Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)^(3)/(1+cos(x))+sin(x)^(3)/(1-sin(x)^(2))
¿Cómo vas a descomponer esta sin(x)^(3)/(1+cos(x))+sin(x)^(3)/(1-sin(x)^(2)) expresión en fracciones?
Solución
Ha introducido
[src]
3 3
sin (x) sin (x)
---------- + -----------
1 + cos(x) 2
1 - sin (x)
$$\frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\sin^{3}{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}}$$
sin(x)^3/(1 + cos(x)) + sin(x)^3/(1 - sin(x)^2)
Simplificación general
[src]
3 / 2 \
sin (x)*\2 - sin (x) + cos(x)/
------------------------------
2
(1 + cos(x))*cos (x)
$$\frac{\left(- \sin^{2}{\left(x \right)} + \cos{\left(x \right)} + 2\right) \sin^{3}{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}$$
sin(x)^3*(2 - sin(x)^2 + cos(x))/((1 + cos(x))*cos(x)^2)
Respuesta numérica
[src]
sin(x)^3/(1.0 - sin(x)^2) + sin(x)^3/(1.0 + cos(x))
sin(x)^3/(1.0 - sin(x)^2) + sin(x)^3/(1.0 + cos(x))
Denominador racional
[src]
3 / 2 \ 3
sin (x)*\1 - sin (x)/ + sin (x)*(1 + cos(x))
--------------------------------------------
/ 2 \
\1 - sin (x)/*(1 + cos(x))
$$\frac{\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{3}{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)}}{\left(1 - \sin^{2}{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)}$$
(sin(x)^3*(1 - sin(x)^2) + sin(x)^3*(1 + cos(x)))/((1 - sin(x)^2)*(1 + cos(x)))
Unión de expresiones racionales
[src]
3 / 2 \ sin (x)*\2 - sin (x) + cos(x)/ ------------------------------ / 2 \ \1 - sin (x)/*(1 + cos(x))
$$\frac{\left(- \sin^{2}{\left(x \right)} + \cos{\left(x \right)} + 2\right) \sin^{3}{\left(x \right)}}{\left(1 - \sin^{2}{\left(x \right)}\right) \left(\cos{\left(x \right)} + 1\right)}$$
sin(x)^3*(2 - sin(x)^2 + cos(x))/((1 - sin(x)^2)*(1 + cos(x)))
Potencias
[src]
3 3
/ -I*x I*x\ / -I*x I*x\
I*\- e + e / I*\- e + e /
------------------------- + --------------------
/ 2\ / I*x -I*x\
| / -I*x I*x\ | | e e |
| \- e + e / | 8*|1 + ---- + -----|
8*|1 + -----------------| \ 2 2 /
\ 4 /
$$\frac{i \left(e^{i x} - e^{- i x}\right)^{3}}{8 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)} + \frac{i \left(e^{i x} - e^{- i x}\right)^{3}}{8 \left(\frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4} + 1\right)}$$
i*(-exp(-i*x) + exp(i*x))^3/(8*(1 + (-exp(-i*x) + exp(i*x))^2/4)) + i*(-exp(-i*x) + exp(i*x))^3/(8*(1 + exp(i*x)/2 + exp(-i*x)/2))
Denominador común
[src]
3 5
sin (x) - sin (x) + cos(x)*sin(x) + sin(x)
-sin(x) - ------------------------------------------
2 2
-1 + sin (x) - cos(x) + sin (x)*cos(x)
$$- \sin{\left(x \right)} - \frac{- \sin^{5}{\left(x \right)} + \sin^{3}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}}{\sin^{2}{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)} - \cos{\left(x \right)} - 1}$$
-sin(x) - (sin(x)^3 - sin(x)^5 + cos(x)*sin(x) + sin(x))/(-1 + sin(x)^2 - cos(x) + sin(x)^2*cos(x))
Combinatoria
[src]
3 / 2 \
-sin (x)*\2 - sin (x) + cos(x)/
---------------------------------------
(1 + cos(x))*(1 + sin(x))*(-1 + sin(x))
$$- \frac{\left(- \sin^{2}{\left(x \right)} + \cos{\left(x \right)} + 2\right) \sin^{3}{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(\sin{\left(x \right)} + 1\right) \left(\cos{\left(x \right)} + 1\right)}$$
-sin(x)^3*(2 - sin(x)^2 + cos(x))/((1 + cos(x))*(1 + sin(x))*(-1 + sin(x)))
Parte trigonométrica
[src]
3/x\ 3/x\
8*cot |-| 8*cot |-|
\2/ \2/
--------------------------------- + -----------------------------------
/ 2/x\\ / 2/x\ \
3 | -1 + cot |-|| 3 | 4*cot |-| |
/ 2/x\\ | \2/| / 2/x\\ | \2/ |
|1 + cot |-|| *|1 + ------------| |1 + cot |-|| *|1 - --------------|
\ \2// | 2/x\ | \ \2// | 2|
| 1 + cot |-| | | / 2/x\\ |
\ \2/ / | |1 + cot |-|| |
\ \ \2// /
$$\frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
6/x\ 6/pi x\
64*sin |-|*sin |-- + -|
-sin(3*x) + 3*sin(x) \2/ \2 2/
-------------------- + -----------------------
/ / pi\\ 3 2/ pi\
4*|1 + sin|x + --|| sin (x)*sin |x + --|
\ \ 2 // \ 2 /
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} + \frac{64 \sin^{6}{\left(\frac{x}{2} \right)} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin^{3}{\left(x \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)}}$$
/3*x\ /x\
2*tan|---| 6*tan|-|
\ 2 / \2/
- ------------- + ----------- 2 6
2/3*x\ 2/x\ / 2/x\\ / 2/x\\ 3/x\
1 + tan |---| 1 + tan |-| 8*|1 + tan |-|| *|1 - tan |-|| *tan |-|
\ 2 / \2/ \ \2// \ \4// \2/
----------------------------- + ---------------------------------------
/ 2/x\\ 6 2
| 1 - tan |-|| / 2/x\\ / 2/x\\
| \2/| |1 + tan |-|| *|1 - tan |-||
4*|1 + -----------| \ \4// \ \2//
| 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} + \frac{- \frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)}$$
3 3
sin (x) sin (x)
----------- + ---------------
2 / pi\
1 - sin (x) 1 + sin|x + --|
\ 2 /
$$\frac{\sin^{3}{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1} + \frac{\sin^{3}{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}}$$
/3*x\ /x\
2*cot|---| 6*cot|-|
\ 2 / \2/
- ------------- + ----------- 2
2/3*x\ 2/x\ / 2/x\\ 3/x\ 6/x\
1 + cot |---| 1 + cot |-| 512*|1 + cot |-|| *cot |-|*cot |-|
\ 2 / \2/ \ \2// \2/ \4/
----------------------------- + ----------------------------------
/ 2/x\\ 6 2
| -1 + cot |-|| / 2/x\\ / 2/x\\
| \2/| |1 + cot |-|| *|-1 + cot |-||
4*|1 + ------------| \ \4// \ \2//
| 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{512 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot^{6}{\left(\frac{x}{4} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}} + \frac{- \frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)}$$
3/ pi\ 3/ pi\
cos |x - --| cos |x - --|
\ 2 / \ 2 /
---------------- + ------------
2/ pi\ 1 + cos(x)
1 - cos |x - --|
\ 2 /
$$\frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1} + \frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{1 - \cos^{2}{\left(x - \frac{\pi}{2} \right)}}$$
1 3 2/pi \ - -------- + ------ 8*csc |-- - x| csc(3*x) csc(x) \2 / ------------------- + -------------------- / 1 \ 3/x\ 3/pi x\ 4*|1 + -----------| csc |-|*csc |-- - -| | /pi \| \2/ \2 2/ | csc|-- - x|| \ \2 //
$$\frac{8 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{- \frac{1}{\csc{\left(3 x \right)}} + \frac{3}{\csc{\left(x \right)}}}{4 \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)}$$
2
1 sec (x)
------------------------- + ------------
/ 1 \ 3/ pi\ 3/ pi\
|1 + ------|*sec |x - --| sec |x - --|
\ sec(x)/ \ 2 / \ 2 /
$$\frac{\sec^{2}{\left(x \right)}}{\sec^{3}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}}$$
3/ pi\ 3/ pi\
cos |x - --| cos |x - --|
\ 2 / \ 2 /
------------ + ------------
1 + cos(x) 2
cos (x)
$$\frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}} + \frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}$$
/3*x\ /x\
2*tan|---| 6*tan|-|
\ 2 / \2/
- ------------- + ----------- 2
2/3*x\ 2/x\ / 2/x\\ 6/x\
1 + tan |---| 1 + tan |-| 512*|1 + tan |-|| *tan |-|
\ 2 / \2/ \ \2// \4/
----------------------------- + -------------------------------------
/ 2/x\\ 6 2
| 1 - tan |-|| / 2/x\\ / 2/x\\ 3/x\
| \2/| |1 + tan |-|| *|1 - tan |-|| *tan |-|
4*|1 + -----------| \ \4// \ \2// \2/
| 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{- \frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)} + \frac{512 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{6}{\left(\frac{x}{4} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \tan^{3}{\left(\frac{x}{2} \right)}}$$
3
sin (x) -sin(3*x) + 3*sin(x)
------------ + --------------------
2/ pi\ / / pi\\
sin |x + --| 4*|1 + sin|x + --||
\ 2 / \ \ 2 //
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} + \frac{\sin^{3}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}}$$
3 sin (x) -sin(3*x) + 3*sin(x) ------- + -------------------- 2 4*(1 + cos(x)) cos (x)
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$
6
/ 2/x\\ 12/x\ 3/x\
8*|-1 + cot |-|| *sin |-|*tan |-|
-sin(3*x) + 3*sin(x) \ \4// \4/ \2/
-------------------- + ----------------------------------
4*(1 + cos(x)) 2
/ 2/x\\ 4/x\
|-1 + cot |-|| *sin |-|
\ \2// \2/
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}$$
-sin(3*x) + 3*sin(x) 3/x\ 2 3/x\ -------------------- + 8*cos |-|*sec (x)*sin |-| 4*(1 + cos(x)) \2/ \2/
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + 8 \sin^{3}{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \sec^{2}{\left(x \right)}$$
3/x\ 3/x\
8*tan |-| 8*tan |-|
\2/ \2/
-------------------------------- + -----------------------------------
/ 2/x\\ / 2/x\ \
3 | 1 - tan |-|| 3 | 4*tan |-| |
/ 2/x\\ | \2/| / 2/x\\ | \2/ |
|1 + tan |-|| *|1 + -----------| |1 + tan |-|| *|1 - --------------|
\ \2// | 2/x\| \ \2// | 2|
| 1 + tan |-|| | / 2/x\\ |
\ \2// | |1 + tan |-|| |
\ \ \2// /
$$\frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
/ pi\ / pi\ 3/x\ 3/x pi\
- cos|3*x - --| + 3*cos|x - --| 8*cos |-|*cos |- - --|
\ 2 / \ 2 / \2/ \2 2 /
------------------------------- + ----------------------
4*(1 + cos(x)) 2
cos (x)
$$\frac{8 \cos^{3}{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \cos{\left(x - \frac{\pi}{2} \right)} - \cos{\left(3 x - \frac{\pi}{2} \right)}}{4 \left(\cos{\left(x \right)} + 1\right)}$$
3 sin (x) 2 3 ---------- + sec (x)*sin (x) 1 + cos(x)
$$\sin^{3}{\left(x \right)} \sec^{2}{\left(x \right)} + \frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 1}$$
6
/ 2/x\\ 12/x\ 3/x\
8*|1 - tan |-|| *cos |-|*tan |-|
-sin(3*x) + 3*sin(x) \ \4// \4/ \2/
-------------------- + ---------------------------------
4*(1 + cos(x)) 2
/ 2/x\\ 4/x\
|1 - tan |-|| *cos |-|
\ \2// \2/
$$\frac{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \cos^{12}{\left(\frac{x}{4} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}} + \frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)}$$
1 1
-------------------- + ---------------------
/ 1 \ 3 / 1 \ 3
|1 + ------|*csc (x) |1 - -------|*csc (x)
\ sec(x)/ | 2 |
\ csc (x)/
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc^{3}{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\csc^{2}{\left(x \right)}}\right) \csc^{3}{\left(x \right)}}$$
3/x\ 6/x\
8*cot |-|*sin |-|
-sin(3*x) + 3*sin(x) \2/ \2/
-------------------- + -----------------
4*(1 + cos(x)) 2
cos (x)
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \sin^{6}{\left(\frac{x}{2} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(x \right)}}$$
3 3
sin (x) sin (x)
---------- + -------
1 + cos(x) 2
cos (x)
$$\frac{\sin^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 1}$$
3/x\ 3/x\
8*tan |-| 8*tan |-|
\2/ \2/
---------------------------- + --------------------------------
2 / 2/x\\
/ 2/x\\ / 2/x\\ 3 | 1 - tan |-||
|1 + tan |-||*|1 - tan |-|| / 2/x\\ | \2/|
\ \2// \ \2// |1 + tan |-|| *|1 + -----------|
\ \2// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
3 3
sin (x) sin (x)
--------------- + ------------
/ pi\ 2/ pi\
1 + sin|x + --| sin |x + --|
\ 2 / \ 2 /
$$\frac{\sin^{3}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} + \frac{\sin^{3}{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}$$
-sin(3*x) + 3*sin(x) -sin(3*x) + 3*sin(x) -------------------- + -------------------- 2*(1 + cos(2*x)) 4*(1 + cos(x))
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{2 \left(\cos{\left(2 x \right)} + 1\right)} + \frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)}$$
1 1
------------------------- + -------------------------------
/ 1 \ 3/ pi\ / 1 \ 3/ pi\
|1 + ------|*sec |x - --| |1 - ------------|*sec |x - --|
\ sec(x)/ \ 2 / | 2/ pi\| \ 2 /
| sec |x - --||
\ \ 2 //
$$\frac{1}{\left(1 - \frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}}$$
3/x\ 3/x\
8*cos |-|*sin |-|
-sin(3*x) + 3*sin(x) \2/ \2/
-------------------- + -----------------
4*(1 + cos(x)) 2
cos (x)
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \sin^{3}{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(x \right)}}$$
/3*x\ /x\
2*cot|---| 6*cot|-|
\ 2 / \2/
- ------------- + ----------- 2 6
2/3*x\ 2/x\ / 2/x\\ / 2/x\\
1 + cot |---| 1 + cot |-| 8*|1 + cot |-|| *|-1 + cot |-||
\ 2 / \2/ \ \2// \ \4//
----------------------------- + --------------------------------------
/ 2/x\\ 6 2
| -1 + cot |-|| / 2/x\\ / 2/x\\ 3/x\
| \2/| |1 + cot |-|| *|-1 + cot |-|| *cot |-|
4*|1 + ------------| \ \4// \ \2// \2/
| 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{8 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \cot^{3}{\left(\frac{x}{2} \right)}} + \frac{- \frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)}$$
3/x\ 6/x\ 12/x\
512*cot |-|*cot |-|*sin |-|
-sin(3*x) + 3*sin(x) \2/ \4/ \4/
-------------------- + ----------------------------
4*(1 + cos(x)) 2
/ 2/x\\ 4/x\
|-1 + cot |-|| *sin |-|
\ \2// \2/
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{512 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{2} \right)}}$$
1 1 ------------------------- + --------------------- / 1 \ 3 / 1 \ 3 |1 + -----------|*csc (x) |1 - -------|*csc (x) | /pi \| | 2 | | csc|-- - x|| \ csc (x)/ \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(x \right)}} + \frac{1}{\left(1 - \frac{1}{\csc^{2}{\left(x \right)}}\right) \csc^{3}{\left(x \right)}}$$
6/x\ 3/x\
8*cos |-|*tan |-|
-sin(3*x) + 3*sin(x) \2/ \2/
-------------------- + -----------------
4*(1 + cos(x)) 2
cos (x)
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{8 \cos^{6}{\left(\frac{x}{2} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(x \right)}}$$
3/x\ 3/x\
8*cot |-| 8*cot |-|
\2/ \2/
----------------------------- + ---------------------------------
2 / 2/x\\
/ 2/x\\ / 2/x\\ 3 | -1 + cot |-||
|1 + cot |-||*|-1 + cot |-|| / 2/x\\ | \2/|
\ \2// \ \2// |1 + cot |-|| *|1 + ------------|
\ \2// | 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
1 3
- ------------- + -----------
/ pi\ / pi\
sec|3*x - --| sec|x - --| 2
\ 2 / \ 2 / 8*sec (x)
----------------------------- + --------------------
/ 1 \ 3/x\ 3/x pi\
4*|1 + ------| sec |-|*sec |- - --|
\ sec(x)/ \2/ \2 2 /
$$\frac{8 \sec^{2}{\left(x \right)}}{\sec^{3}{\left(\frac{x}{2} \right)} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{- \frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(x - \frac{\pi}{2} \right)}}}{4 \left(1 + \frac{1}{\sec{\left(x \right)}}\right)}$$
2/pi \
csc |-- - x|
1 \2 /
------------------------- + ------------
/ 1 \ 3 3
|1 + -----------|*csc (x) csc (x)
| /pi \|
| csc|-- - x||
\ \2 //
$$\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(x \right)}} + \frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(x \right)}}$$
1/((1 + 1/csc(pi/2 - x))*csc(x)^3) + csc(pi/2 - x)^2/csc(x)^3