Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)^(1/(x^2))*(-2*log(sin(x))/x^3+cos(x)/(x^2*sin(x)))
¿Cómo vas a descomponer esta sin(x)^(1/(x^2))*(-2*log(sin(x))/x^3+cos(x)/(x^2*sin(x))) expresión en fracciones?
Solución
Ha introducido
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1
--
2
x /-2*log(sin(x)) cos(x) \
(sin(x)) *|-------------- + ---------|
| 3 2 |
\ x x *sin(x)/
$$\left(\frac{\cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} + \frac{\left(-1\right) 2 \log{\left(\sin{\left(x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}$$
sin(x)^(1/(x^2))*((-2*log(sin(x)))/x^3 + cos(x)/((x^2*sin(x))))
Simplificación general
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1
-1 + --
2
x
(sin(x)) *(x*cos(x) - 2*log(sin(x))*sin(x))
-------------------------------------------------
3
x
$$\frac{\left(x \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) \sin^{-1 + \frac{1}{x^{2}}}{\left(x \right)}}{x^{3}}$$
sin(x)^(-1 + x^(-2))*(x*cos(x) - 2*log(sin(x))*sin(x))/x^3
Respuesta numérica
[src]
sin(x)^(x^(-2))*(-2.0*log(sin(x))/x^3 + cos(x)/(x^2*sin(x)))
sin(x)^(x^(-2))*(-2.0*log(sin(x))/x^3 + cos(x)/(x^2*sin(x)))
Combinatoria
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1
--
2
x
(sin(x)) *(x*cos(x) - 2*log(sin(x))*sin(x))
--------------------------------------------
3
x *sin(x)
$$\frac{\left(x \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{3} \sin{\left(x \right)}}$$
sin(x)^(x^(-2))*(x*cos(x) - 2*log(sin(x))*sin(x))/(x^3*sin(x))
Potencias
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1
--
2 / / / -I*x I*x\ \ / I*x -I*x\\
x | |-I*\- e + e / | |e e ||
/ / -I*x I*x\ \ | 2*log|--------------------| 2*I*|---- + -----||
|-I*\- e + e / | | \ 2 / \ 2 2 /|
|--------------------| *|- --------------------------- + -------------------|
\ 2 / | 3 2 / -I*x I*x\|
\ x x *\- e + e //
$$\left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)^{\frac{1}{x^{2}}} \left(\frac{2 i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}{x^{2} \left(e^{i x} - e^{- i x}\right)} - \frac{2 \log{\left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} \right)}}{x^{3}}\right)$$
1
--
2
x / 2*log(sin(x)) cos(x) \
(sin(x)) *|- ------------- + ---------|
| 3 2 |
\ x x *sin(x)/
$$\left(\frac{\cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}$$
sin(x)^(x^(-2))*(-2*log(sin(x))/x^3 + cos(x)/(x^2*sin(x)))
Unión de expresiones racionales
[src]
1
--
2
x
(sin(x)) *(x*cos(x) - 2*log(sin(x))*sin(x))
--------------------------------------------
3
x *sin(x)
$$\frac{\left(x \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{3} \sin{\left(x \right)}}$$
sin(x)^(x^(-2))*(x*cos(x) - 2*log(sin(x))*sin(x))/(x^3*sin(x))
Denominador común
[src]
1 1
-- --
2 2
x x
x*(sin(x)) *cos(x) - 2*(sin(x)) *log(sin(x))*sin(x)
-----------------------------------------------------
3
x *sin(x)
$$\frac{x \sin^{\frac{1}{x^{2}}}{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{3} \sin{\left(x \right)}}$$
(x*sin(x)^(x^(-2))*cos(x) - 2*sin(x)^(x^(-2))*log(sin(x))*sin(x))/(x^3*sin(x))
Denominador racional
[src]
1 1
-1 + -- -1 + --
2 2
3 x 2 x
x *(sin(x)) *cos(x) - 2*x *(sin(x)) *log(sin(x))*sin(x)
-------------------------------------------------------------------
5
x
$$\frac{x^{3} \sin^{-1 + \frac{1}{x^{2}}}{\left(x \right)} \cos{\left(x \right)} - 2 x^{2} \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \sin^{-1 + \frac{1}{x^{2}}}{\left(x \right)}}{x^{5}}$$
(x^3*sin(x)^(-1 + x^(-2))*cos(x) - 2*x^2*sin(x)^(-1 + x^(-2))*log(sin(x))*sin(x))/x^5
Parte trigonométrica
[src]
1
-- / /x\ /x\ \
2 |cot|-| tan|-| |
x | \2/ \2/ 2*log(sin(x))|
(sin(x)) *|------ - ------ - -------------|
\ 2 2 x /
--------------------------------------------
2
x
$$\frac{\left(- \frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{\cot{\left(\frac{x}{2} \right)}}{2} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{2}}$$
1
-- / /x\ \
2 | tan|-| |
x | 1 \2/ 2*log(sin(x))|
(sin(x)) *|-------- - ------ - -------------|
| /x\ 2 x |
|2*tan|-| |
\ \2/ /
----------------------------------------------
2
x
$$\frac{\left(- \frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2 \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{2}}$$
1
--
2
x / 2*log(sin(x)) cos(x) \
(sin(x)) *|- ------------- + ---------|
| 3 2 |
\ x x *sin(x)/
$$\left(\frac{\cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}$$
1 / / /x\ \ \
-- | | 2*cot|-| | |
2 | | \2/ | |
x | 2*log|-----------| |
/ /x\ \ | | 2/x\| 2/x\|
| 2*cot|-| | | |1 + cot |-|| -1 + cot |-||
| \2/ | | \ \2// \2/|
|-----------| *|- ------------------ + ------------|
| 2/x\| | 3 2 /x\ |
|1 + cot |-|| | x 2*x *cot|-| |
\ \2// \ \2/ /
$$\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{2 x^{2} \cot{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{3}}\right)$$
1 / / 1 \ \
-- | 2*log|-----------| |
2 | /x pi\ | / pi\| /x\ |
x |sec|- - --| |sec|x - --|| sec|-| |
/ 1 \ | \2 2 / \ \ 2 // \2/ |
|-----------| *|----------- - ------------------ - -------------|
| / pi\| | /x\ x /x pi\|
|sec|x - --|| | 2*sec|-| 2*sec|- - --||
\ \ 2 // \ \2/ \2 2 //
------------------------------------------------------------------
2
x
$$\frac{\left(- \frac{\sec{\left(\frac{x}{2} \right)}}{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{x}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{x^{2}}}}{x^{2}}$$
1
--
2 / / / pi\\ \
x | 2*log|cos|x - --|| |
/ / pi\\ | \ \ 2 // cos(x) |
|cos|x - --|| *|- ------------------ + --------------|
\ \ 2 // | 3 2 / pi\|
| x x *cos|x - --||
\ \ 2 //
$$\left(\frac{\cos{\left(x \right)}}{x^{2} \cos{\left(x - \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)}}{x^{3}}\right) \cos^{\frac{1}{x^{2}}}{\left(x - \frac{\pi}{2} \right)}$$
1
--
2 / / 1 \ \
x | 2*log|------| |
/ 1 \ | \csc(x)/ csc(x) |
|------| *|- ------------- + --------------|
\csc(x)/ | 3 2 /pi \|
| x x *csc|-- - x||
\ \2 //
$$\left(\frac{\csc{\left(x \right)}}{x^{2} \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{x^{2}}}$$
1
--
2 / / 1 \ \
x | 2*log|------| |
/ 1 \ | \csc(x)/ csc(x) |
|------| *|- ------------- + ---------|
\csc(x)/ | 3 2 |
\ x x *sec(x)/
$$\left(\frac{\csc{\left(x \right)}}{x^{2} \sec{\left(x \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{x^{2}}}$$
1 / / 1 \ \
-- | 2*log|-----------| |
2 | | / pi\| / pi\|
x | |sec|x - --|| sec|x - --||
/ 1 \ | \ \ 2 // \ 2 /|
|-----------| *|- ------------------ + -----------|
| / pi\| | 3 2 |
|sec|x - --|| \ x x *sec(x) /
\ \ 2 //
$$\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{x^{2} \sec{\left(x \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{x^{3}}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{x^{2}}}$$
1
--
2
x / 1 2*log(sin(x))\
(sin(x)) *|------ - -------------|
\tan(x) x /
-----------------------------------
2
x
$$\frac{\left(\frac{1}{\tan{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{2}}$$
1
--
2
x / 2*log(sin(x)) \
(sin(x)) *|- ------------- + cot(x)|
\ x /
-------------------------------------
2
x
$$\frac{\left(\cot{\left(x \right)} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{2}}$$
1
-- / 2/x\ \
2 | sin |-| |
x | sin(x) \2/ 2*log(sin(x))|
(sin(x)) *|-------------- - ------- - -------------|
\2*(1 - cos(x)) sin(x) x /
-----------------------------------------------------
2
x
$$\frac{\left(- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + \frac{\sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right)} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{2}}$$
1
--
2 / /x\ / / pi\\ /x pi\\
x | cos|-| 2*log|cos|x - --|| cos|- - --||
/ / pi\\ | \2/ \ \ 2 // \2 2 /|
|cos|x - --|| *|------------- - ------------------ - -----------|
\ \ 2 // | /x pi\ x /x\ |
|2*cos|- - --| 2*cos|-| |
\ \2 2 / \2/ /
------------------------------------------------------------------
2
x
$$\frac{\left(\frac{\cos{\left(\frac{x}{2} \right)}}{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)}}{x}\right) \cos^{\frac{1}{x^{2}}}{\left(x - \frac{\pi}{2} \right)}}{x^{2}}$$
1
--
2
x / sin(2*x) 2*log(sin(x))\
(sin(x)) *|--------- - -------------|
| 2 x |
\2*sin (x) /
--------------------------------------
2
x
$$\frac{\left(\frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{2}}$$
1 / / /x\ \ \
-- | | 2*tan|-| | |
2 | | \2/ | |
x | 2*log|-----------| |
/ /x\ \ | | 2/x\| 2/x\|
| 2*tan|-| | | |1 + tan |-|| 1 - tan |-||
| \2/ | | \ \2// \2/|
|-----------| *|- ------------------ + -----------|
| 2/x\| | 3 2 /x\|
|1 + tan |-|| | x 2*x *tan|-||
\ \2// \ \2//
$$\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{2 x^{2} \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x^{3}}\right)$$
1 / / /x\ \\
-- | | 2*tan|-| ||
2 | | \2/ ||
x | 2*log|-----------||
/ /x\ \ | | 2/x\||
| 2*tan|-| | | |1 + tan |-|||
| \2/ | | 1 \ \2//|
|-----------| *|------ - ------------------|
| 2/x\| \tan(x) x /
|1 + tan |-||
\ \2//
---------------------------------------------
2
x
$$\frac{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{1}{\tan{\left(x \right)}} - \frac{2 \log{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x^{2}}$$
1 / / 1 \\
-- | 2*log|-----------||
2 | / pi\ | / pi\||
x |sec|x - --| |sec|x - --|||
/ 1 \ | \ 2 / \ \ 2 //|
|-----------| *|----------- - ------------------|
| / pi\| \ sec(x) x /
|sec|x - --||
\ \ 2 //
--------------------------------------------------
2
x
$$\frac{\left(\frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} - \frac{2 \log{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{x}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{x^{2}}}}{x^{2}}$$
1 / / /x\ \\
-- | | 2*tan|-| ||
2 | | \2/ ||
x | 2*log|-----------||
/ /x\ \ | /x\ | 2/x\||
| 2*tan|-| | | tan|-| |1 + tan |-|||
| \2/ | | 1 \2/ \ \2//|
|-----------| *|-------- - ------ - ------------------|
| 2/x\| | /x\ 2 x |
|1 + tan |-|| |2*tan|-| |
\ \2// \ \2/ /
--------------------------------------------------------
2
x
$$\frac{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(- \frac{\tan{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2 \tan{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x^{2}}$$
1 / / /x\ \\
-- | | 2*cot|-| ||
2 | | \2/ ||
x | 2*log|-----------||
/ /x\ \ | /x\ | 2/x\||
| 2*cot|-| | |cot|-| |1 + cot |-|||
| \2/ | | \2/ 1 \ \2//|
|-----------| *|------ - -------- - ------------------|
| 2/x\| | 2 /x\ x |
|1 + cot |-|| | 2*cot|-| |
\ \2// \ \2/ /
--------------------------------------------------------
2
x
$$\frac{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\frac{\cot{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2 \cot{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x^{2}}$$
1 / / /x\ \ \
-- | | 2*cot|-| | |
2 | | \2/ | |
x | 2*log|-----------| |
/ /x\ \ | | 2/x\| |
| 2*cot|-| | | |1 + cot |-|| |
| \2/ | | \ \2// |
|-----------| *|- ------------------ + cot(x)|
| 2/x\| \ x /
|1 + cot |-||
\ \2//
-----------------------------------------------
2
x
$$\frac{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1}{x^{2}}} \left(\cot{\left(x \right)} - \frac{2 \log{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{x}\right)}{x^{2}}$$
1
--
2 / / / pi\\\
x | 2*log|cos|x - --|||
/ / pi\\ | cos(x) \ \ 2 //|
|cos|x - --|| *|----------- - ------------------|
\ \ 2 // | / pi\ x |
|cos|x - --| |
\ \ 2 / /
--------------------------------------------------
2
x
$$\frac{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)}}{x}\right) \cos^{\frac{1}{x^{2}}}{\left(x - \frac{\pi}{2} \right)}}{x^{2}}$$
1
-- / / pi\\
2 | sin|x + --||
x | 2*log(sin(x)) \ 2 /|
(sin(x)) *|- ------------- + -----------|
| 3 2 |
\ x x *sin(x) /
$$\left(\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{x^{2} \sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}$$
1
-- / 2/x\ \
2 | sin |-| |
x | \2/ 2*log(sin(x)) sin(x) |
(sin(x)) *|- ------- - ------------- + ---------|
| sin(x) x 2/x\|
| 4*sin |-||
\ \2//
--------------------------------------------------
2
x
$$\frac{\left(- \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + \frac{\sin{\left(x \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}}{x^{2}}$$
1
--
2 / / 1 \\
x | 2*log|------||
/ 1 \ | csc(x) \csc(x)/|
|------| *|----------- - -------------|
\csc(x)/ | /pi \ x |
|csc|-- - x| |
\ \2 / /
----------------------------------------
2
x
$$\frac{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{x}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{x^{2}}}}{x^{2}}$$
1
--
2 / /x\ / 1 \ /pi x\\
x | csc|-| 2*log|------| csc|-- - -||
/ 1 \ | \2/ \csc(x)/ \2 2/|
|------| *|------------- - ------------- - -----------|
\csc(x)/ | /pi x\ x /x\ |
|2*csc|-- - -| 2*csc|-| |
\ \2 2/ \2/ /
--------------------------------------------------------
2
x
$$\frac{\left(\frac{\csc{\left(\frac{x}{2} \right)}}{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{x}{2} \right)}} - \frac{2 \log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{x}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{x^{2}}}}{x^{2}}$$
(1/csc(x))^(x^(-2))*(csc(x/2)/(2*csc(pi/2 - x/2)) - 2*log(1/csc(x))/x - csc(pi/2 - x/2)/(2*csc(x/2)))/x^2
Compilar la expresión
[src]
1
--
2
x / 2*log(sin(x)) cos(x) \
(sin(x)) *|- ------------- + ---------|
| 3 2 |
\ x x *sin(x)/
$$\left(\frac{\cos{\left(x \right)}}{x^{2} \sin{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(x \right)} \right)}}{x^{3}}\right) \sin^{\frac{1}{x^{2}}}{\left(x \right)}$$
sin(x)^(1/(x^2))*(-2*log(sin(x))/x^3 + cos(x)/(x^2*sin(x)))