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