁡(x)(3x2+3)atan⁡23(x)\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
            / I*x    -I*x\        /   -I*x    I*x\  
3 _________ |e      e    |      I*\- e     + e   /  
\/ atan(x) *|---- + -----| - -----------------------
            \ 2       2  /     /       2\     2/3   
                             2*\3 + 3*x /*atan   (x)
(eix2+e−ix2)atan⁡(x)3−i(eix−e−ix)2(3x2+3)atan⁡23(x)\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \sqrt[3]{\operatorname{atan}{\left(x \right)}} - \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
atan(x)^(1/3)*(exp(i*x)/2 + exp(-i*x)/2) - i*(-exp(-i*x) + exp(i*x))/(2*(3 + 3*x^2)*atan(x)^(2/3))
Combinatoria [src] 
                      2                        
3*atan(x)*cos(x) + 3*x *atan(x)*cos(x) + sin(x)
-----------------------------------------------
               /     2\     2/3                
             3*\1 + x /*atan   (x)
3x2cos⁡(x)atan⁡(x)+sin⁡(x)+3cos⁡(x)atan⁡(x)3(x2+1)atan⁡23(x)\frac{3 x^{2} \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)} + 3 \cos{\left(x \right)} \operatorname{atan}{\left(x \right)}}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
(3*atan(x)*cos(x) + 3*x^2*atan(x)*cos(x) + sin(x))/(3*(1 + x^2)*atan(x)^(2/3))
Denominador racional [src] 
/       2\                        
\3 + 3*x /*atan(x)*cos(x) + sin(x)
----------------------------------
      /       2\     2/3          
      \3 + 3*x /*atan   (x)
(3x2+3)cos⁡(x)atan⁡(x)+sin⁡(x)(3x2+3)atan⁡23(x)\frac{\left(3 x^{2} + 3\right) \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
((3 + 3*x^2)*atan(x)*cos(x) + sin(x))/((3 + 3*x^2)*atan(x)^(2/3))
Abrimos la expresión [src] 
            sin(x)               3 _________       
------------------------------ + \/ atan(x) *cos(x)
      2/3         2     2/3                        
3*atan   (x) + 3*x *atan   (x)
cos⁡(x)atan⁡(x)3+sin⁡(x)3x2atan⁡23(x)+3atan⁡23(x)\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{3 x^{2} \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} + 3 \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
sin(x)/(3*atan(x)^(2/3) + 3*x^2*atan(x)^(2/3)) + atan(x)^(1/3)*cos(x)
Compilar la expresión [src] 
3 _________                  sin(x)       
\/ atan(x) *cos(x) + ---------------------
                     /       2\     2/3   
                     \3 + 3*x /*atan   (x)
cos⁡(x)atan⁡(x)3+sin⁡(x)(3x2+3)atan⁡23(x)\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
atan(x)^(1/3)*cos(x) + sin(x)/((3 + 3*x^2)*atan(x)^(2/3))
Parte trigonométrica [src] 
3 _________                    1                 
\/ atan(x) *cos(x) + ---------------------*sin(x)
                       /     2\     2/3          
                     3*\1 + x /*atan   (x)
13(x2+1)atan⁡23(x)sin⁡(x)+cos⁡(x)atan⁡(x)3\frac{1}{3 \left(x^{2} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} \sin{\left(x \right)} + \cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} 
3 _________                  sin(x)       
\/ atan(x) *cos(x) + ---------------------
                     /       2\     2/3   
                     \3 + 3*x /*atan   (x)
cos⁡(x)atan⁡(x)3+sin⁡(x)(3x2+3)atan⁡23(x)\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
3 _________                               
\/ atan(x)                 1              
----------- + ----------------------------
   sec(x)     /       2\     2/3          
              \3 + 3*x /*atan   (x)*csc(x)
atan⁡(x)3sec⁡(x)+1(3x2+3)atan⁡23(x)csc⁡(x)\frac{\sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\sec{\left(x \right)}} + \frac{1}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} \csc{\left(x \right)}} 
3 _________    /    pi\           sin(x)       
\/ atan(x) *sin|x + --| + ---------------------
               \    2 /   /       2\     2/3   
                          \3 + 3*x /*atan   (x)
sin⁡(x+π2)atan⁡(x)3+sin⁡(x)(3x2+3)atan⁡23(x)\sin{\left(x + \frac{\pi}{2} \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
                             /    pi\     
                          cos|x - --|     
3 _________                  \    2 /     
\/ atan(x) *cos(x) + ---------------------
                     /       2\     2/3   
                     \3 + 3*x /*atan   (x)
cos⁡(x)atan⁡(x)3+cos⁡(x−π2)(3x2+3)atan⁡23(x)\cos{\left(x \right)} \sqrt[3]{\operatorname{atan}{\left(x \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
3 _________ /        2/x\\                      /x\             
\/ atan(x) *|-1 + cot |-||                 2*cot|-|             
            \         \2//                      \2/             
-------------------------- + -----------------------------------
              2/x\           /       2/x\\ /       2\     2/3   
       1 + cot |-|           |1 + cot |-||*\3 + 3*x /*atan   (x)
               \2/           \        \2//
(cot⁡2(x2)−1)atan⁡(x)3cot⁡2(x2)+1+2cot⁡(x2)(3x2+3)(cot⁡2(x2)+1)atan⁡23(x)\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(3 x^{2} + 3\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
3 _________                                    
\/ atan(x)                    1                
----------- + ---------------------------------
   sec(x)     /       2\     2/3       /    pi\
              \3 + 3*x /*atan   (x)*sec|x - --|
                                       \    2 /
atan⁡(x)3sec⁡(x)+1(3x2+3)atan⁡23(x)sec⁡(x−π2)\frac{\sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\sec{\left(x \right)}} + \frac{1}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} 
3 _________                               
\/ atan(x)                 1              
----------- + ----------------------------
   /pi    \   /       2\     2/3          
csc|-- - x|   \3 + 3*x /*atan   (x)*csc(x)
   \2     /
atan⁡(x)3csc⁡(−x+π2)+1(3x2+3)atan⁡23(x)csc⁡(x)\frac{\sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\left(3 x^{2} + 3\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)} \csc{\left(x \right)}} 
3 _________ /       2/x\\                      /x\             
\/ atan(x) *|1 - tan |-||                 2*tan|-|             
            \        \2//                      \2/             
------------------------- + -----------------------------------
              2/x\          /       2/x\\ /       2\     2/3   
       1 + tan |-|          |1 + tan |-||*\3 + 3*x /*atan   (x)
               \2/          \        \2//
(1−tan⁡2(x2))atan⁡(x)3tan⁡2(x2)+1+2tan⁡(x2)(3x2+3)(tan⁡2(x2)+1)atan⁡23(x)\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sqrt[3]{\operatorname{atan}{\left(x \right)}}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(3 x^{2} + 3\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \operatorname{atan}^{\frac{2}{3}}{\left(x \right)}} 
atan(x)^(1/3)*(1 - tan(x/2)^2)/(1 + tan(x/2)^2) + 2*tan(x/2)/((1 + tan(x/2)^2)*(3 + 3*x^2)*atan(x)^(2/3))
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