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Simplificación de expresiones/ Descomposición en fracciones simples/ cot(4*x)^2*(-12-12*cot(4*x)^2)*atan(x^3)+3*x^2*cot(4*x)^3/(1+x^6)

¿Cómo vas a descomponer esta cot(4*x)^2*(-12-12*cot(4*x)^2)*atan(x^3)+3*x^2*cot(4*x)^3/(1+x^6) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
                                             2    3     
   2      /            2     \     / 3\   3*x *cot (4*x)
cot (4*x)*\-12 - 12*cot (4*x)/*atan\x / + --------------
                                                   6    
                                              1 + x
$$\frac{3 x^{2} \cot^{3}{\left(4 x \right)}}{x^{6} + 1} + \left(- 12 \cot^{2}{\left(4 x \right)} - 12\right) \cot^{2}{\left(4 x \right)} \operatorname{atan}{\left(x^{3} \right)}$$ 
(cot(4*x)^2*(-12 - 12*cot(4*x)^2))*atan(x^3) + ((3*x^2)*cot(4*x)^3)/(1 + x^6)
Simplificación general [src] 
            /                /     6\     / 3\\
     2      | 2            4*\1 + x /*atan\x /|
3*cot (4*x)*|x *cot(4*x) - -------------------|
            |                      2          |
            \                   sin (4*x)     /
-----------------------------------------------
                          6                    
                     1 + x
$$\frac{3 \left(x^{2} \cot{\left(4 x \right)} - \frac{4 \left(x^{6} + 1\right) \operatorname{atan}{\left(x^{3} \right)}}{\sin^{2}{\left(4 x \right)}}\right) \cot^{2}{\left(4 x \right)}}{x^{6} + 1}$$ 
3*cot(4*x)^2*(x^2*cot(4*x) - 4*(1 + x^6)*atan(x^3)/sin(4*x)^2)/(1 + x^6)
Respuesta numérica [src] 
cot(4*x)^2*(-12.0 - 12.0*cot(4*x)^2)*atan(x^3) + 3.0*x^2*cot(4*x)^3/(1.0 + x^6)
cot(4*x)^2*(-12.0 - 12.0*cot(4*x)^2)*atan(x^3) + 3.0*x^2*cot(4*x)^3/(1.0 + x^6)
Unión de expresiones racionales [src] 
     2      / 2              /     6\ /        2     \     / 3\\
3*cot (4*x)*\x *cot(4*x) + 4*\1 + x /*\-1 - cot (4*x)/*atan\x //
----------------------------------------------------------------
                                  6                             
                             1 + x
$$\frac{3 \left(x^{2} \cot{\left(4 x \right)} + 4 \left(x^{6} + 1\right) \left(- \cot^{2}{\left(4 x \right)} - 1\right) \operatorname{atan}{\left(x^{3} \right)}\right) \cot^{2}{\left(4 x \right)}}{x^{6} + 1}$$ 
3*cot(4*x)^2*(x^2*cot(4*x) + 4*(1 + x^6)*(-1 - cot(4*x)^2)*atan(x^3))/(1 + x^6)
Denominador común [src] 
                                                     2    3     
        2          / 3\         4          / 3\   3*x *cot (4*x)
- 12*cot (4*x)*atan\x / - 12*cot (4*x)*atan\x / + --------------
                                                           6    
                                                      1 + x
$$\frac{3 x^{2} \cot^{3}{\left(4 x \right)}}{x^{6} + 1} - 12 \cot^{4}{\left(4 x \right)} \operatorname{atan}{\left(x^{3} \right)} - 12 \cot^{2}{\left(4 x \right)} \operatorname{atan}{\left(x^{3} \right)}$$ 
-12*cot(4*x)^2*atan(x^3) - 12*cot(4*x)^4*atan(x^3) + 3*x^2*cot(4*x)^3/(1 + x^6)
Combinatoria [src] 
      2      /      / 3\    2               6     / 3\        2          / 3\      6    2          / 3\\
-3*cot (4*x)*\4*atan\x / - x *cot(4*x) + 4*x *atan\x / + 4*cot (4*x)*atan\x / + 4*x *cot (4*x)*atan\x //
--------------------------------------------------------------------------------------------------------
                                         /     2\ /     4    2\                                         
                                         \1 + x /*\1 + x  - x /
$$- \frac{3 \left(4 x^{6} \cot^{2}{\left(4 x \right)} \operatorname{atan}{\left(x^{3} \right)} + 4 x^{6} \operatorname{atan}{\left(x^{3} \right)} - x^{2} \cot{\left(4 x \right)} + 4 \cot^{2}{\left(4 x \right)} \operatorname{atan}{\left(x^{3} \right)} + 4 \operatorname{atan}{\left(x^{3} \right)}\right) \cot^{2}{\left(4 x \right)}}{\left(x^{2} + 1\right) \left(x^{4} - x^{2} + 1\right)}$$ 
-3*cot(4*x)^2*(4*atan(x^3) - x^2*cot(4*x) + 4*x^6*atan(x^3) + 4*cot(4*x)^2*atan(x^3) + 4*x^6*cot(4*x)^2*atan(x^3))/((1 + x^2)*(1 + x^4 - x^2))
Denominador racional [src] 
   2    3           2      /     6\ /            2     \     / 3\
3*x *cot (4*x) + cot (4*x)*\1 + x /*\-12 - 12*cot (4*x)/*atan\x /
-----------------------------------------------------------------
                                   6                             
                              1 + x
$$\frac{3 x^{2} \cot^{3}{\left(4 x \right)} + \left(x^{6} + 1\right) \left(- 12 \cot^{2}{\left(4 x \right)} - 12\right) \cot^{2}{\left(4 x \right)} \operatorname{atan}{\left(x^{3} \right)}}{x^{6} + 1}$$ 
(3*x^2*cot(4*x)^3 + cot(4*x)^2*(1 + x^6)*(-12 - 12*cot(4*x)^2)*atan(x^3))/(1 + x^6)
Abrimos la expresión [src] 
                      / 3\                                                       / 3\                                                                                                2                                                                                               8        / 3\                                                               4        / 3\                                                                12        / 3\                                                                                       2    6                                                                          4        / 3\                                                                     2    2                                                                                                                    2    10                                                                         8        / 3\                                              16        / 3\                                                                                      2    12                                                                           2        / 3\                            6        / 3\                                              2        / 3\                                                               14        / 3\                                                                                        2    4                                                                                                                    2    8                                                                                             6        / 3\                                                               10        / 3\                          
               12*atan\x /                                                12*atan\x /                                                                                             3*x                                                                                       20808*cot (x)*atan\x /                                                       2640*cot (x)*atan\x /                                                        2640*cot  (x)*atan\x /                                                                                  756*x *cot (x)                                                                456*cot (x)*atan\x /                                                                 54*x *cot (x)                                                                                                             54*x *cot  (x)                                                                12*cot (x)*atan\x /                                        12*cot  (x)*atan\x /                                                                                   3*x *cot  (x)                                                                 144*cot (x)*atan\x /                     144*cot (x)*atan\x /                                       288*cot (x)*atan\x /                                                        288*cot  (x)*atan\x /                                                                                   333*x *cot (x)                                                                                                            333*x *cot (x)                                                                                 11232*cot (x)*atan\x /                                                      11232*cot  (x)*atan\x /                          
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          4            2            6                6              10             4             12              8               7            3            9             5           6    7          6    3          6    9           6    5                6              10             4             12              8                6              10             4             12              8                6              10             4             12              8               7            3            9             5           6    7          6    3          6    9           6    5              4            2            6               7            3            9             5           6    7          6    3          6    9           6    5               7            3            9             5           6    7          6    3          6    9           6    5              4            2            6                6              10             4             12              8               7            3            9             5           6    7          6    3          6    9           6    5              4            2            6              4            2            6                6              10             4             12              8                6              10             4             12              8               7            3            9             5           6    7          6    3          6    9           6    5               7            3            9             5           6    7          6    3          6    9           6    5                6              10             4             12              8                6              10             4             12              8   
  - 32*cot (x) + 16*cot (x) + 16*cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 192*cot (x) - 64*cot (x) + 64*cot (x) + 192*cot (x) - 192*x *cot (x) - 64*x *cot (x) + 64*x *cot (x) + 192*x *cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 192*cot (x) - 64*cot (x) + 64*cot (x) + 192*cot (x) - 192*x *cot (x) - 64*x *cot (x) + 64*x *cot (x) + 192*x *cot (x)   - 32*cot (x) + 16*cot (x) + 16*cot (x)   - 192*cot (x) - 64*cot (x) + 64*cot (x) + 192*cot (x) - 192*x *cot (x) - 64*x *cot (x) + 64*x *cot (x) + 192*x *cot (x)   - 192*cot (x) - 64*cot (x) + 64*cot (x) + 192*cot (x) - 192*x *cot (x) - 64*x *cot (x) + 64*x *cot (x) + 192*x *cot (x)   - 32*cot (x) + 16*cot (x) + 16*cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 192*cot (x) - 64*cot (x) + 64*cot (x) + 192*cot (x) - 192*x *cot (x) - 64*x *cot (x) + 64*x *cot (x) + 192*x *cot (x)   - 32*cot (x) + 16*cot (x) + 16*cot (x)   - 32*cot (x) + 16*cot (x) + 16*cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 192*cot (x) - 64*cot (x) + 64*cot (x) + 192*cot (x) - 192*x *cot (x) - 64*x *cot (x) + 64*x *cot (x) + 192*x *cot (x)   - 192*cot (x) - 64*cot (x) + 64*cot (x) + 192*cot (x) - 192*x *cot (x) - 64*x *cot (x) + 64*x *cot (x) + 192*x *cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)   - 1024*cot (x) - 1024*cot  (x) + 256*cot (x) + 256*cot  (x) + 1536*cot (x)
$$\frac{3 x^{2} \cot^{12}{\left(x \right)}}{64 x^{6} \cot^{9}{\left(x \right)} - 192 x^{6} \cot^{7}{\left(x \right)} + 192 x^{6} \cot^{5}{\left(x \right)} - 64 x^{6} \cot^{3}{\left(x \right)} + 64 \cot^{9}{\left(x \right)} - 192 \cot^{7}{\left(x \right)} + 192 \cot^{5}{\left(x \right)} - 64 \cot^{3}{\left(x \right)}} - \frac{54 x^{2} \cot^{10}{\left(x \right)}}{64 x^{6} \cot^{9}{\left(x \right)} - 192 x^{6} \cot^{7}{\left(x \right)} + 192 x^{6} \cot^{5}{\left(x \right)} - 64 x^{6} \cot^{3}{\left(x \right)} + 64 \cot^{9}{\left(x \right)} - 192 \cot^{7}{\left(x \right)} + 192 \cot^{5}{\left(x \right)} - 64 \cot^{3}{\left(x \right)}} + \frac{333 x^{2} \cot^{8}{\left(x \right)}}{64 x^{6} \cot^{9}{\left(x \right)} - 192 x^{6} \cot^{7}{\left(x \right)} + 192 x^{6} \cot^{5}{\left(x \right)} - 64 x^{6} \cot^{3}{\left(x \right)} + 64 \cot^{9}{\left(x \right)} - 192 \cot^{7}{\left(x \right)} + 192 \cot^{5}{\left(x \right)} - 64 \cot^{3}{\left(x \right)}} - \frac{756 x^{2} \cot^{6}{\left(x \right)}}{64 x^{6} \cot^{9}{\left(x \right)} - 192 x^{6} \cot^{7}{\left(x \right)} + 192 x^{6} \cot^{5}{\left(x \right)} - 64 x^{6} \cot^{3}{\left(x \right)} + 64 \cot^{9}{\left(x \right)} - 192 \cot^{7}{\left(x \right)} + 192 \cot^{5}{\left(x \right)} - 64 \cot^{3}{\left(x \right)}} + \frac{333 x^{2} \cot^{4}{\left(x \right)}}{64 x^{6} \cot^{9}{\left(x \right)} - 192 x^{6} \cot^{7}{\left(x \right)} + 192 x^{6} \cot^{5}{\left(x \right)} - 64 x^{6} \cot^{3}{\left(x \right)} + 64 \cot^{9}{\left(x \right)} - 192 \cot^{7}{\left(x \right)} + 192 \cot^{5}{\left(x \right)} - 64 \cot^{3}{\left(x \right)}} - \frac{54 x^{2} \cot^{2}{\left(x \right)}}{64 x^{6} \cot^{9}{\left(x \right)} - 192 x^{6} \cot^{7}{\left(x \right)} + 192 x^{6} \cot^{5}{\left(x \right)} - 64 x^{6} \cot^{3}{\left(x \right)} + 64 \cot^{9}{\left(x \right)} - 192 \cot^{7}{\left(x \right)} + 192 \cot^{5}{\left(x \right)} - 64 \cot^{3}{\left(x \right)}} + \frac{3 x^{2}}{64 x^{6} \cot^{9}{\left(x \right)} - 192 x^{6} \cot^{7}{\left(x \right)} + 192 x^{6} \cot^{5}{\left(x \right)} - 64 x^{6} \cot^{3}{\left(x \right)} + 64 \cot^{9}{\left(x \right)} - 192 \cot^{7}{\left(x \right)} + 192 \cot^{5}{\left(x \right)} - 64 \cot^{3}{\left(x \right)}} - \frac{12 \cot^{16}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} + \frac{288 \cot^{14}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} - \frac{2640 \cot^{12}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} + \frac{11232 \cot^{10}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} - \frac{20808 \cot^{8}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} + \frac{11232 \cot^{6}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} - \frac{2640 \cot^{4}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} + \frac{288 \cot^{2}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} - \frac{12 \operatorname{atan}{\left(x^{3} \right)}}{256 \cot^{12}{\left(x \right)} - 1024 \cot^{10}{\left(x \right)} + 1536 \cot^{8}{\left(x \right)} - 1024 \cot^{6}{\left(x \right)} + 256 \cot^{4}{\left(x \right)}} - \frac{12 \cot^{8}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{16 \cot^{6}{\left(x \right)} - 32 \cot^{4}{\left(x \right)} + 16 \cot^{2}{\left(x \right)}} + \frac{144 \cot^{6}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{16 \cot^{6}{\left(x \right)} - 32 \cot^{4}{\left(x \right)} + 16 \cot^{2}{\left(x \right)}} - \frac{456 \cot^{4}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{16 \cot^{6}{\left(x \right)} - 32 \cot^{4}{\left(x \right)} + 16 \cot^{2}{\left(x \right)}} + \frac{144 \cot^{2}{\left(x \right)} \operatorname{atan}{\left(x^{3} \right)}}{16 \cot^{6}{\left(x \right)} - 32 \cot^{4}{\left(x \right)} + 16 \cot^{2}{\left(x \right)}} - \frac{12 \operatorname{atan}{\left(x^{3} \right)}}{16 \cot^{6}{\left(x \right)} - 32 \cot^{4}{\left(x \right)} + 16 \cot^{2}{\left(x \right)}}$$ 
-12*atan(x^3)/(-32*cot(x)^4 + 16*cot(x)^2 + 16*cot(x)^6) - 12*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) + 3*x^2/(-192*cot(x)^7 - 64*cot(x)^3 + 64*cot(x)^9 + 192*cot(x)^5 - 192*x^6*cot(x)^7 - 64*x^6*cot(x)^3 + 64*x^6*cot(x)^9 + 192*x^6*cot(x)^5) - 20808*cot(x)^8*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) - 2640*cot(x)^4*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) - 2640*cot(x)^12*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) - 756*x^2*cot(x)^6/(-192*cot(x)^7 - 64*cot(x)^3 + 64*cot(x)^9 + 192*cot(x)^5 - 192*x^6*cot(x)^7 - 64*x^6*cot(x)^3 + 64*x^6*cot(x)^9 + 192*x^6*cot(x)^5) - 456*cot(x)^4*atan(x^3)/(-32*cot(x)^4 + 16*cot(x)^2 + 16*cot(x)^6) - 54*x^2*cot(x)^2/(-192*cot(x)^7 - 64*cot(x)^3 + 64*cot(x)^9 + 192*cot(x)^5 - 192*x^6*cot(x)^7 - 64*x^6*cot(x)^3 + 64*x^6*cot(x)^9 + 192*x^6*cot(x)^5) - 54*x^2*cot(x)^10/(-192*cot(x)^7 - 64*cot(x)^3 + 64*cot(x)^9 + 192*cot(x)^5 - 192*x^6*cot(x)^7 - 64*x^6*cot(x)^3 + 64*x^6*cot(x)^9 + 192*x^6*cot(x)^5) - 12*cot(x)^8*atan(x^3)/(-32*cot(x)^4 + 16*cot(x)^2 + 16*cot(x)^6) - 12*cot(x)^16*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) + 3*x^2*cot(x)^12/(-192*cot(x)^7 - 64*cot(x)^3 + 64*cot(x)^9 + 192*cot(x)^5 - 192*x^6*cot(x)^7 - 64*x^6*cot(x)^3 + 64*x^6*cot(x)^9 + 192*x^6*cot(x)^5) + 144*cot(x)^2*atan(x^3)/(-32*cot(x)^4 + 16*cot(x)^2 + 16*cot(x)^6) + 144*cot(x)^6*atan(x^3)/(-32*cot(x)^4 + 16*cot(x)^2 + 16*cot(x)^6) + 288*cot(x)^2*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) + 288*cot(x)^14*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) + 333*x^2*cot(x)^4/(-192*cot(x)^7 - 64*cot(x)^3 + 64*cot(x)^9 + 192*cot(x)^5 - 192*x^6*cot(x)^7 - 64*x^6*cot(x)^3 + 64*x^6*cot(x)^9 + 192*x^6*cot(x)^5) + 333*x^2*cot(x)^8/(-192*cot(x)^7 - 64*cot(x)^3 + 64*cot(x)^9 + 192*cot(x)^5 - 192*x^6*cot(x)^7 - 64*x^6*cot(x)^3 + 64*x^6*cot(x)^9 + 192*x^6*cot(x)^5) + 11232*cot(x)^6*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8) + 11232*cot(x)^10*atan(x^3)/(-1024*cot(x)^6 - 1024*cot(x)^10 + 256*cot(x)^4 + 256*cot(x)^12 + 1536*cot(x)^8)
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