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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(5*x)/(x+3)-5*log((x+3)/2)*sin(5*x)

¿Cómo vas a descomponer esta cos(5*x)/(x+3)-5*log((x+3)/2)*sin(5*x) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
cos(5*x)        /x + 3\         
-------- - 5*log|-----|*sin(5*x)
 x + 3          \  2  /
−5log⁡(x+32)sin⁡(5x)+cos⁡(5x)x+3- 5 \log{\left(\frac{x + 3}{2} \right)} \sin{\left(5 x \right)} + \frac{\cos{\left(5 x \right)}}{x + 3} 
cos(5*x)/(x + 3) - 5*log((x + 3)/2)*sin(5*x)
Simplificación general [src] 
               /3   x\                    
- 5*(3 + x)*log|- + -|*sin(5*x) + cos(5*x)
               \2   2/                    
------------------------------------------
                  3 + x
−5(x+3)log⁡(x2+32)sin⁡(5x)+cos⁡(5x)x+3\frac{- 5 \left(x + 3\right) \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} + \cos{\left(5 x \right)}}{x + 3} 
(-5*(3 + x)*log(3/2 + x/2)*sin(5*x) + cos(5*x))/(3 + x)
Denominador racional [src] 
        /3   x\                   /3   x\                    
- 15*log|- + -|*sin(5*x) - 5*x*log|- + -|*sin(5*x) + cos(5*x)
        \2   2/                   \2   2/                    
-------------------------------------------------------------
                            3 + x
−5xlog⁡(x2+32)sin⁡(5x)−15log⁡(x2+32)sin⁡(5x)+cos⁡(5x)x+3\frac{- 5 x \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} - 15 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} + \cos{\left(5 x \right)}}{x + 3} 
(-15*log(3/2 + x/2)*sin(5*x) - 5*x*log(3/2 + x/2)*sin(5*x) + cos(5*x))/(3 + x)
Denominador común [src] 
cos(5*x)                                            
-------- - 5*log(3 + x)*sin(5*x) + 5*log(2)*sin(5*x)
 3 + x
−5log⁡(x+3)sin⁡(5x)+5log⁡(2)sin⁡(5x)+cos⁡(5x)x+3- 5 \log{\left(x + 3 \right)} \sin{\left(5 x \right)} + 5 \log{\left(2 \right)} \sin{\left(5 x \right)} + \frac{\cos{\left(5 x \right)}}{x + 3} 
cos(5*x)/(3 + x) - 5*log(3 + x)*sin(5*x) + 5*log(2)*sin(5*x)
Abrimos la expresión [src] 
                                                                            3                       5                                                                   
         3                   5                                        20*cos (x)   5*cos(x)   16*cos (x)                            5                    3              
- 100*sin (x)*log(2) - 80*sin (x)*log(x + 3) - 25*log(x + 3)*sin(x) - ---------- + -------- + ---------- + 25*log(2)*sin(x) + 80*sin (x)*log(2) + 100*sin (x)*log(x + 3)
                                                                        x + 3       x + 3       x + 3
−80log⁡(x+3)sin⁡5(x)+100log⁡(x+3)sin⁡3(x)−25log⁡(x+3)sin⁡(x)+80log⁡(2)sin⁡5(x)−100log⁡(2)sin⁡3(x)+25log⁡(2)sin⁡(x)+16cos⁡5(x)x+3−20cos⁡3(x)x+3+5cos⁡(x)x+3- 80 \log{\left(x + 3 \right)} \sin^{5}{\left(x \right)} + 100 \log{\left(x + 3 \right)} \sin^{3}{\left(x \right)} - 25 \log{\left(x + 3 \right)} \sin{\left(x \right)} + 80 \log{\left(2 \right)} \sin^{5}{\left(x \right)} - 100 \log{\left(2 \right)} \sin^{3}{\left(x \right)} + 25 \log{\left(2 \right)} \sin{\left(x \right)} + \frac{16 \cos^{5}{\left(x \right)}}{x + 3} - \frac{20 \cos^{3}{\left(x \right)}}{x + 3} + \frac{5 \cos{\left(x \right)}}{x + 3} 
-100*sin(x)^3*log(2) - 80*sin(x)^5*log(x + 3) - 25*log(x + 3)*sin(x) - 20*cos(x)^3/(x + 3) + 5*cos(x)/(x + 3) + 16*cos(x)^5/(x + 3) + 25*log(2)*sin(x) + 80*sin(x)^5*log(2) + 100*sin(x)^3*log(x + 3)
Potencias [src] 
cos(5*x)        /3   x\         
-------- - 5*log|- + -|*sin(5*x)
 3 + x          \2   2/
−5log⁡(x2+32)sin⁡(5x)+cos⁡(5x)x+3- 5 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} + \frac{\cos{\left(5 x \right)}}{x + 3} 
 -5*I*x    5*I*x                                      
e         e            /   -5*I*x    5*I*x\    /3   x\
------- + ------   5*I*\- e       + e     /*log|- + -|
   2        2                                  \2   2/
---------------- + -----------------------------------
     3 + x                          2
5i(e5ix−e−5ix)log⁡(x2+32)2+e5ix2+e−5ix2x+3\frac{5 i \left(e^{5 i x} - e^{- 5 i x}\right) \log{\left(\frac{x}{2} + \frac{3}{2} \right)}}{2} + \frac{\frac{e^{5 i x}}{2} + \frac{e^{- 5 i x}}{2}}{x + 3} 
(exp(-5*i*x)/2 + exp(5*i*x)/2)/(3 + x) + 5*i*(-exp(-5*i*x) + exp(5*i*x))*log(3/2 + x/2)/2
Respuesta numérica [src] 
cos(5*x)/(3.0 + x) - 5.0*log((x + 3)/2)*sin(5*x)
cos(5*x)/(3.0 + x) - 5.0*log((x + 3)/2)*sin(5*x)
Combinatoria [src] 
 /                  /3   x\                   /3   x\         \ 
-|-cos(5*x) + 15*log|- + -|*sin(5*x) + 5*x*log|- + -|*sin(5*x)| 
 \                  \2   2/                   \2   2/         / 
----------------------------------------------------------------
                             3 + x
−5xlog⁡(x2+32)sin⁡(5x)+15log⁡(x2+32)sin⁡(5x)−cos⁡(5x)x+3- \frac{5 x \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} + 15 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} - \cos{\left(5 x \right)}}{x + 3} 
-(-cos(5*x) + 15*log(3/2 + x/2)*sin(5*x) + 5*x*log(3/2 + x/2)*sin(5*x))/(3 + x)
Parte trigonométrica [src] 
   /pi      \                        
sin|-- + 5*x|                        
   \2       /        /3   x\         
------------- - 5*log|- + -|*sin(5*x)
    3 + x            \2   2/
−5log⁡(x2+32)sin⁡(5x)+sin⁡(5x+π2)x+3- 5 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} + \frac{\sin{\left(5 x + \frac{\pi}{2} \right)}}{x + 3} 
             2/5*x\             /5*x\    /3   x\
     -1 + cot |---|       10*cot|---|*log|- + -|
              \ 2 /             \ 2 /    \2   2/
----------------------- - ----------------------
/       2/5*x\\                      2/5*x\     
|1 + cot |---||*(3 + x)       1 + cot |---|     
\        \ 2 //                       \ 2 /
−10log⁡(x2+32)cot⁡(5x2)cot⁡2(5x2)+1+cot⁡2(5x2)−1(x+3)(cot⁡2(5x2)+1)- \frac{10 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \cot{\left(\frac{5 x}{2} \right)}}{\cot^{2}{\left(\frac{5 x}{2} \right)} + 1} + \frac{\cot^{2}{\left(\frac{5 x}{2} \right)} - 1}{\left(x + 3\right) \left(\cot^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} 
                        /3   x\
                   5*log|- + -|
       1                \2   2/
---------------- - ------------
(3 + x)*sec(5*x)     csc(5*x)
−5log⁡(x2+32)csc⁡(5x)+1(x+3)sec⁡(5x)- \frac{5 \log{\left(\frac{x}{2} + \frac{3}{2} \right)}}{\csc{\left(5 x \right)}} + \frac{1}{\left(x + 3\right) \sec{\left(5 x \right)}} 
cos(5*x)        /      pi\    /3   x\
-------- - 5*cos|5*x - --|*log|- + -|
 3 + x          \      2 /    \2   2/
−5log⁡(x2+32)cos⁡(5x−π2)+cos⁡(5x)x+3- 5 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \cos{\left(5 x - \frac{\pi}{2} \right)} + \frac{\cos{\left(5 x \right)}}{x + 3} 
            2/5*x\              /3   x\    /5*x\
     1 - tan |---|        10*log|- + -|*tan|---|
             \ 2 /              \2   2/    \ 2 /
----------------------- - ----------------------
/       2/5*x\\                      2/5*x\     
|1 + tan |---||*(3 + x)       1 + tan |---|     
\        \ 2 //                       \ 2 /
1−tan⁡2(5x2)(x+3)(tan⁡2(5x2)+1)−10log⁡(x2+32)tan⁡(5x2)tan⁡2(5x2)+1\frac{1 - \tan^{2}{\left(\frac{5 x}{2} \right)}}{\left(x + 3\right) \left(\tan^{2}{\left(\frac{5 x}{2} \right)} + 1\right)} - \frac{10 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \tan{\left(\frac{5 x}{2} \right)}}{\tan^{2}{\left(\frac{5 x}{2} \right)} + 1} 
                             /3   x\
                        5*log|- + -|
          1                  \2   2/
--------------------- - ------------
           /pi      \     csc(5*x)  
(3 + x)*csc|-- - 5*x|               
           \2       /
−5log⁡(x2+32)csc⁡(5x)+1(x+3)csc⁡(−5x+π2)- \frac{5 \log{\left(\frac{x}{2} + \frac{3}{2} \right)}}{\csc{\left(5 x \right)}} + \frac{1}{\left(x + 3\right) \csc{\left(- 5 x + \frac{\pi}{2} \right)}} 
cos(5*x)        /3   x\         
-------- - 5*log|- + -|*sin(5*x)
 3 + x          \2   2/
−5log⁡(x2+32)sin⁡(5x)+cos⁡(5x)x+3- 5 \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} + \frac{\cos{\left(5 x \right)}}{x + 3} 
                         /3   x\
                    5*log|- + -|
       1                 \2   2/
---------------- - -------------
(3 + x)*sec(5*x)      /      pi\
                   sec|5*x - --|
                      \      2 /
−5log⁡(x2+32)sec⁡(5x−π2)+1(x+3)sec⁡(5x)- \frac{5 \log{\left(\frac{x}{2} + \frac{3}{2} \right)}}{\sec{\left(5 x - \frac{\pi}{2} \right)}} + \frac{1}{\left(x + 3\right) \sec{\left(5 x \right)}} 
1/((3 + x)*sec(5*x)) - 5*log(3/2 + x/2)/sec(5*x - pi/2)
Unión de expresiones racionales [src] 
               /3   x\                    
- 5*(3 + x)*log|- + -|*sin(5*x) + cos(5*x)
               \2   2/                    
------------------------------------------
                  3 + x
−5(x+3)log⁡(x2+32)sin⁡(5x)+cos⁡(5x)x+3\frac{- 5 \left(x + 3\right) \log{\left(\frac{x}{2} + \frac{3}{2} \right)} \sin{\left(5 x \right)} + \cos{\left(5 x \right)}}{x + 3} 
(-5*(3 + x)*log(3/2 + x/2)*sin(5*x) + cos(5*x))/(3 + x)
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