Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)^2*(1/(3*cos(x)^2)+2*(x+2)*sin(x)/(3*cos(x)^3))/(x+2)
¿Cómo vas a descomponer esta cos(x)^2*(1/(3*cos(x)^2)+2*(x+2)*sin(x)/(3*cos(x)^3))/(x+2) expresión en fracciones?
Solución
Ha introducido
[src]
2 / 1 2*(x + 2)*sin(x)\
cos (x)*|--------- + ----------------|
| 2 3 |
\3*cos (x) 3*cos (x) /
--------------------------------------
x + 2
$$\frac{\left(\frac{2 \left(x + 2\right) \sin{\left(x \right)}}{3 \cos^{3}{\left(x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{x + 2}$$
(cos(x)^2*(1/(3*cos(x)^2) + ((2*(x + 2))*sin(x))/((3*cos(x)^3))))/(x + 2)
Simplificación general
[src]
1 + 4*tan(x) + 2*x*tan(x)
-------------------------
3*(2 + x)
$$\frac{2 x \tan{\left(x \right)} + 4 \tan{\left(x \right)} + 1}{3 \left(x + 2\right)}$$
(1 + 4*tan(x) + 2*x*tan(x))/(3*(2 + x))
Unión de expresiones racionales
[src]
2*(2 + x)*sin(x) + cos(x)
-------------------------
3*(2 + x)*cos(x)
$$\frac{2 \left(x + 2\right) \sin{\left(x \right)} + \cos{\left(x \right)}}{3 \left(x + 2\right) \cos{\left(x \right)}}$$
(2*(2 + x)*sin(x) + cos(x))/(3*(2 + x)*cos(x))
Denominador racional
[src]
3 2
3*cos (x) + 3*cos (x)*(4 + 2*x)*sin(x)
--------------------------------------
3
9*(2 + x)*cos (x)
$$\frac{3 \left(2 x + 4\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 3 \cos^{3}{\left(x \right)}}{9 \left(x + 2\right) \cos^{3}{\left(x \right)}}$$
(3*cos(x)^3 + 3*cos(x)^2*(4 + 2*x)*sin(x))/(9*(2 + x)*cos(x)^3)
Compilar la expresión
[src]
2 / 1 (4 + 2*x)*sin(x)\
cos (x)*|--------- + ----------------|
| 2 3 |
\3*cos (x) 3*cos (x) /
--------------------------------------
2 + x
$$\frac{\left(\frac{\left(2 x + 4\right) \sin{\left(x \right)}}{3 \cos^{3}{\left(x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{x + 2}$$
cos(x)^2*(1/(3*cos(x)^2) + (4 + 2*x)*sin(x)/(3*cos(x)^3))/(2 + x)
Combinatoria
[src]
4*sin(x) + 2*x*sin(x) + cos(x)
------------------------------
3*(2 + x)*cos(x)
$$\frac{2 x \sin{\left(x \right)} + 4 \sin{\left(x \right)} + \cos{\left(x \right)}}{3 \left(x + 2\right) \cos{\left(x \right)}}$$
(4*sin(x) + 2*x*sin(x) + cos(x))/(3*(2 + x)*cos(x))
Denominador común
[src]
4*sin(x) + 2*x*sin(x) + cos(x)
------------------------------
6*cos(x) + 3*x*cos(x)
$$\frac{2 x \sin{\left(x \right)} + 4 \sin{\left(x \right)} + \cos{\left(x \right)}}{3 x \cos{\left(x \right)} + 6 \cos{\left(x \right)}}$$
(4*sin(x) + 2*x*sin(x) + cos(x))/(6*cos(x) + 3*x*cos(x))
Respuesta numérica
[src]
cos(x)^2*(0.333333333333333/cos(x)^2 + 0.333333333333333*(4.0 + 2.0*x)*sin(x)/cos(x)^3)/(2.0 + x)
cos(x)^2*(0.333333333333333/cos(x)^2 + 0.333333333333333*(4.0 + 2.0*x)*sin(x)/cos(x)^3)/(2.0 + x)
Potencias
[src]
2
/ I*x -I*x\ / / -I*x I*x\\
|e e | | 1 I*(4 + 2*x)*\- e + e /|
|---- + -----| *|----------------- - ----------------------------|
\ 2 2 / | 2 3 |
| / I*x -I*x\ / I*x -I*x\ |
| |e e | |e e | |
|3*|---- + -----| 6*|---- + -----| |
\ \ 2 2 / \ 2 2 / /
------------------------------------------------------------------
2 + x
$$\frac{\left(- \frac{i \left(2 x + 4\right) \left(e^{i x} - e^{- i x}\right)}{6 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{3}} + \frac{1}{3 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}{x + 2}$$
2 / 1 (4 + 2*x)*sin(x)\
cos (x)*|--------- + ----------------|
| 2 3 |
\3*cos (x) 3*cos (x) /
--------------------------------------
2 + x
$$\frac{\left(\frac{\left(2 x + 4\right) \sin{\left(x \right)}}{3 \cos^{3}{\left(x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{x + 2}$$
cos(x)^2*(1/(3*cos(x)^2) + (4 + 2*x)*sin(x)/(3*cos(x)^3))/(2 + x)
Parte trigonométrica
[src]
/ / pi\\
| (4 + 2*x)*cos|x - --||
2 | 1 \ 2 /|
cos (x)*|--------- + ---------------------|
| 2 3 |
\3*cos (x) 3*cos (x) /
-------------------------------------------
2 + x
$$\frac{\left(\frac{\left(2 x + 4\right) \cos{\left(x - \frac{\pi}{2} \right)}}{3 \cos^{3}{\left(x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{x + 2}$$
2 / /8 4*x\ \
/ 2 \ | |- + ---|*sin(x) |
| sin (x) | 4/x\ | 2 \3 3 / |
|-1 + ---------| *sin |-|*|--------------------- + ---------------------------|
| 4/x\| \2/ | / /pi \\ 3 |
| 4*sin |-|| |3*|1 + sin|-- + 2*x|| / 2 \ |
\ \2// | \ \2 // | sin (x) | 6/x\|
| 2*|-1 + ---------| *sin |-||
| | 4/x\| \2/|
| | 4*sin |-|| |
\ \ \2// /
-------------------------------------------------------------------------------
2 + x
$$\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2}{3 \left(\sin{\left(2 x + \frac{\pi}{2} \right)} + 1\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \sin{\left(x \right)}}{2 \left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)}}\right) \sin^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
/ 4/x\ /1 x \ /x\\
2 | tan |-|*|- + --|*cot|-||
/ 2/x\\ 4/x\ 8/x\ | 1 \4/ \6 12/ \2/|
16*|-1 + cot |-|| *cot |-|*sin |-|*|--------- + -----------------------|
\ \2// \4/ \4/ | 2 3 |
|3*cos (x) / 2/x\\ 8/x\|
| |-1 + cot |-|| *sin |-||
\ \ \2// \4//
------------------------------------------------------------------------
2 + x
$$\frac{16 \left(\frac{\left(\frac{x}{12} + \frac{1}{6}\right) \tan^{4}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \sin^{8}{\left(\frac{x}{4} \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{8}{\left(\frac{x}{4} \right)} \cot^{4}{\left(\frac{x}{4} \right)}}{x + 2}$$
/ /8 4*x\ /x\ \
2 4 | |- + ---|*tan|-| |
/ 2/x\\ / 2/x\\ 8/x\ | 1 \3 3 / \2/ |
|1 - tan |-|| *|1 - tan |-|| *cos |-|*|--------- + -------------------------------------|
\ \2// \ \4// \4/ | 2 3 4 |
|3*cos (x) / 2/x\\ / 2/x\\ 8/x\|
| |1 - tan |-|| *|1 - tan |-|| *cos |-||
\ \ \2// \ \4// \4//
-----------------------------------------------------------------------------------------
2 + x
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\frac{1}{3 \cos^{2}{\left(x \right)}} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3} \cos^{8}{\left(\frac{x}{4} \right)}}\right) \cos^{8}{\left(\frac{x}{4} \right)}}{x + 2}$$
2/pi \ 3/pi \
csc |-- - x| csc |-- - x|*(4 + 2*x)
\2 / \2 /
------------ + ----------------------
3 3*csc(x)
-------------------------------------
2/pi \
(2 + x)*csc |-- - x|
\2 /
$$\frac{\frac{\left(2 x + 4\right) \csc^{3}{\left(- x + \frac{\pi}{2} \right)}}{3 \csc{\left(x \right)}} + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{3}}{\left(x + 2\right) \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}$$
/ /8 4*x\ /x\ \
2 | |- + ---|*tan|-| |
/ 2/x\\ 4/x\ | 2 \3 3 / \2/ |
|1 - tan |-|| *cos |-|*|---------------- + ----------------------|
\ \2// \2/ |3*(1 + cos(2*x)) 3 |
| / 2/x\\ 4/x\|
| |1 - tan |-|| *cos |-||
\ \ \2// \2//
------------------------------------------------------------------
2 + x
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\frac{2}{3 \left(\cos{\left(2 x \right)} + 1\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3} \cos^{4}{\left(\frac{x}{2} \right)}}\right) \cos^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
2/ pi\ / 1 (4 + 2*x)*sin(x)\
sin |x + --|*|-------------- + ----------------|
\ 2 / | 2/ pi\ 3/ pi\ |
|3*sin |x + --| 3*sin |x + --| |
\ \ 2 / \ 2 / /
------------------------------------------------
2 + x
$$\frac{\left(\frac{\left(2 x + 4\right) \sin{\left(x \right)}}{3 \sin^{3}{\left(x + \frac{\pi}{2} \right)}} + \frac{1}{3 \sin^{2}{\left(x + \frac{\pi}{2} \right)}}\right) \sin^{2}{\left(x + \frac{\pi}{2} \right)}}{x + 2}$$
2
/ 4/x\\ / /16 8*x\ \
| 4*sin |-|| | |-- + ---|*sin(x) |
| \2/| 4/x\ | 1 \3 3 / |
|1 - ---------| *cos |-|*|--------- + -------------------|
| 2 | \2/ | 2 3*cos(x) + cos(3*x)|
\ sin (x) / \3*cos (x) /
----------------------------------------------------------
2 + x
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \left(\frac{\left(\frac{8 x}{3} + \frac{16}{3}\right) \sin{\left(x \right)}}{3 \cos{\left(x \right)} + \cos{\left(3 x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \cos^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
4 2*x
1 + ------ + ------
cot(x) cot(x)
-------------------
3*(2 + x)
$$\frac{\frac{2 x}{\cot{\left(x \right)}} + 1 + \frac{4}{\cot{\left(x \right)}}}{3 \left(x + 2\right)}$$
4*sec(x) 2*x*sec(x)
1 + ----------- + -----------
/ pi\ / pi\
sec|x - --| sec|x - --|
\ 2 / \ 2 /
-----------------------------
3*(2 + x)
$$\frac{\frac{2 x \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{4 \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1}{3 \left(x + 2\right)}$$
2 / 1 (4 + 2*x)*sin(x)\
cos (x)*|--------- + ----------------|
| 2 3 |
\3*cos (x) 3*cos (x) /
--------------------------------------
2 + x
$$\frac{\left(\frac{\left(2 x + 4\right) \sin{\left(x \right)}}{3 \cos^{3}{\left(x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{x + 2}$$
1 + 4*tan(x) + 2*x*tan(x)
-------------------------
3*(2 + x)
$$\frac{2 x \tan{\left(x \right)} + 4 \tan{\left(x \right)} + 1}{3 \left(x + 2\right)}$$
/ 4 \
| / 2/x\\ /8 4*x\ |
2 4 | |1 + cot |-|| *|- + ---| |
/ 1 \ / 2/x\\ | 2 \ \4// \3 3 / |
|1 - -------| *|-1 + cot |-|| *|-------------------- + -------------------------------------|
| 2/x\| \ \4// | / 2 \ 3 4 |
| cot |-|| | | -1 + cot (x)| / 1 \ / 2/x\\ /x\|
\ \2// |3*|1 + ------------| |1 - -------| *|-1 + cot |-|| *cot|-||
| | 2 | | 2/x\| \ \4// \2/|
| \ 1 + cot (x) / | cot |-|| |
\ \ \2// /
---------------------------------------------------------------------------------------------
4
/ 2/x\\
|1 + cot |-|| *(2 + x)
\ \4//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2}{3 \left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \cot{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}{\left(x + 2\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}$$
2 3
sec (x) sec (x)*(4 + 2*x)
------- + -----------------
3 / pi\
3*sec|x - --|
\ 2 /
---------------------------
2
(2 + x)*sec (x)
$$\frac{\frac{\left(2 x + 4\right) \sec^{3}{\left(x \right)}}{3 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\sec^{2}{\left(x \right)}}{3}}{\left(x + 2\right) \sec^{2}{\left(x \right)}}$$
2
/ 2/pi x\\ / 5/pi x\ /8 4*x\ \
| csc |-- - -|| | csc |-- - -|*|- + ---| |
| \2 2/| | 2 \2 2/ \3 3 / |
|1 - ------------| *|--------------------- + --------------------------|
| 2/x\ | | / 1 \ 3 |
| csc |-| | |3*|1 + -------------| / 2/pi x\\ |
\ \2/ / | | /pi \| | csc |-- - -|| |
| | csc|-- - 2*x|| | \2 2/| /x\|
| \ \2 // |1 - ------------| *csc|-||
| | 2/x\ | \2/|
| | csc |-| | |
\ \ \2/ / /
------------------------------------------------------------------------
4/pi x\
(2 + x)*csc |-- - -|
\2 2/
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2}{3 \left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \csc^{5}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \csc{\left(\frac{x}{2} \right)}}\right)}{\left(x + 2\right) \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
/ 2 2 \
| / 2/x\\ / 2/x\\ /x\|
2 | |1 + tan |-|| 2*|1 + tan |-|| *(4 + 2*x)*tan|-||
/ 2/x\\ | \ \2// \ \2// \2/|
|1 - tan |-|| *|---------------- + ---------------------------------|
\ \2// | 2 3 |
| / 2/x\\ / 2/x\\ |
|3*|1 - tan |-|| 3*|1 - tan |-|| |
\ \ \2// \ \2// /
---------------------------------------------------------------------
2
/ 2/x\\
|1 + tan |-|| *(2 + x)
\ \2//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}} + \frac{2 \left(2 x + 4\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3}}\right)}{\left(x + 2\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
/ /8 4*x\ /x\ \
2 4 | |- + ---|*tan|-| |
/ 2/x\\ / 2/x\\ 8/x\ | 1 \3 3 / \2/ |
|1 - tan |-|| *|-1 + cot |-|| *sin |-|*|--------- + --------------------------------------|
\ \2// \ \4// \4/ | 2 3 4 |
|3*cos (x) / 2/x\\ / 2/x\\ 8/x\|
| |1 - tan |-|| *|-1 + cot |-|| *sin |-||
\ \ \2// \ \4// \4//
-------------------------------------------------------------------------------------------
2 + x
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\frac{1}{3 \cos^{2}{\left(x \right)}} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)}}{x + 2}$$
/ /8 4*x\ /x\ \
2 | |- + ---|*cot|-| |
/ 2/x\\ 4/x\ | 2 \3 3 / \2/ |
|-1 + cot |-|| *sin |-|*|---------------- + -----------------------|
\ \2// \2/ |3*(1 + cos(2*x)) 3 |
| / 2/x\\ 4/x\|
| |-1 + cot |-|| *sin |-||
\ \ \2// \2//
--------------------------------------------------------------------
2 + x
$$\frac{\left(\frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{2}{3 \left(\cos{\left(2 x \right)} + 1\right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
2
/ 4/x\\ / 2/x\ /8 4*x\ \
| 4*sin |-|| | 2*sin |-|*|- + ---| |
| \2/| 4/pi x\ | 2 \2/ \3 3 / |
|1 - ---------| *sin |-- + -|*|--------------------- + ------------------------------------|
| 2 | \2 2/ | / /pi \\ 3 |
\ sin (x) / |3*|1 + sin|-- + 2*x|| / 4/x\\ |
| \ \2 // | 4*sin |-|| |
| | \2/| 4/pi x\|
| |1 - ---------| *sin(x)*sin |-- + -||
| | 2 | \2 2/|
\ \ sin (x) / /
--------------------------------------------------------------------------------------------
2 + x
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{2} \left(\frac{2 \left(\frac{4 x}{3} + \frac{8}{3}\right) \sin^{2}{\left(\frac{x}{2} \right)}}{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{3} \sin{\left(x \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{2}{3 \left(\sin{\left(2 x + \frac{\pi}{2} \right)} + 1\right)}\right) \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{x + 2}$$
2 3
sec (x) sec (x)*(4 + 2*x)
------- + -----------------
3 3*csc(x)
---------------------------
2
(2 + x)*sec (x)
$$\frac{\frac{\left(2 x + 4\right) \sec^{3}{\left(x \right)}}{3 \csc{\left(x \right)}} + \frac{\sec^{2}{\left(x \right)}}{3}}{\left(x + 2\right) \sec^{2}{\left(x \right)}}$$
/ pi\ / pi\
4*cos|x - --| 2*x*cos|x - --|
\ 2 / \ 2 /
1 + ------------- + ---------------
cos(x) cos(x)
-----------------------------------
3*(2 + x)
$$\frac{\frac{2 x \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} + 1 + \frac{4 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}{3 \left(x + 2\right)}$$
2 2
8*sin (x) 4*x*sin (x)
1 + --------- + -----------
sin(2*x) sin(2*x)
---------------------------
3*(2 + x)
$$\frac{\frac{4 x \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{8 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1}{3 \left(x + 2\right)}$$
/pi \ /pi \
4*csc|-- - x| 2*x*csc|-- - x|
\2 / \2 /
1 + ------------- + ---------------
csc(x) csc(x)
-----------------------------------
3*(2 + x)
$$\frac{\frac{2 x \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + 1 + \frac{4 \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}}{3 \left(x + 2\right)}$$
2
/ 2/x pi\\ / 5/x pi\ /8 4*x\ \
| sec |- - --|| | sec |- - --|*|- + ---| |
| \2 2 /| | 2 \2 2 / \3 3 / |
|-1 + ------------| *|---------------- + ---------------------------|
| 2/x\ | | / 1 \ 3 |
| sec |-| | |3*|1 + --------| / 2/x pi\\ |
\ \2/ / | \ sec(2*x)/ | sec |- - --|| |
| | \2 2 /| /x\|
| |-1 + ------------| *sec|-||
| | 2/x\ | \2/|
| | sec |-| | |
\ \ \2/ / /
---------------------------------------------------------------------
4/x pi\
(2 + x)*sec |- - --|
\2 2 /
$$\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2}{3 \left(1 + \frac{1}{\sec{\left(2 x \right)}}\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \sec^{5}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \sec{\left(\frac{x}{2} \right)}}\right)}{\left(x + 2\right) \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 4 \
| / 2/x\\ /8 4*x\ /x\|
2 4 | |1 + tan |-|| *|- + ---|*tan|-||
/ 2/x\\ / 2/x\\ | 2 \ \4// \3 3 / \2/|
|1 - tan |-|| *|1 - tan |-|| *|------------------- + -------------------------------|
\ \2// \ \4// | / 2 \ 3 4 |
| | 1 - tan (x)| / 2/x\\ / 2/x\\ |
|3*|1 + -----------| |1 - tan |-|| *|1 - tan |-|| |
| | 2 | \ \2// \ \4// |
\ \ 1 + tan (x)/ /
-------------------------------------------------------------------------------------
4
/ 2/x\\
|1 + tan |-|| *(2 + x)
\ \4//
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\frac{2}{3 \left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3}}\right)}{\left(x + 2\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}$$
/ 5/x\ /8 4*x\ /x\\
2 | sec |-|*|- + ---|*sin|-||
/ 2/x\\ 4/x\ | 1 \2/ \3 3 / \2/|
|1 - tan |-|| *cos |-|*|--------- + ------------------------|
\ \2// \2/ | 2 3 |
|3*cos (x) / 2/x\\ |
| |1 - tan |-|| |
\ \ \2// /
-------------------------------------------------------------
2 + x
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\frac{1}{3 \cos^{2}{\left(x \right)}} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \sin{\left(\frac{x}{2} \right)} \sec^{5}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3}}\right) \cos^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
/ 3 / 1 x\ /x\\
| (-1 + cos(x)) *|- - - -|*cos|-||
2 4/x\ | 1 \ 3 6/ \2/|
4*cos (x)*sin |-|*|--------- + -------------------------------|
\2/ | 2 3 5/x\ |
|3*cos (x) cos (x)*sin |-| |
\ \2/ /
---------------------------------------------------------------
2
(-1 + cos(x)) *(2 + x)
$$\frac{4 \left(\frac{\left(- \frac{x}{6} - \frac{1}{3}\right) \left(\cos{\left(x \right)} - 1\right)^{3} \cos{\left(\frac{x}{2} \right)}}{\sin^{5}{\left(\frac{x}{2} \right)} \cos^{3}{\left(x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \sin^{4}{\left(\frac{x}{2} \right)} \cos^{2}{\left(x \right)}}{\left(x + 2\right) \left(\cos{\left(x \right)} - 1\right)^{2}}$$
2
/ 2/x\ \ / /8 4*x\ /x\ \
| cos |-| | | |- + ---|*cos|-| |
| \2/ | 4/x pi\ | 2 \3 3 / \2/ |
|-1 + ------------| *cos |- - --|*|---------------- + ---------------------------------|
| 2/x pi\| \2 2 / |3*(1 + cos(2*x)) 3 |
| cos |- - --|| | / 2/x\ \ |
\ \2 2 // | | cos |-| | |
| | \2/ | 5/x pi\|
| |-1 + ------------| *cos |- - --||
| | 2/x pi\| \2 2 /|
| | cos |- - --|| |
\ \ \2 2 // /
----------------------------------------------------------------------------------------
2 + x
$$\frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2} \left(\frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{3} \cos^{5}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{2}{3 \left(\cos{\left(2 x \right)} + 1\right)}\right) \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{x + 2}$$
/ 2 2 \
| / 2/x\\ / 2/x\\ /x\|
2 | |1 + cot |-|| 2*|1 + cot |-|| *(4 + 2*x)*cot|-||
/ 2/x\\ | \ \2// \ \2// \2/|
|-1 + cot |-|| *|----------------- + ---------------------------------|
\ \2// | 2 3 |
| / 2/x\\ / 2/x\\ |
|3*|-1 + cot |-|| 3*|-1 + cot |-|| |
\ \ \2// \ \2// /
-----------------------------------------------------------------------
2
/ 2/x\\
|1 + cot |-|| *(2 + x)
\ \2//
$$\frac{\left(\frac{2 \left(2 x + 4\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}}{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3}} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(x + 2\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
/ 5/x\ /8 4*x\ /x\\
2 | csc |-|*|- + ---|*cos|-||
/ 1 \ 4/x\ | 1 \2/ \3 3 / \2/|
|-1 + -------| *sin |-|*|--------- + ------------------------|
| 2/x\| \2/ | 2 3 |
| tan |-|| |3*cos (x) / 1 \ |
\ \2// | |-1 + -------| |
| | 2/x\| |
| | tan |-|| |
\ \ \2// /
--------------------------------------------------------------
2 + x
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{1}{3 \cos^{2}{\left(x \right)}} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \cos{\left(\frac{x}{2} \right)} \csc^{5}{\left(\frac{x}{2} \right)}}{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{3}}\right) \sin^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
/ 4 \
| / 2/x\\ /8 4*x\ |
2 | |1 + tan |-|| *|- + ---| |
/ 1 \ 4/x\ | 2 \ \4// \3 3 / |
16*|-1 + -------| *tan |-|*|------------------- + ---------------------------------|
| 2/x\| \4/ | / 2 \ 3 |
| tan |-|| | | 1 - tan (x)| / 1 \ /x\ 4/x\|
\ \2// |3*|1 + -----------| 16*|-1 + -------| *tan|-|*tan |-||
| | 2 | | 2/x\| \2/ \4/|
| \ 1 + tan (x)/ | tan |-|| |
\ \ \2// /
------------------------------------------------------------------------------------
4
/ 2/x\\
|1 + tan |-|| *(2 + x)
\ \4//
$$\frac{16 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2}{3 \left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}{16 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \tan^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}\right) \tan^{4}{\left(\frac{x}{4} \right)}}{\left(x + 2\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}$$
/ 4 \
| / 2/x\\ /8 4*x\ /x\|
2 | |1 + cot |-|| *|- + ---|*cot|-||
/ 2/x\\ 4/x\ | 2 \ \4// \3 3 / \2/|
16*|-1 + cot |-|| *cot |-|*|-------------------- + -------------------------------|
\ \2// \4/ | / 2 \ 3 |
| | -1 + cot (x)| / 2/x\\ 4/x\ |
|3*|1 + ------------| 16*|-1 + cot |-|| *cot |-| |
| | 2 | \ \2// \4/ |
\ \ 1 + cot (x) / /
-----------------------------------------------------------------------------------
4
/ 2/x\\
|1 + cot |-|| *(2 + x)
\ \4//
$$\frac{16 \left(\frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} \right)}}{16 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \cot^{4}{\left(\frac{x}{4} \right)}} + \frac{2}{3 \left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1\right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \cot^{4}{\left(\frac{x}{4} \right)}}{\left(x + 2\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}$$
2
/ 2/x pi\\ / /8 4*x\ /x pi\ \
| cos |- - --|| | |- + ---|*cos|- - --| |
| \2 2 /| 4/x\ | 2 \3 3 / \2 2 / |
|1 - ------------| *cos |-|*|---------------- + ---------------------------|
| 2/x\ | \2/ |3*(1 + cos(2*x)) 3 |
| cos |-| | | / 2/x pi\\ |
\ \2/ / | | cos |- - --|| |
| | \2 2 /| 5/x\|
| |1 - ------------| *cos |-||
| | 2/x\ | \2/|
| | cos |-| | |
\ \ \2/ / /
----------------------------------------------------------------------------
2 + x
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2}{3 \left(\cos{\left(2 x \right)} + 1\right)} + \frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \cos^{5}{\left(\frac{x}{2} \right)}}\right) \cos^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
/ 1 x \
2 | - + -- |
/ 1 \ 8/x\ 4/x\ | 1 6 12 |
16*|-1 + -------| *cos |-|*tan |-|*|--------- + --------------------------------------|
| 2/x\| \4/ \4/ | 2 3 |
| tan |-|| |3*cos (x) / 1 \ 8/x\ /x\ 4/x\|
\ \2// | |-1 + -------| *cos |-|*tan|-|*tan |-||
| | 2/x\| \4/ \2/ \4/|
| | tan |-|| |
\ \ \2// /
---------------------------------------------------------------------------------------
2 + x
$$\frac{16 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{1}{3 \cos^{2}{\left(x \right)}} + \frac{\frac{x}{12} + \frac{1}{6}}{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}\right) \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)}}{x + 2}$$
2 / /4 2*x\ \
/ 2 \ | |- + ---|*sin(x) |
| sin (x) | 4/x\ | 1 \3 3 / |
|-1 + ---------| *sin |-|*|--------- + -------------------------|
| 4/x\| \2/ | 2 3 |
| 4*sin |-|| |3*cos (x) / 2 \ |
\ \2// | | sin (x) | 6/x\|
| |-1 + ---------| *sin |-||
| | 4/x\| \2/|
| | 4*sin |-|| |
\ \ \2// /
-----------------------------------------------------------------
2 + x
$$\frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{1}{3 \cos^{2}{\left(x \right)}} + \frac{\left(\frac{2 x}{3} + \frac{4}{3}\right) \sin{\left(x \right)}}{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)}}\right) \sin^{4}{\left(\frac{x}{2} \right)}}{x + 2}$$
1 + 4*tan(x) + 2*x*tan(x)
-------------------------
3*(x + 2)
$$\frac{2 x \tan{\left(x \right)} + 4 \tan{\left(x \right)} + 1}{3 \left(x + 2\right)}$$
2
/ 2/x\ \ / 5/x\ /8 4*x\ \
| sec |-| | | sec |-|*|- + ---| |
| \2/ | | 2 \2/ \3 3 / |
|1 - ------------| *|---------------- + -------------------------------|
| 2/x pi\| | / 1 \ 3 |
| sec |- - --|| |3*|1 + --------| / 2/x\ \ |
\ \2 2 // | \ sec(2*x)/ | sec |-| | |
| | \2/ | /x pi\|
| |1 - ------------| *sec|- - --||
| | 2/x pi\| \2 2 /|
| | sec |- - --|| |
\ \ \2 2 // /
------------------------------------------------------------------------
4/x\
(2 + x)*sec |-|
\2/
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \left(\frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \sec^{5}{\left(\frac{x}{2} \right)}}{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{3} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{2}{3 \left(1 + \frac{1}{\sec{\left(2 x \right)}}\right)}\right)}{\left(x + 2\right) \sec^{4}{\left(\frac{x}{2} \right)}}$$
2
/ 2/x\ \ / 5/x\ /8 4*x\ \
| csc |-| | | csc |-|*|- + ---| |
| \2/ | | 2 \2/ \3 3 / |
|-1 + ------------| *|--------------------- + --------------------------------|
| 2/pi x\| | / 1 \ 3 |
| csc |-- - -|| |3*|1 + -------------| / 2/x\ \ |
\ \2 2// | | /pi \| | csc |-| | |
| | csc|-- - 2*x|| | \2/ | /pi x\|
| \ \2 // |-1 + ------------| *csc|-- - -||
| | 2/pi x\| \2 2/|
| | csc |-- - -|| |
\ \ \2 2// /
-------------------------------------------------------------------------------
4/x\
(2 + x)*csc |-|
\2/
$$\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2} \left(\frac{\left(\frac{4 x}{3} + \frac{8}{3}\right) \csc^{5}{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{3} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{2}{3 \left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right)}\right)}{\left(x + 2\right) \csc^{4}{\left(\frac{x}{2} \right)}}$$
/ 3 /1 x\ /x\\
| (1 + cos(x)) *|- + -|*sin|-||
2 4/x\ | 1 \3 6/ \2/|
4*cos (x)*cos |-|*|--------- + ----------------------------|
\2/ | 2 3 5/x\ |
|3*cos (x) cos (x)*cos |-| |
\ \2/ /
------------------------------------------------------------
2
(1 + cos(x)) *(2 + x)
$$\frac{4 \left(\frac{\left(\frac{x}{6} + \frac{1}{3}\right) \left(\cos{\left(x \right)} + 1\right)^{3} \sin{\left(\frac{x}{2} \right)}}{\cos^{5}{\left(\frac{x}{2} \right)} \cos^{3}{\left(x \right)}} + \frac{1}{3 \cos^{2}{\left(x \right)}}\right) \cos^{4}{\left(\frac{x}{2} \right)} \cos^{2}{\left(x \right)}}{\left(x + 2\right) \left(\cos{\left(x \right)} + 1\right)^{2}}$$
4*cos(x)^2*cos(x/2)^4*(1/(3*cos(x)^2) + (1 + cos(x))^3*(1/3 + x/6)*sin(x/2)/(cos(x)^3*cos(x/2)^5))/((1 + cos(x))^2*(2 + x))