Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt(sin(x)+1)*(-3*x^2+3*sin(x))/(x^3+3*cos(x))^2+cos(x)/(2*(x^3+3*cos(x))*sqrt(sin(x)+1))
¿Cómo vas a descomponer esta sqrt(sin(x)+1)*(-3*x^2+3*sin(x))/(x^3+3*cos(x))^2+cos(x)/(2*(x^3+3*cos(x))*sqrt(sin(x)+1)) expresión en fracciones?
Solución
Ha introducido
[src]
____________ / 2 \
\/ sin(x) + 1 *\- 3*x + 3*sin(x)/ cos(x)
---------------------------------- + --------------------------------
2 / 3 \ ____________
/ 3 \ 2*\x + 3*cos(x)/*\/ sin(x) + 1
\x + 3*cos(x)/
$$\frac{\left(- 3 x^{2} + 3 \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{2 \left(x^{3} + 3 \cos{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
(sqrt(sin(x) + 1)*(-3*x^2 + 3*sin(x)))/(x^3 + 3*cos(x))^2 + cos(x)/(((2*(x^3 + 3*cos(x)))*sqrt(sin(x) + 1)))
Simplificación general
[src]
/ 3 \ / 2 \
\x + 3*cos(x)/*cos(x) - 6*(1 + sin(x))*\x - sin(x)/
-----------------------------------------------------
2
____________ / 3 \
2*\/ 1 + sin(x) *\x + 3*cos(x)/
$$\frac{- 6 \left(x^{2} - \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right) + \left(x^{3} + 3 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{2 \left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \sqrt{\sin{\left(x \right)} + 1}}$$
((x^3 + 3*cos(x))*cos(x) - 6*(1 + sin(x))*(x^2 - sin(x)))/(2*sqrt(1 + sin(x))*(x^3 + 3*cos(x))^2)
Denominador común
[src]
2 2 2 3 2
- 6*x + 3*cos (x) + 6*sin (x) + 6*sin(x) + x *cos(x) - 6*x *sin(x)
-----------------------------------------------------------------------------
6 ____________ ____________ 2 3 ____________
2*x *\/ 1 + sin(x) + 18*\/ 1 + sin(x) *cos (x) + 12*x *\/ 1 + sin(x) *cos(x)
$$\frac{x^{3} \cos{\left(x \right)} - 6 x^{2} \sin{\left(x \right)} - 6 x^{2} + 6 \sin^{2}{\left(x \right)} + 6 \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}}{2 x^{6} \sqrt{\sin{\left(x \right)} + 1} + 12 x^{3} \sqrt{\sin{\left(x \right)} + 1} \cos{\left(x \right)} + 18 \sqrt{\sin{\left(x \right)} + 1} \cos^{2}{\left(x \right)}}$$
(-6*x^2 + 3*cos(x)^2 + 6*sin(x)^2 + 6*sin(x) + x^3*cos(x) - 6*x^2*sin(x))/(2*x^6*sqrt(1 + sin(x)) + 18*sqrt(1 + sin(x))*cos(x)^2 + 12*x^3*sqrt(1 + sin(x))*cos(x))
Denominador racional
[src]
2
5 / 3 \ 2 5 3 2 3 2 2
- 6*x + \x + 3*cos(x)/ *cos(x) - 18*x *cos(x) - 6*x *sin(x) + 6*x *sin (x) + 6*x *sin(x) + 18*sin (x)*cos(x) + 18*cos(x)*sin(x) - 18*x *cos(x)*sin(x)
-------------------------------------------------------------------------------------------------------------------------------------------------------
2
____________ / 3 \ / 3 \
\/ 1 + sin(x) *\x + 3*cos(x)/ *\2*x + 6*cos(x)/
$$\frac{- 6 x^{5} \sin{\left(x \right)} - 6 x^{5} + 6 x^{3} \sin^{2}{\left(x \right)} + 6 x^{3} \sin{\left(x \right)} - 18 x^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 18 x^{2} \cos{\left(x \right)} + \left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \cos{\left(x \right)} + 18 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 18 \sin{\left(x \right)} \cos{\left(x \right)}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \left(2 x^{3} + 6 \cos{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
(-6*x^5 + (x^3 + 3*cos(x))^2*cos(x) - 18*x^2*cos(x) - 6*x^5*sin(x) + 6*x^3*sin(x)^2 + 6*x^3*sin(x) + 18*sin(x)^2*cos(x) + 18*cos(x)*sin(x) - 18*x^2*cos(x)*sin(x))/(sqrt(1 + sin(x))*(x^3 + 3*cos(x))^2*(2*x^3 + 6*cos(x)))
Respuesta numérica
[src]
(1.0 + sin(x))^(-0.5)*cos(x)/(2.0*x^3 + 6.0*cos(x)) + 0.111111111111111*(1.0 + sin(x))^0.5*(3.0*sin(x) - 3.0*x^2)/(0.333333333333333*x^3 + cos(x))^2
(1.0 + sin(x))^(-0.5)*cos(x)/(2.0*x^3 + 6.0*cos(x)) + 0.111111111111111*(1.0 + sin(x))^0.5*(3.0*sin(x) - 3.0*x^2)/(0.333333333333333*x^3 + cos(x))^2
Unión de expresiones racionales
[src]
/ 3 \ / 2 \
\x + 3*cos(x)/*cos(x) + 6*(1 + sin(x))*\- x + sin(x)/
-------------------------------------------------------
2
____________ / 3 \
2*\/ 1 + sin(x) *\x + 3*cos(x)/
$$\frac{6 \left(- x^{2} + \sin{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right) + \left(x^{3} + 3 \cos{\left(x \right)}\right) \cos{\left(x \right)}}{2 \left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \sqrt{\sin{\left(x \right)} + 1}}$$
((x^3 + 3*cos(x))*cos(x) + 6*(1 + sin(x))*(-x^2 + sin(x)))/(2*sqrt(1 + sin(x))*(x^3 + 3*cos(x))^2)
Potencias
[src]
________________________
/ / -I*x I*x\ / / -I*x I*x\\ I*x -I*x
/ I*\- e + e / | 2 3*I*\- e + e /| e e
/ 1 - ------------------ *|- 3*x - --------------------| ---- + -----
\/ 2 \ 2 / 2 2
------------------------------------------------------------- + -------------------------------------------------------
2 ________________________
/ I*x -I*x\ / / -I*x I*x\
| 3 3*e 3*e | / I*\- e + e / / 3 I*x -I*x\
|x + ------ + -------| / 1 - ------------------ *\2*x + 3*e + 3*e /
\ 2 2 / \/ 2
$$\frac{\left(- 3 x^{2} - \frac{3 i \left(e^{i x} - e^{- i x}\right)}{2}\right) \sqrt{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1}}{\left(x^{3} + \frac{3 e^{i x}}{2} + \frac{3 e^{- i x}}{2}\right)^{2}} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\sqrt{- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1} \left(2 x^{3} + 3 e^{i x} + 3 e^{- i x}\right)}$$
____________ / 2 \
\/ 1 + sin(x) *\- 3*x + 3*sin(x)/ cos(x)
---------------------------------- + --------------------------------
2 ____________ / 3 \
/ 3 \ \/ 1 + sin(x) *\2*x + 6*cos(x)/
\x + 3*cos(x)/
$$\frac{\left(- 3 x^{2} + 3 \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{\left(2 x^{3} + 6 \cos{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
sqrt(1 + sin(x))*(-3*x^2 + 3*sin(x))/(x^3 + 3*cos(x))^2 + cos(x)/(sqrt(1 + sin(x))*(2*x^3 + 6*cos(x)))
Compilar la expresión
[src]
____________ / 2 \
\/ 1 + sin(x) *\- 3*x + 3*sin(x)/ cos(x)
---------------------------------- + --------------------------------
2 ____________ / 3 \
/ 3 \ \/ 1 + sin(x) *\2*x + 6*cos(x)/
\x + 3*cos(x)/
$$\frac{\left(- 3 x^{2} + 3 \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{\left(2 x^{3} + 6 \cos{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
sqrt(1 + sin(x))*(-3*x^2 + 3*sin(x))/(x^3 + 3*cos(x))^2 + cos(x)/(sqrt(1 + sin(x))*(2*x^3 + 6*cos(x)))
Combinatoria
[src]
2 2 2 3 2
- 6*x + 3*cos (x) + 6*sin (x) + 6*sin(x) + x *cos(x) - 6*x *sin(x)
-------------------------------------------------------------------
2
____________ / 3 \
2*\/ 1 + sin(x) *\x + 3*cos(x)/
$$\frac{x^{3} \cos{\left(x \right)} - 6 x^{2} \sin{\left(x \right)} - 6 x^{2} + 6 \sin^{2}{\left(x \right)} + 6 \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}}{2 \left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \sqrt{\sin{\left(x \right)} + 1}}$$
(-6*x^2 + 3*cos(x)^2 + 6*sin(x)^2 + 6*sin(x) + x^3*cos(x) - 6*x^2*sin(x))/(2*sqrt(1 + sin(x))*(x^3 + 3*cos(x))^2)
Parte trigonométrica
[src]
_________________
/ 1 / 2 3 \
/ 1 + ----------- *|- 3*x + -----------|
/ / pi\ | / pi\|
/ sec|x - --| | sec|x - --||
\/ \ 2 / \ \ 2 // 1
---------------------------------------------- + ----------------------------------------------
2 _________________
/ 3 3 \ / 1 / 3 6 \
|x + ------| / 1 + ----------- *|2*x + ------|*sec(x)
\ sec(x)/ / / pi\ \ sec(x)/
/ sec|x - --|
\/ \ 2 /
$$\frac{\sqrt{1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} \left(- 3 x^{2} + \frac{3}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)}{\left(x^{3} + \frac{3}{\sec{\left(x \right)}}\right)^{2}} + \frac{1}{\sqrt{1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} \left(2 x^{3} + \frac{6}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}}$$
____________
/ 1 / 2 3 \
/ 1 + ------ *|- 3*x + ------|
\/ csc(x) \ csc(x)/ 1
---------------------------------- + -------------------------------------------------
2 ____________
/ 3 3 \ / 1 / 3 6 \ /pi \
|x + -----------| / 1 + ------ *|2*x + -----------|*csc|-- - x|
| /pi \| \/ csc(x) | /pi \| \2 /
| csc|-- - x|| | csc|-- - x||
\ \2 // \ \2 //
$$\frac{\sqrt{1 + \frac{1}{\csc{\left(x \right)}}} \left(- 3 x^{2} + \frac{3}{\csc{\left(x \right)}}\right)}{\left(x^{3} + \frac{3}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2}} + \frac{1}{\sqrt{1 + \frac{1}{\csc{\left(x \right)}}} \left(2 x^{3} + \frac{6}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/pi \ 3 / pi\
3*sin|-- + 2*x| x *sin|x + --|
9 2 \2 / \ 2 / 2
- - 3*x + 3*sin(x) - --------------- + -------------- - 3*x *sin(x)
4 4 2
--------------------------------------------------------------------
2
____________ / 3 / pi\\
\/ 1 + sin(x) *|x + 3*sin|x + --||
\ \ 2 //
$$\frac{\frac{x^{3} \sin{\left(x + \frac{\pi}{2} \right)}}{2} - 3 x^{2} \sin{\left(x \right)} - 3 x^{2} + 3 \sin{\left(x \right)} - \frac{3 \sin{\left(2 x + \frac{\pi}{2} \right)}}{4} + \frac{9}{4}}{\left(x^{3} + 3 \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2} \sqrt{\sin{\left(x \right)} + 1}}$$
3 2
9 2 3 3 x 3*x
- - 3*x + ------ - --------------- + ------------- - ------
4 csc(x) /pi \ /pi \ csc(x)
4*csc|-- - 2*x| 2*csc|-- - x|
\2 / \2 /
------------------------------------------------------------
____________ 2
/ 1 / 3 3 \
/ 1 + ------ *|x + -----------|
\/ csc(x) | /pi \|
| csc|-- - x||
\ \2 //
$$\frac{\frac{x^{3}}{2 \csc{\left(- x + \frac{\pi}{2} \right)}} - 3 x^{2} - \frac{3 x^{2}}{\csc{\left(x \right)}} + \frac{9}{4} - \frac{3}{4 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} + \frac{3}{\csc{\left(x \right)}}}{\sqrt{1 + \frac{1}{\csc{\left(x \right)}}} \left(x^{3} + \frac{3}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2}}$$
3
9 2 3*cos(2*x) x *cos(x) 2
- - 3*x + 3*sin(x) - ---------- + --------- - 3*x *sin(x)
4 4 2
----------------------------------------------------------
2
____________ / 3 \
\/ 1 + sin(x) *\x + 3*cos(x)/
$$\frac{\frac{x^{3} \cos{\left(x \right)}}{2} - 3 x^{2} \sin{\left(x \right)} - 3 x^{2} + 3 \sin{\left(x \right)} - \frac{3 \cos{\left(2 x \right)}}{4} + \frac{9}{4}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \sqrt{\sin{\left(x \right)} + 1}}$$
_________________
/ /x\ / /x\ \
/ 2*tan|-| | 6*tan|-| |
/ \2/ | 2 \2/ |
/ 1 + ----------- *|- 3*x + -----------|
/ 2/x\ | 2/x\| 2/x\
/ 1 + tan |-| | 1 + tan |-|| 1 - tan |-|
\/ \2/ \ \2// \2/
------------------------------------------------ + ----------------------------------------------------------------
2 _________________
/ / 2/x\\\ / /x\ / / 2/x\\\
| 3*|1 - tan |-||| / 2*tan|-| | 6*|1 - tan |-|||
| 3 \ \2//| / 2/x\\ / \2/ | 3 \ \2//|
|x + ---------------| |1 + tan |-||* / 1 + ----------- *|2*x + ---------------|
| 2/x\ | \ \2// / 2/x\ | 2/x\ |
| 1 + tan |-| | / 1 + tan |-| | 1 + tan |-| |
\ \2/ / \/ \2/ \ \2/ /
$$\frac{\sqrt{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- 3 x^{2} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{\left(x^{3} + \frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\sqrt{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(2 x^{3} + \frac{6 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
_________________
/ / pi\ / 2 / pi\\
3* / 1 + cos|x - --| *|- x + cos|x - --||
cos(x) \/ \ 2 / \ \ 2 //
--------------------------------------- + --------------------------------------------
_________________ 2
/ / pi\ / 3 \ / 3 \
2* / 1 + cos|x - --| *\x + 3*cos(x)/ \x + 3*cos(x)/
\/ \ 2 /
$$\frac{3 \left(- x^{2} + \cos{\left(x - \frac{\pi}{2} \right)}\right) \sqrt{\cos{\left(x - \frac{\pi}{2} \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{2 \left(x^{3} + 3 \cos{\left(x \right)}\right) \sqrt{\cos{\left(x - \frac{\pi}{2} \right)} + 1}}$$
/x\ 3 / 2/x\\ 2 /x\
6*cot|-| / 2 \ x *|-1 + cot |-|| 6*x *cot|-|
9 2 \2/ 3*\-1 + cot (x)/ \ \2// \2/
- - 3*x + ----------- - ---------------- + ----------------- - -----------
4 2/x\ / 2 \ / 2/x\\ 2/x\
1 + cot |-| 4*\1 + cot (x)/ 2*|1 + cot |-|| 1 + cot |-|
\2/ \ \2// \2/
---------------------------------------------------------------------------
_________________ 2
/ /x\ / / 2/x\\\
/ 2*cot|-| | 3*|-1 + cot |-|||
/ \2/ | 3 \ \2//|
/ 1 + ----------- *|x + ----------------|
/ 2/x\ | 2/x\ |
/ 1 + cot |-| | 1 + cot |-| |
\/ \2/ \ \2/ /
$$\frac{\frac{x^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - 3 x^{2} - \frac{6 x^{2} \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{3 \left(\cot^{2}{\left(x \right)} - 1\right)}{4 \left(\cot^{2}{\left(x \right)} + 1\right)} + \frac{9}{4} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{\sqrt{1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(x^{3} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
_________________
/ /x\ / /x\ \
/ 2*cot|-| | 2*cot|-| |
/ \2/ | 2 \2/ |
3* / 1 + ----------- *|- x + -----------|
/ 2/x\ | 2/x\| 2/x\
/ 1 + cot |-| | 1 + cot |-|| -1 + cot |-|
\/ \2/ \ \2// \2/
------------------------------------------------ + -----------------------------------------------------------------
2 _________________
/ / 2/x\\\ / /x\ / / 2/x\\\
| 3*|-1 + cot |-||| / 2*cot|-| | 3*|-1 + cot |-|||
| 3 \ \2//| / 2/x\\ / \2/ | 3 \ \2//|
|x + ----------------| 2*|1 + cot |-||* / 1 + ----------- *|x + ----------------|
| 2/x\ | \ \2// / 2/x\ | 2/x\ |
| 1 + cot |-| | / 1 + cot |-| | 1 + cot |-| |
\ \2/ / \/ \2/ \ \2/ /
$$\frac{3 \sqrt{1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- x^{2} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{\left(x^{3} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{2 \sqrt{1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(x^{3} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
____________ / 2 \
\/ 1 + sin(x) *\- 3*x + 3*sin(x)/ cos(x)
---------------------------------- + --------------------------------
2 ____________ / 3 \
/ 3 \ \/ 1 + sin(x) *\2*x + 6*cos(x)/
\x + 3*cos(x)/
$$\frac{\left(- 3 x^{2} + 3 \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{\left(2 x^{3} + 6 \cos{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
/ pi\
____________ / 2 \ sin|x + --|
\/ 1 + sin(x) *\- 3*x + 3*sin(x)/ \ 2 /
---------------------------------- + -------------------------------------
2 ____________ / 3 / pi\\
/ 3 / pi\\ \/ 1 + sin(x) *|2*x + 6*sin|x + --||
|x + 3*sin|x + --|| \ \ 2 //
\ \ 2 //
$$\frac{\left(- 3 x^{2} + 3 \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2}} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\left(2 x^{3} + 6 \sin{\left(x + \frac{\pi}{2} \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
____________ / 2 \
cos(x) 3*\/ sin(x) + 1 *\- x + sin(x)/
-------------------------------- + --------------------------------
____________ / 3 \ 2
2*\/ 1 + sin(x) *\x + 3*cos(x)/ / 3 \
\x + 3*cos(x)/
$$\frac{3 \left(- x^{2} + \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{2 \left(x^{3} + 3 \cos{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
3 2
9 2 3 3 x 3*x
- - 3*x + ----------- - ---------- + -------- - -----------
4 / pi\ 4*sec(2*x) 2*sec(x) / pi\
sec|x - --| sec|x - --|
\ 2 / \ 2 /
------------------------------------------------------------
_________________ 2
/ 1 / 3 3 \
/ 1 + ----------- *|x + ------|
/ / pi\ \ sec(x)/
/ sec|x - --|
\/ \ 2 /
$$\frac{\frac{x^{3}}{2 \sec{\left(x \right)}} - 3 x^{2} - \frac{3 x^{2}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{9}{4} + \frac{3}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{3}{4 \sec{\left(2 x \right)}}}{\sqrt{1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} \left(x^{3} + \frac{3}{\sec{\left(x \right)}}\right)^{2}}$$
/x\ 3 / 2/x\\ 2 /x\
6*tan|-| / 2 \ x *|1 - tan |-|| 6*x *tan|-|
9 2 \2/ 3*\1 - tan (x)/ \ \2// \2/
- - 3*x + ----------- - --------------- + ---------------- - -----------
4 2/x\ / 2 \ / 2/x\\ 2/x\
1 + tan |-| 4*\1 + tan (x)/ 2*|1 + tan |-|| 1 + tan |-|
\2/ \ \2// \2/
-------------------------------------------------------------------------
_________________ 2
/ /x\ / / 2/x\\\
/ 2*tan|-| | 3*|1 - tan |-|||
/ \2/ | 3 \ \2//|
/ 1 + ----------- *|x + ---------------|
/ 2/x\ | 2/x\ |
/ 1 + tan |-| | 1 + tan |-| |
\/ \2/ \ \2/ /
$$\frac{\frac{x^{3} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - 3 x^{2} - \frac{6 x^{2} \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{3 \left(1 - \tan^{2}{\left(x \right)}\right)}{4 \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{9}{4} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{\sqrt{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(x^{3} + \frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}}$$
3
9 2 / pi\ 3*cos(2*x) x *cos(x) 2 / pi\
- - 3*x + 3*cos|x - --| - ---------- + --------- - 3*x *cos|x - --|
4 \ 2 / 4 2 \ 2 /
--------------------------------------------------------------------
_________________ 2
/ / pi\ / 3 \
/ 1 + cos|x - --| *\x + 3*cos(x)/
\/ \ 2 /
$$\frac{\frac{x^{3} \cos{\left(x \right)}}{2} - 3 x^{2} \cos{\left(x - \frac{\pi}{2} \right)} - 3 x^{2} - \frac{3 \cos{\left(2 x \right)}}{4} + 3 \cos{\left(x - \frac{\pi}{2} \right)} + \frac{9}{4}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \sqrt{\cos{\left(x - \frac{\pi}{2} \right)} + 1}}$$
____________
/ 1 / 2 3 \
/ 1 + ------ *|- 3*x + ------|
\/ csc(x) \ csc(x)/ 1
---------------------------------- + ---------------------------------------
2 ____________
/ 3 3 \ / 1 / 3 6 \
|x + ------| / 1 + ------ *|2*x + ------|*sec(x)
\ sec(x)/ \/ csc(x) \ sec(x)/
$$\frac{\sqrt{1 + \frac{1}{\csc{\left(x \right)}}} \left(- 3 x^{2} + \frac{3}{\csc{\left(x \right)}}\right)}{\left(x^{3} + \frac{3}{\sec{\left(x \right)}}\right)^{2}} + \frac{1}{\sqrt{1 + \frac{1}{\csc{\left(x \right)}}} \left(2 x^{3} + \frac{6}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}}$$
____________
/ 1 / 1 2\
3* / 1 + ------ *|------ - x |
1 \/ csc(x) \csc(x) /
------------------------------------------------- + --------------------------------
____________ 2
/ 1 / 3 3 \ /pi \ / 3 3 \
2* / 1 + ------ *|x + -----------|*csc|-- - x| |x + -----------|
\/ csc(x) | /pi \| \2 / | /pi \|
| csc|-- - x|| | csc|-- - x||
\ \2 // \ \2 //
$$\frac{3 \sqrt{1 + \frac{1}{\csc{\left(x \right)}}} \left(- x^{2} + \frac{1}{\csc{\left(x \right)}}\right)}{\left(x^{3} + \frac{3}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2}} + \frac{1}{2 \sqrt{1 + \frac{1}{\csc{\left(x \right)}}} \left(x^{3} + \frac{3}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
_________________
/ / pi\ / 2 / pi\\
/ 1 + cos|x - --| *|- 3*x + 3*cos|x - --||
\/ \ 2 / \ \ 2 // cos(x)
---------------------------------------------- + ---------------------------------------
2 _________________
/ 3 \ / / pi\ / 3 \
\x + 3*cos(x)/ / 1 + cos|x - --| *\2*x + 6*cos(x)/
\/ \ 2 /
$$\frac{\left(- 3 x^{2} + 3 \cos{\left(x - \frac{\pi}{2} \right)}\right) \sqrt{\cos{\left(x - \frac{\pi}{2} \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{\left(2 x^{3} + 6 \cos{\left(x \right)}\right) \sqrt{\cos{\left(x - \frac{\pi}{2} \right)} + 1}}$$
2 3
2 3*cos (x) x *cos(x) 2
3 - 3*x + 3*sin(x) - --------- + --------- - 3*x *sin(x)
2 2
---------------------------------------------------------
2
____________ / 3 \
\/ 1 + sin(x) *\x + 3*cos(x)/
$$\frac{\frac{x^{3} \cos{\left(x \right)}}{2} - 3 x^{2} \sin{\left(x \right)} - 3 x^{2} + 3 \sin{\left(x \right)} - \frac{3 \cos^{2}{\left(x \right)}}{2} + 3}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2} \sqrt{\sin{\left(x \right)} + 1}}$$
_________________
/ /x\ / /x\ \
/ 2*cot|-| | 6*cot|-| |
/ \2/ | 2 \2/ |
/ 1 + ----------- *|- 3*x + -----------|
/ 2/x\ | 2/x\| 2/x\
/ 1 + cot |-| | 1 + cot |-|| -1 + cot |-|
\/ \2/ \ \2// \2/
------------------------------------------------ + -----------------------------------------------------------------
2 _________________
/ / 2/x\\\ / /x\ / / 2/x\\\
| 3*|-1 + cot |-||| / 2*cot|-| | 6*|-1 + cot |-|||
| 3 \ \2//| / 2/x\\ / \2/ | 3 \ \2//|
|x + ----------------| |1 + cot |-||* / 1 + ----------- *|2*x + ----------------|
| 2/x\ | \ \2// / 2/x\ | 2/x\ |
| 1 + cot |-| | / 1 + cot |-| | 1 + cot |-| |
\ \2/ / \/ \2/ \ \2/ /
$$\frac{\sqrt{1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- 3 x^{2} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{\left(x^{3} + \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\sqrt{1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(2 x^{3} + \frac{6 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
_________________
/ /x\ / /x\ \
/ 2*tan|-| | 2*tan|-| |
/ \2/ | 2 \2/ |
3* / 1 + ----------- *|- x + -----------|
/ 2/x\ | 2/x\| 2/x\
/ 1 + tan |-| | 1 + tan |-|| 1 - tan |-|
\/ \2/ \ \2// \2/
------------------------------------------------ + ----------------------------------------------------------------
2 _________________
/ / 2/x\\\ / /x\ / / 2/x\\\
| 3*|1 - tan |-||| / 2*tan|-| | 3*|1 - tan |-|||
| 3 \ \2//| / 2/x\\ / \2/ | 3 \ \2//|
|x + ---------------| 2*|1 + tan |-||* / 1 + ----------- *|x + ---------------|
| 2/x\ | \ \2// / 2/x\ | 2/x\ |
| 1 + tan |-| | / 1 + tan |-| | 1 + tan |-| |
\ \2/ / \/ \2/ \ \2/ /
$$\frac{3 \sqrt{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- x^{2} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{\left(x^{3} + \frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{2 \sqrt{1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(x^{3} + \frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
____________ / 2 \
cos(x) 3*\/ 1 + sin(x) *\- x + sin(x)/
-------------------------------- + --------------------------------
____________ / 3 \ 2
2*\/ 1 + sin(x) *\x + 3*cos(x)/ / 3 \
\x + 3*cos(x)/
$$\frac{3 \left(- x^{2} + \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \cos{\left(x \right)}\right)^{2}} + \frac{\cos{\left(x \right)}}{2 \left(x^{3} + 3 \cos{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
/ pi\
sin|x + --| ____________ / 2 \
\ 2 / 3*\/ 1 + sin(x) *\- x + sin(x)/
------------------------------------- + --------------------------------
____________ / 3 / pi\\ 2
2*\/ 1 + sin(x) *|x + 3*sin|x + --|| / 3 / pi\\
\ \ 2 // |x + 3*sin|x + --||
\ \ 2 //
$$\frac{3 \left(- x^{2} + \sin{\left(x \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}{\left(x^{3} + 3 \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2}} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2 \left(x^{3} + 3 \sin{\left(x + \frac{\pi}{2} \right)}\right) \sqrt{\sin{\left(x \right)} + 1}}$$
_________________
/ 1 / 1 2\
3* / 1 + ----------- *|----------- - x |
/ / pi\ | / pi\ |
/ sec|x - --| |sec|x - --| |
1 \/ \ 2 / \ \ 2 / /
---------------------------------------------- + --------------------------------------------
_________________ 2
/ 1 / 3 3 \ / 3 3 \
2* / 1 + ----------- *|x + ------|*sec(x) |x + ------|
/ / pi\ \ sec(x)/ \ sec(x)/
/ sec|x - --|
\/ \ 2 /
$$\frac{3 \sqrt{1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} \left(- x^{2} + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)}{\left(x^{3} + \frac{3}{\sec{\left(x \right)}}\right)^{2}} + \frac{1}{2 \sqrt{1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}} \left(x^{3} + \frac{3}{\sec{\left(x \right)}}\right) \sec{\left(x \right)}}$$
1/(2*sqrt(1 + 1/sec(x - pi/2))*(x^3 + 3/sec(x))*sec(x)) + 3*sqrt(1 + 1/sec(x - pi/2))*(1/sec(x - pi/2) - x^2)/(x^3 + 3/sec(x))^2