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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
               /  ___\ /       /  ___\          /  ___\                \
            sin\\/ x / |    sin\\/ x /       cos\\/ x /*log(log(x + 3))|
(log(x + 3))          *|------------------ + --------------------------|
                       |(x + 3)*log(x + 3)                ___          |
                       \                              2*\/ x           /
$$\left(\frac{\sin{\left(\sqrt{x} \right)}}{\left(x + 3\right) \log{\left(x + 3 \right)}} + \frac{\log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}\right) \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}}$$ 
log(x + 3)^sin(sqrt(x))*(sin(sqrt(x))/(((x + 3)*log(x + 3))) + (cos(sqrt(x))*log(log(x + 3)))/((2*sqrt(x))))
Simplificación general [src] 
                    /  ___\ /                              /  ___\                           \
            -1 + sin\\/ x / |  ___    /  ___\   (3 + x)*cos\\/ x /*log(3 + x)*log(log(3 + x))|
(log(3 + x))               *|\/ x *sin\\/ x / + ---------------------------------------------|
                            \                                         2                      /
----------------------------------------------------------------------------------------------
                                          ___                                                 
                                        \/ x *(3 + x)
$$\frac{\left(\sqrt{x} \sin{\left(\sqrt{x} \right)} + \frac{\left(x + 3\right) \log{\left(x + 3 \right)} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}}{2}\right) \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)} - 1}}{\sqrt{x} \left(x + 3\right)}$$ 
log(3 + x)^(-1 + sin(sqrt(x)))*(sqrt(x)*sin(sqrt(x)) + (3 + x)*cos(sqrt(x))*log(3 + x)*log(log(3 + x))/2)/(sqrt(x)*(3 + x))
Potencias [src] 
               /        ___        ___\  //     ___         ___\                                             \
               |   -I*\/ x     I*\/ x |  || I*\/ x     -I*\/ x |                                             |
            -I*\- e         + e       /  ||e          e        |                     /        ___        ___\|
            ---------------------------- ||-------- + ---------|*log(log(3 + x))     |   -I*\/ x     I*\/ x ||
                         2               |\   2           2    /                   I*\- e         + e       /|
(log(3 + x))                            *|-------------------------------------- - --------------------------|
                                         |                   ___                      2*(3 + x)*log(3 + x)   |
                                         \               2*\/ x                                              /
$$\left(- \frac{i \left(e^{i \sqrt{x}} - e^{- i \sqrt{x}}\right)}{2 \left(x + 3\right) \log{\left(x + 3 \right)}} + \frac{\left(\frac{e^{i \sqrt{x}}}{2} + \frac{e^{- i \sqrt{x}}}{2}\right) \log{\left(\log{\left(x + 3 \right)} \right)}}{2 \sqrt{x}}\right) \log{\left(x + 3 \right)}^{- \frac{i \left(e^{i \sqrt{x}} - e^{- i \sqrt{x}}\right)}{2}}$$ 
log(3 + x)^(-i*(-exp(-i*sqrt(x)) + exp(i*sqrt(x)))/2)*((exp(i*sqrt(x))/2 + exp(-i*sqrt(x))/2)*log(log(3 + x))/(2*sqrt(x)) - i*(-exp(-i*sqrt(x)) + exp(i*sqrt(x)))/(2*(3 + x)*log(3 + x)))
Denominador racional [src] 
                            /  ___\                                    /  ___\                                                               /  ___\                                      
    ___             -1 + sin\\/ x /    /  ___\                 -1 + sin\\/ x /    /  ___\                                            -1 + sin\\/ x /    /  ___\                           
2*\/ x *(log(3 + x))               *sin\\/ x / + 3*(log(3 + x))               *cos\\/ x /*log(3 + x)*log(log(3 + x)) + x*(log(3 + x))               *cos\\/ x /*log(3 + x)*log(log(3 + x))
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                         ___                                                                                              
                                                                                     2*\/ x *(3 + x)
$$\frac{2 \sqrt{x} \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)} - 1} \sin{\left(\sqrt{x} \right)} + x \log{\left(x + 3 \right)} \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)} - 1} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)} + 3 \log{\left(x + 3 \right)} \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)} - 1} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x} \left(x + 3\right)}$$ 
(2*sqrt(x)*log(3 + x)^(-1 + sin(sqrt(x)))*sin(sqrt(x)) + 3*log(3 + x)^(-1 + sin(sqrt(x)))*cos(sqrt(x))*log(3 + x)*log(log(3 + x)) + x*log(3 + x)^(-1 + sin(sqrt(x)))*cos(sqrt(x))*log(3 + x)*log(log(3 + x)))/(2*sqrt(x)*(3 + x))
Unión de expresiones racionales [src] 
               /  ___\                                                                     
            sin\\/ x / /    ___    /  ___\              /  ___\                           \
(log(3 + x))          *\2*\/ x *sin\\/ x / + (3 + x)*cos\\/ x /*log(3 + x)*log(log(3 + x))/
-------------------------------------------------------------------------------------------
                                     ___                                                   
                                 2*\/ x *(3 + x)*log(3 + x)
$$\frac{\left(2 \sqrt{x} \sin{\left(\sqrt{x} \right)} + \left(x + 3\right) \log{\left(x + 3 \right)} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}\right) \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}}}{2 \sqrt{x} \left(x + 3\right) \log{\left(x + 3 \right)}}$$ 
log(3 + x)^sin(sqrt(x))*(2*sqrt(x)*sin(sqrt(x)) + (3 + x)*cos(sqrt(x))*log(3 + x)*log(log(3 + x)))/(2*sqrt(x)*(3 + x)*log(3 + x))
Respuesta numérica [src] 
log(x + 3)^sin(sqrt(x))*(sin(sqrt(x))/((3.0 + x)*log(x + 3)) + 0.5*x^(-0.5)*cos(sqrt(x))*log(log(x + 3)))
log(x + 3)^sin(sqrt(x))*(sin(sqrt(x))/((3.0 + x)*log(x + 3)) + 0.5*x^(-0.5)*cos(sqrt(x))*log(log(x + 3)))
Combinatoria [src] 
               /  ___\                                                                                                         
            sin\\/ x / /    ___    /  ___\        /  ___\                                   /  ___\                           \
(log(3 + x))          *\2*\/ x *sin\\/ x / + 3*cos\\/ x /*log(3 + x)*log(log(3 + x)) + x*cos\\/ x /*log(3 + x)*log(log(3 + x))/
-------------------------------------------------------------------------------------------------------------------------------
                                                       ___                                                                     
                                                   2*\/ x *(3 + x)*log(3 + x)
$$\frac{\left(2 \sqrt{x} \sin{\left(\sqrt{x} \right)} + x \log{\left(x + 3 \right)} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)} + 3 \log{\left(x + 3 \right)} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}\right) \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}}}{2 \sqrt{x} \left(x + 3\right) \log{\left(x + 3 \right)}}$$ 
log(3 + x)^sin(sqrt(x))*(2*sqrt(x)*sin(sqrt(x)) + 3*cos(sqrt(x))*log(3 + x)*log(log(3 + x)) + x*cos(sqrt(x))*log(3 + x)*log(log(3 + x)))/(2*sqrt(x)*(3 + x)*log(3 + x))
Denominador común [src] 
                       /  ___\                               /  ___\                                                          /  ___\                                      
    ___             sin\\/ x /    /  ___\                 sin\\/ x /    /  ___\                                            sin\\/ x /    /  ___\                           
2*\/ x *(log(3 + x))          *sin\\/ x / + 3*(log(3 + x))          *cos\\/ x /*log(3 + x)*log(log(3 + x)) + x*(log(3 + x))          *cos\\/ x /*log(3 + x)*log(log(3 + x))
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                      3/2                  ___                                                                             
                                                                   2*x   *log(3 + x) + 6*\/ x *log(3 + x)
$$\frac{2 \sqrt{x} \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}} \sin{\left(\sqrt{x} \right)} + x \log{\left(x + 3 \right)} \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)} + 3 \log{\left(x + 3 \right)} \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}}{2 x^{\frac{3}{2}} \log{\left(x + 3 \right)} + 6 \sqrt{x} \log{\left(x + 3 \right)}}$$ 
(2*sqrt(x)*log(3 + x)^sin(sqrt(x))*sin(sqrt(x)) + 3*log(3 + x)^sin(sqrt(x))*cos(sqrt(x))*log(3 + x)*log(log(3 + x)) + x*log(3 + x)^sin(sqrt(x))*cos(sqrt(x))*log(3 + x)*log(log(3 + x)))/(2*x^(3/2)*log(3 + x) + 6*sqrt(x)*log(3 + x))
Abrimos la expresión [src] 
               /  ___\                             /  ___\                           
            sin\\/ x /    /  ___\               sin\\/ x /    /  ___\                
(log(x + 3))          *sin\\/ x /   (log(x + 3))          *cos\\/ x /*log(log(x + 3))
--------------------------------- + -------------------------------------------------
   3*log(x + 3) + x*log(x + 3)                               ___                     
                                                         2*\/ x
$$\frac{\log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}} \sin{\left(\sqrt{x} \right)}}{x \log{\left(x + 3 \right)} + 3 \log{\left(x + 3 \right)}} + \frac{\log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$$ 
log(x + 3)^sin(sqrt(x))*sin(sqrt(x))/(3*log(x + 3) + x*log(x + 3)) + log(x + 3)^sin(sqrt(x))*cos(sqrt(x))*log(log(x + 3))/(2*sqrt(x))
Parte trigonométrica [src] 
                   1                                                                 
            ---------------                                                          
               /  ___   pi\                                                          
            sec|\/ x  - --|                                                          
               \        2 / /                1                     log(log(3 + x))  \
(log(3 + x))               *|---------------------------------- + ------------------|
                            |                      /  ___   pi\       ___    /  ___\|
                            |(3 + x)*log(3 + x)*sec|\/ x  - --|   2*\/ x *sec\\/ x /|
                            \                      \        2 /                     /
$$\left(\frac{1}{\left(x + 3\right) \log{\left(x + 3 \right)} \sec{\left(\sqrt{x} - \frac{\pi}{2} \right)}} + \frac{\log{\left(\log{\left(x + 3 \right)} \right)}}{2 \sqrt{x} \sec{\left(\sqrt{x} \right)}}\right) \log{\left(x + 3 \right)}^{\frac{1}{\sec{\left(\sqrt{x} - \frac{\pi}{2} \right)}}}$$ 
                       /                                        /  ___   pi\\
               /  ___\ |       /  ___\       log(log(3 + x))*sin|\/ x  + --||
            sin\\/ x / |    sin\\/ x /                          \        2 /|
(log(3 + x))          *|------------------ + -------------------------------|
                       |(3 + x)*log(3 + x)                   ___            |
                       \                                 2*\/ x             /
$$\left(\frac{\sin{\left(\sqrt{x} \right)}}{\left(x + 3\right) \log{\left(x + 3 \right)}} + \frac{\log{\left(\log{\left(x + 3 \right)} \right)} \sin{\left(\sqrt{x} + \frac{\pi}{2} \right)}}{2 \sqrt{x}}\right) \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}}$$ 
               /  ___   pi\ /    /  ___   pi\                               \
            cos|\/ x  - --| | cos|\/ x  - --|        /  ___\                |
               \        2 / |    \        2 /     cos\\/ x /*log(log(3 + x))|
(log(3 + x))               *|------------------ + --------------------------|
                            |(3 + x)*log(3 + x)                ___          |
                            \                              2*\/ x           /
$$\left(\frac{\cos{\left(\sqrt{x} - \frac{\pi}{2} \right)}}{\left(x + 3\right) \log{\left(x + 3 \right)}} + \frac{\log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}\right) \log{\left(x + 3 \right)}^{\cos{\left(\sqrt{x} - \frac{\pi}{2} \right)}}$$ 
                1                                                          
            ----------                                                     
               /  ___\                                                     
            csc\\/ x / /              1                  log(log(3 + x))  \
(log(3 + x))          *|----------------------------- + ------------------|
                       |           /  ___\                  ___    /  ___\|
                       \(3 + x)*csc\\/ x /*log(3 + x)   2*\/ x *sec\\/ x //
$$\left(\frac{1}{\left(x + 3\right) \log{\left(x + 3 \right)} \csc{\left(\sqrt{x} \right)}} + \frac{\log{\left(\log{\left(x + 3 \right)} \right)}}{2 \sqrt{x} \sec{\left(\sqrt{x} \right)}}\right) \log{\left(x + 3 \right)}^{\frac{1}{\csc{\left(\sqrt{x} \right)}}}$$ 
               /  ___\                                                                     
            sin\\/ x / /        1             /  ___\      1       /  ___\                \
(log(x + 3))          *|------------------*sin\\/ x / + -------*cos\\/ x /*log(log(x + 3))|
                       |(x + 3)*log(x + 3)                  ___                           |
                       \                                2*\/ x                            /
$$\left(\frac{1}{2 \sqrt{x}} \log{\left(\log{\left(x + 3 \right)} \right)} \cos{\left(\sqrt{x} \right)} + \frac{1}{\left(x + 3\right) \log{\left(x + 3 \right)}} \sin{\left(\sqrt{x} \right)}\right) \log{\left(x + 3 \right)}^{\sin{\left(\sqrt{x} \right)}}$$ 
                1                                                               
            ----------                                                          
               /  ___\                                                          
            csc\\/ x / /              1                     log(log(3 + x))    \
(log(3 + x))          *|----------------------------- + -----------------------|
                       |           /  ___\                  ___    /pi     ___\|
                       |(3 + x)*csc\\/ x /*log(3 + x)   2*\/ x *csc|-- - \/ x ||
                       \                                           \2         //
$$\left(\frac{1}{\left(x + 3\right) \log{\left(x + 3 \right)} \csc{\left(\sqrt{x} \right)}} + \frac{\log{\left(\log{\left(x + 3 \right)} \right)}}{2 \sqrt{x} \csc{\left(- \sqrt{x} + \frac{\pi}{2} \right)}}\right) \log{\left(x + 3 \right)}^{\frac{1}{\csc{\left(\sqrt{x} \right)}}}$$ 
                   /  ___\                                                                             
                   |\/ x |                                                                             
              2*cot|-----|                                                                             
                   \  2  /                                                                             
            ---------------                                                                            
                    /  ___\ //         /  ___\\                                    /  ___\            \
                   2|\/ x | ||        2|\/ x ||                                    |\/ x |            |
            1 + cot |-----| ||-1 + cot |-----||*log(log(3 + x))               2*cot|-----|            |
                    \  2  / |\         \  2  //                                    \  2  /            |
(log(3 + x))               *|---------------------------------- + ------------------------------------|
                            |            /        /  ___\\        /        /  ___\\                   |
                            |        ___ |       2|\/ x ||        |       2|\/ x ||                   |
                            |    2*\/ x *|1 + cot |-----||        |1 + cot |-----||*(3 + x)*log(3 + x)|
                            \            \        \  2  //        \        \  2  //                   /
$$\left(\frac{2 \cot{\left(\frac{\sqrt{x}}{2} \right)}}{\left(x + 3\right) \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right) \log{\left(x + 3 \right)}} + \frac{\left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} - 1\right) \log{\left(\log{\left(x + 3 \right)} \right)}}{2 \sqrt{x} \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)}\right) \log{\left(x + 3 \right)}^{\frac{2 \cot{\left(\frac{\sqrt{x}}{2} \right)}}{\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}}$$ 
                   /  ___\                                                                            
                   |\/ x |                                                                            
              2*tan|-----|                                                                            
                   \  2  /                                                                            
            ---------------                                                                           
                    /  ___\ //        /  ___\\                                    /  ___\            \
                   2|\/ x | ||       2|\/ x ||                                    |\/ x |            |
            1 + tan |-----| ||1 - tan |-----||*log(log(3 + x))               2*tan|-----|            |
                    \  2  / |\        \  2  //                                    \  2  /            |
(log(3 + x))               *|--------------------------------- + ------------------------------------|
                            |            /        /  ___\\       /        /  ___\\                   |
                            |        ___ |       2|\/ x ||       |       2|\/ x ||                   |
                            |    2*\/ x *|1 + tan |-----||       |1 + tan |-----||*(3 + x)*log(3 + x)|
                            \            \        \  2  //       \        \  2  //                   /
$$\left(\frac{2 \tan{\left(\frac{\sqrt{x}}{2} \right)}}{\left(x + 3\right) \left(\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right) \log{\left(x + 3 \right)}} + \frac{\left(1 - \tan^{2}{\left(\frac{\sqrt{x}}{2} \right)}\right) \log{\left(\log{\left(x + 3 \right)} \right)}}{2 \sqrt{x} \left(\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)}\right) \log{\left(x + 3 \right)}^{\frac{2 \tan{\left(\frac{\sqrt{x}}{2} \right)}}{\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1}}$$ 
log(3 + x)^(2*tan(sqrt(x)/2)/(1 + tan(sqrt(x)/2)^2))*((1 - tan(sqrt(x)/2)^2)*log(log(3 + x))/(2*sqrt(x)*(1 + tan(sqrt(x)/2)^2)) + 2*tan(sqrt(x)/2)/((1 + tan(sqrt(x)/2)^2)*(3 + x)*log(3 + x)))
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