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Simplificación de expresiones/ Descomposición en fracciones simples/ atan((2*sin(15*x))/(1+cos(15*x)))/30

¿Cómo vas a descomponer esta atan((2*sin(15*x))/(1+cos(15*x)))/30 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    / 2*sin(15*x) \
atan|-------------|
    \1 + cos(15*x)/
-------------------
         30
atan⁡(2sin⁡(15x)cos⁡(15x)+1)30\frac{\operatorname{atan}{\left(\frac{2 \sin{\left(15 x \right)}}{\cos{\left(15 x \right)} + 1} \right)}}{30} 
atan((2*sin(15*x))/(1 + cos(15*x)))/30
Descomposición de una fracción [src] 
atan((2*sin(15*x))/(1 + cos(15*x)))/30
atan⁡(2sin⁡(15x)cos⁡(15x)+1)30\frac{\operatorname{atan}{\left(\frac{2 \sin{\left(15 x \right)}}{\cos{\left(15 x \right)} + 1} \right)}}{30} 
    / 2*sin(15*x) \
atan|-------------|
    \1 + cos(15*x)/
-------------------
         30
Respuesta numérica [src] 
0.0333333333333333*atan((2*sin(15*x))/(1 + cos(15*x)))
0.0333333333333333*atan((2*sin(15*x))/(1 + cos(15*x)))
Potencias [src] 
     /  /   -15*I*x    15*I*x\\ 
     |I*\- e        + e      /| 
-atan|------------------------| 
     |      -15*I*x    15*I*x | 
     |     e          e       | 
     | 1 + -------- + ------- | 
     \        2          2    / 
--------------------------------
               30
−atan⁡(i(e15ix−e−15ix)e15ix2+1+e−15ix2)30- \frac{\operatorname{atan}{\left(\frac{i \left(e^{15 i x} - e^{- 15 i x}\right)}{\frac{e^{15 i x}}{2} + 1 + \frac{e^{- 15 i x}}{2}} \right)}}{30} 
-atan(i*(-exp(-15*i*x) + exp(15*i*x))/(1 + exp(-15*i*x)/2 + exp(15*i*x)/2))/30
Abrimos la expresión [src] 
     /                                                                    9                                                                                                                              13                                                                                                                              5                                                                                                                                                                                                                                                               3                                                                                                                               15                                                                                                                              7                                                                                                                               11                                                          \ 
     |                                                          140800*sin (x)                                                                                                                 122880*sin  (x)                                                                                                                  12096*sin (x)                                                                                                                     30*sin(x)                                                                                                                      1120*sin (x)                                                                                                                   32768*sin  (x)                                                                                                                  57600*sin (x)                                                                                                                  184320*sin  (x)                                                       | 
-atan|- ----------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------------------------| 
     |               9               13              5                         3               15               7               11                   9               13              5                         3               15               7               11                   9               13              5                         3               15               7               11                   9               13              5                         3               15               7               11                   9               13              5                         3               15               7               11                   9               13              5                         3               15               7               11                   9               13              5                         3               15               7               11                   9               13              5                         3               15               7               11   | 
     \  1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)   1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)   1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)   1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)   1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)   1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)   1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)   1 - 70400*cos (x) - 61440*cos  (x) - 6048*cos (x) - 15*cos(x) + 560*cos (x) + 16384*cos  (x) + 28800*cos (x) + 92160*cos  (x)/ 
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   30
−atan⁡(32768sin⁡15(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1−122880sin⁡13(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1+184320sin⁡11(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1−140800sin⁡9(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1+57600sin⁡7(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1−12096sin⁡5(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1+1120sin⁡3(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1−30sin⁡(x)16384cos⁡15(x)−61440cos⁡13(x)+92160cos⁡11(x)−70400cos⁡9(x)+28800cos⁡7(x)−6048cos⁡5(x)+560cos⁡3(x)−15cos⁡(x)+1)30- \frac{\operatorname{atan}{\left(\frac{32768 \sin^{15}{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} - \frac{122880 \sin^{13}{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} + \frac{184320 \sin^{11}{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} - \frac{140800 \sin^{9}{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} + \frac{57600 \sin^{7}{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} - \frac{12096 \sin^{5}{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} + \frac{1120 \sin^{3}{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} - \frac{30 \sin{\left(x \right)}}{16384 \cos^{15}{\left(x \right)} - 61440 \cos^{13}{\left(x \right)} + 92160 \cos^{11}{\left(x \right)} - 70400 \cos^{9}{\left(x \right)} + 28800 \cos^{7}{\left(x \right)} - 6048 \cos^{5}{\left(x \right)} + 560 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)} + 1} \right)}}{30} 
-atan(-140800*sin(x)^9/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11) - 122880*sin(x)^13/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11) - 12096*sin(x)^5/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11) - 30*sin(x)/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11) + 1120*sin(x)^3/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11) + 32768*sin(x)^15/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11) + 57600*sin(x)^7/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11) + 184320*sin(x)^11/(1 - 70400*cos(x)^9 - 61440*cos(x)^13 - 6048*cos(x)^5 - 15*cos(x) + 560*cos(x)^3 + 16384*cos(x)^15 + 28800*cos(x)^7 + 92160*cos(x)^11))/30
Parte trigonométrica [src] 
    /              2               \
atan|------------------------------|
    |/          1       \          |
    ||1 + --------------|*csc(15*x)|
    ||       /pi       \|          |
    ||    csc|-- - 15*x||          |
    \\       \2        //          /
------------------------------------
                 30
atan⁡(2(1+1csc⁡(−15x+π2))csc⁡(15x))30\frac{\operatorname{atan}{\left(\frac{2}{\left(1 + \frac{1}{\csc{\left(- 15 x + \frac{\pi}{2} \right)}}\right) \csc{\left(15 x \right)}} \right)}}{30} 
    /   2*sin(15*x)    \
atan|------------------|
    |       /pi       \|
    |1 + sin|-- + 15*x||
    \       \2        //
------------------------
           30
atan⁡(2sin⁡(15x)sin⁡(15x+π2)+1)30\frac{\operatorname{atan}{\left(\frac{2 \sin{\left(15 x \right)}}{\sin{\left(15 x + \frac{\pi}{2} \right)} + 1} \right)}}{30} 
    /     /       pi\\
    |2*cos|15*x - --||
    |     \       2 /|
atan|----------------|
    \ 1 + cos(15*x)  /
----------------------
          30
atan⁡(2cos⁡(15x−π2)cos⁡(15x)+1)30\frac{\operatorname{atan}{\left(\frac{2 \cos{\left(15 x - \frac{\pi}{2} \right)}}{\cos{\left(15 x \right)} + 1} \right)}}{30} 
    /                  /15*x\              \
    |             4*cot|----|              |
    |                  \ 2  /              |
atan|--------------------------------------|
    |                 /            2/15*x\\|
    |                 |    -1 + cot |----|||
    |/       2/15*x\\ |             \ 2  /||
    ||1 + cot |----||*|1 + ---------------||
    |\        \ 2  // |            2/15*x\||
    |                 |     1 + cot |----|||
    \                 \             \ 2  ///
--------------------------------------------
                     30
atan⁡(4cot⁡(15x2)(cot⁡2(15x2)−1cot⁡2(15x2)+1+1)(cot⁡2(15x2)+1))30\frac{\operatorname{atan}{\left(\frac{4 \cot{\left(\frac{15 x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{15 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{15 x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{15 x}{2} \right)} + 1\right)} \right)}}{30} 
    /              2               \
atan|------------------------------|
    |/        1    \    /       pi\|
    ||1 + ---------|*sec|15*x - --||
    \\    sec(15*x)/    \       2 //
------------------------------------
                 30
atan⁡(2(1+1sec⁡(15x))sec⁡(15x−π2))30\frac{\operatorname{atan}{\left(\frac{2}{\left(1 + \frac{1}{\sec{\left(15 x \right)}}\right) \sec{\left(15 x - \frac{\pi}{2} \right)}} \right)}}{30} 
    /                  /15*x\             \
    |             4*tan|----|             |
    |                  \ 2  /             |
atan|-------------------------------------|
    |                 /           2/15*x\\|
    |                 |    1 - tan |----|||
    |/       2/15*x\\ |            \ 2  /||
    ||1 + tan |----||*|1 + --------------||
    |\        \ 2  // |           2/15*x\||
    |                 |    1 + tan |----|||
    \                 \            \ 2  ///
-------------------------------------------
                     30
atan⁡(4tan⁡(15x2)(1−tan⁡2(15x2)tan⁡2(15x2)+1+1)(tan⁡2(15x2)+1))30\frac{\operatorname{atan}{\left(\frac{4 \tan{\left(\frac{15 x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{15 x}{2} \right)}}{\tan^{2}{\left(\frac{15 x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{15 x}{2} \right)} + 1\right)} \right)}}{30} 
    /            2            \
atan|-------------------------|
    |/        1    \          |
    ||1 + ---------|*csc(15*x)|
    \\    sec(15*x)/          /
-------------------------------
               30
atan⁡(2(1+1sec⁡(15x))csc⁡(15x))30\frac{\operatorname{atan}{\left(\frac{2}{\left(1 + \frac{1}{\sec{\left(15 x \right)}}\right) \csc{\left(15 x \right)}} \right)}}{30} 
atan(2/((1 + 1/sec(15*x))*csc(15*x)))/30
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