Parte trigonométrica
[src]
/ 2/x\\ /x\
2*|1 - tan |-||*tan|-|
\ \2// \2/
-----------------------------------------------------------------------
/ / 2/x\\ /x\\ / 2/x\ /x\ \
2 | 2*|1 - tan |-||*tan|-|| |1 - tan |-| 2*tan|-| |
/ 2/x\\ | \ \2// \2/| | \2/ \2/ |
|1 + tan |-|| *|1 - ----------------------|*|----------- + -----------|
\ \2// | 2 | | 2/x\ 2/x\|
| / 2/x\\ | |1 + tan |-| 1 + tan |-||
| |1 + tan |-|| | \ \2/ \2//
\ \ \2// /
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
___
2*\/ 2 *sin(2*x)
-----------------------------------------------------------------------------------------------------------------
___________ ___________ ___________ ___________
___ ___ / ___ / ___ / ___ / ___
\/ 2 *cos(3*x) + 3*\/ 2 *sin(x) - \/ 2 + \/ 2 *\/ 2 - \/ 2 *sin(3*x) + 3*\/ 2 + \/ 2 *\/ 2 - \/ 2 *cos(x)
$$\frac{2 \sqrt{2} \sin{\left(2 x \right)}}{3 \sqrt{2} \sin{\left(x \right)} - \sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(3 x \right)} + 3 \sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \cos{\left(x \right)} + \sqrt{2} \cos{\left(3 x \right)}}$$
1
------------------------------------------------------------------
/ 1 \ / 1 1 \ / pi\
|1 - ------------------|*|------ + -----------|*sec(x)*sec|x - --|
| / pi\| |sec(x) / pi\| \ 2 /
| sec(x)*sec|x - --|| | sec|x - --||
\ \ 2 // \ \ 2 //
$$\frac{1}{\left(1 - \frac{1}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
___ 4/x\ / 2/x\\ /x\
\/ 2 *cos |-|*|1 - tan |-||*tan|-|
\2/ \ \2// \2/
----------------------------------
/ sin(2*x)\ / pi\
|1 - --------|*sin|x + --|
\ 2 / \ 4 /
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{4}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}$$
/ 2 \
___ 2/x\ | sin (x) |
\/ 2 *sin |-|*|-1 + ---------|*sin(x)
\2/ | 4/x\|
| 4*sin |-||
\ \2//
-------------------------------------
/ sin(2*x)\ / pi\
2*|1 - --------|*sin|x + --|
\ 2 / \ 4 /
$$\frac{\sqrt{2} \left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{2 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}$$
___ 3/x\ / 1 \ /x\ / pi\
\/ 2 *sin |-|*|-1 + -------|*cos|-|*csc|x + --|
\2/ | 2/x\| \2/ \ 4 /
| tan |-||
\ \2//
-----------------------------------------------
sin(2*x)
1 - --------
2
$$\frac{\sqrt{2} \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \csc{\left(x + \frac{\pi}{4} \right)}}{1 - \frac{\sin{\left(2 x \right)}}{2}}$$
___
\/ 2 *sin(2*x)
----------------------------
/ sin(2*x)\ / pi\
4*|1 - --------|*sin|x + --|
\ 2 / \ 4 /
$$\frac{\sqrt{2} \sin{\left(2 x \right)}}{4 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}$$
/ pi\
sin(x)*sin|x + --|
\ 2 /
-----------------------------------------------
/ / pi\\ / / pi\\
|1 - sin(x)*sin|x + --||*|sin(x) + sin|x + --||
\ \ 2 // \ \ 2 //
$$\frac{\sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\left(- \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)} + 1\right) \left(\sin{\left(x \right)} + \sin{\left(x + \frac{\pi}{2} \right)}\right)}$$
___ 3/x\ /x\
-2*\/ 2 *sin |-|*cos(x)*cos|-|
\2/ \2/
----------------------------------------
/ sin(2*x)\ / pi\
|1 - --------|*(-1 + cos(x))*cos|x - --|
\ 2 / \ 4 /
$$- \frac{2 \sqrt{2} \sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} - 1\right) \cos{\left(x - \frac{\pi}{4} \right)}}$$
/ 2/x\ \
| cos |-| |
___ 3/x pi\ | \2/ | /x\
\/ 2 *cos |- - --|*|-1 + ------------|*cos|-|
\2 2 / | 2/x pi\| \2/
| cos |- - --||
\ \2 2 //
---------------------------------------------
/ / pi\\
| cos|2*x - --||
| \ 2 /| / pi\
|1 - -------------|*cos|x - --|
\ 2 / \ 4 /
$$\frac{\sqrt{2} \left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right) \cos{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{2}\right) \cos{\left(x - \frac{\pi}{4} \right)}}$$
/ 2/x pi\\
| cos |- - --||
___ 3/x\ | \2 2 /| /x pi\
\/ 2 *cos |-|*|1 - ------------|*cos|- - --|
\2/ | 2/x\ | \2 2 /
| cos |-| |
\ \2/ /
--------------------------------------------
/ / pi\\
| cos|2*x - --||
| \ 2 /| / pi\
|1 - -------------|*cos|x - --|
\ 2 / \ 4 /
$$\frac{\sqrt{2} \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{2}\right) \cos{\left(x - \frac{\pi}{4} \right)}}$$
___ 8/x\ 4/x\ / 2/x pi\\ / 1 \
8*\/ 2 *cos |-|*tan |-|*|1 + tan |- + --||*|-1 + -------|
\4/ \4/ \ \2 8 // | 2/x\|
| tan |-||
\ \2//
---------------------------------------------------------
/ sin(2*x)\ /x\ /x pi\
|1 - --------|*tan|-|*tan|- + --|
\ 2 / \2/ \2 8 /
$$\frac{8 \sqrt{2} \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
___ / pi\
\/ 2 *cos|2*x - --|
\ 2 /
-----------------------------
/ pi\ / pi\
3*cos|x - --| + cos|3*x + --|
\ 4 / \ 4 /
$$\frac{\sqrt{2} \cos{\left(2 x - \frac{\pi}{2} \right)}}{3 \cos{\left(x - \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}}$$
___ 4/x\ / 2/x pi\\ / 1 \
8*\/ 2 *tan |-|*|1 + tan |- + --||*|-1 + -------|
\4/ \ \2 8 // | 2/x\|
| tan |-||
\ \2//
---------------------------------------------------
4
/ 2/x\\ / tan(x) \ /x\ /x pi\
|1 + tan |-|| *|1 - -----------|*tan|-|*tan|- + --|
\ \4// | 2 | \2/ \2 8 /
\ 1 + tan (x)/
$$\frac{8 \sqrt{2} \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \tan^{4}{\left(\frac{x}{4} \right)}}{\left(1 - \frac{\tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
___ 4/x\ / 2/x\\ /x\
\/ 2 *sin |-|*|-1 + cot |-||*cot|-|
\2/ \ \2// \2/
-----------------------------------
/ sin(2*x)\ / pi\
|1 - --------|*sin|x + --|
\ 2 / \ 4 /
$$\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{4}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x + \frac{\pi}{4} \right)}}$$
cos(x)*sin(x)
-------------------------------------
(1 - cos(x)*sin(x))*(cos(x) + sin(x))
$$\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{\left(- \sin{\left(x \right)} \cos{\left(x \right)} + 1\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}$$
___
2*\/ 2 *tan(x)
-----------------------------------------------------
/ 2/pi 3*x\ /x pi\ \
|1 - tan |-- + ---| 6*tan|- + --| |
/ 2 \ | \8 2 / \2 8 / |
\1 + tan (x)/*|------------------ + ----------------|
| 2/pi 3*x\ 2/x pi\|
|1 + tan |-- + ---| 1 + tan |- + --||
\ \8 2 / \2 8 //
$$\frac{2 \sqrt{2} \tan{\left(x \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
___
\/ 2
---------------------------------------
/ 1 3 \
|-------------- + -----------|*csc(2*x)
| / pi\ / pi\|
|csc|-3*x + --| csc|x + --||
\ \ 4 / \ 4 //
$$\frac{\sqrt{2}}{\left(\frac{3}{\csc{\left(x + \frac{\pi}{4} \right)}} + \frac{1}{\csc{\left(- 3 x + \frac{\pi}{4} \right)}}\right) \csc{\left(2 x \right)}}$$
___ 2
2*\/ 2 *cos (x)*tan(x)
-------------------------------------
2/pi 3*x\ /x pi\
1 - tan |-- + ---| 6*tan|- + --|
\8 2 / \2 8 /
------------------ + ----------------
2/pi 3*x\ 2/x pi\
1 + tan |-- + ---| 1 + tan |- + --|
\8 2 / \2 8 /
$$\frac{2 \sqrt{2} \cos^{2}{\left(x \right)} \tan{\left(x \right)}}{\frac{1 - \tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}$$
/ 2/x\ \
| csc |-| |
___ | \2/ | / pi\
\/ 2 *|-1 + ------------|*csc|x + --|
| 2/pi x\| \ 4 /
| csc |-- - -||
\ \2 2//
-------------------------------------
/ 1 \ 3/x\ /pi x\
|1 - ----------|*csc |-|*csc|-- - -|
\ 2*csc(2*x)/ \2/ \2 2/
$$\frac{\sqrt{2} \left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc{\left(x + \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \csc{\left(2 x \right)}}\right) \csc^{3}{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
4
___ / 2/x\\ 8/x\ / 2/x pi\\ / 2/x\\ /x\
\/ 2 *|1 - tan |-|| *cos |-|*|1 + tan |- + --||*|1 - tan |-||*tan|-|
\ \4// \4/ \ \2 8 // \ \2// \2/
--------------------------------------------------------------------
/ sin(2*x)\ /x pi\
2*|1 - --------|*tan|- + --|
\ 2 / \2 8 /
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{2 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
___
\/ 2 *sin(2*x)
-----------------------------
/ pi\ / pi\
3*sin|x + --| + cos|3*x + --|
\ 4 / \ 4 /
$$\frac{\sqrt{2} \sin{\left(2 x \right)}}{3 \sin{\left(x + \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}}$$
___ 3/x\ / pi\ /x\
2*\/ 2 *cos |-|*cos(x)*csc|x + --|*sin|-|
\2/ \ 4 / \2/
-----------------------------------------
/ sin(2*x)\
|1 - --------|*(1 + cos(x))
\ 2 /
$$\frac{2 \sqrt{2} \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(x \right)} \csc{\left(x + \frac{\pi}{4} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} + 1\right)}$$
___ 4/x\ 8/x\ / 2/x\\ /x\ /x pi\
8*\/ 2 *cot |-|*sin |-|*|-1 + cot |-||*cot|-|*tan|- + --|
\4/ \4/ \ \2// \2/ \2 8 /
---------------------------------------------------------
/ sin(2*x)\ 2/x pi\
|1 - --------|*sin |- + --|
\ 2 / \2 8 /
$$\frac{8 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot^{4}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
___ 3/x\ /x\ / pi\
-2*\/ 2 *sin |-|*cos(x)*cos|-|*sec|x - --|
\2/ \2/ \ 4 /
------------------------------------------
/ sin(2*x)\
|1 - --------|*(-1 + cos(x))
\ 2 /
$$- \frac{2 \sqrt{2} \sin^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)} \sec{\left(x - \frac{\pi}{4} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} - 1\right)}$$
___ 3/x\ / 2/x\\ / pi\ /x\
\/ 2 *cos |-|*|1 - tan |-||*sec|x - --|*sin|-|
\2/ \ \2// \ 4 / \2/
----------------------------------------------
sin(2*x)
1 - --------
2
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \sec{\left(x - \frac{\pi}{4} \right)}}{1 - \frac{\sin{\left(2 x \right)}}{2}}$$
___ 3/x\ /x\
2*\/ 2 *cos |-|*cos(x)*sin|-|
\2/ \2/
---------------------------------------
/ sin(2*x)\ / pi\
|1 - --------|*(1 + cos(x))*cos|x - --|
\ 2 / \ 4 /
$$\frac{2 \sqrt{2} \sin{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \left(\cos{\left(x \right)} + 1\right) \cos{\left(x - \frac{\pi}{4} \right)}}$$
4
___ / 2/x\\ / 2/x pi\\ / 2/x\\ /x\
\/ 2 *|1 - tan |-|| *|1 + tan |- + --||*|1 - tan |-||*tan|-|
\ \4// \ \2 8 // \ \2// \2/
------------------------------------------------------------
4
/ 2/x\\ / tan(x) \ /x pi\
2*|1 + tan |-|| *|1 - -----------|*tan|- + --|
\ \4// | 2 | \2 8 /
\ 1 + tan (x)/
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2 \left(1 - \frac{\tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
/ pi\
cos(x)*cos|x - --|
\ 2 /
-----------------------------------------------
/ / pi\\ / / pi\\
|1 - cos(x)*cos|x - --||*|cos(x) + cos|x - --||
\ \ 2 // \ \ 2 //
$$\frac{\cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\left(- \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \left(\cos{\left(x \right)} + \cos{\left(x - \frac{\pi}{2} \right)}\right)}$$
___
\/ 2
-------------------------------------------
/ 1 3 \ / pi\
|------------- + -----------|*sec|2*x - --|
| / pi\ / pi\| \ 2 /
|sec|3*x + --| sec|x - --||
\ \ 4 / \ 4 //
$$\frac{\sqrt{2}}{\left(\frac{1}{\sec{\left(3 x + \frac{\pi}{4} \right)}} + \frac{3}{\sec{\left(x - \frac{\pi}{4} \right)}}\right) \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
/ 2/x\ \
| sec |-| |
___ | \2/ | / pi\
\/ 2 *|1 - ------------|*sec|x - --|
| 2/x pi\| \ 4 /
| sec |- - --||
\ \2 2 //
-----------------------------------------
/ 1 \ 3/x\ /x pi\
|1 - ---------------|*sec |-|*sec|- - --|
| / pi\| \2/ \2 2 /
| 2*sec|2*x - --||
\ \ 2 //
$$\frac{\sqrt{2} \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec^{3}{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
4
___ / 2/x\\ 8/x\ / 2/x\\ /x\ /x pi\
\/ 2 *|-1 + cot |-|| *sin |-|*|1 - tan |-||*tan|-|*tan|- + --|
\ \4// \4/ \ \2// \2/ \2 8 /
--------------------------------------------------------------
/ sin(2*x)\ 2/x pi\
2*|1 - --------|*sin |- + --|
\ 2 / \2 8 /
$$\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{2 \left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
/ 2/x pi\\
| sec |- - --||
___ | \2 2 /| / pi\
\/ 2 *|-1 + ------------|*sec|x - --|
| 2/x\ | \ 4 /
| sec |-| |
\ \2/ /
-----------------------------------------
/ 1 \ /x\ 3/x pi\
|1 - ---------------|*sec|-|*sec |- - --|
| / pi\| \2/ \2 2 /
| 2*sec|2*x - --||
\ \ 2 //
$$\frac{\sqrt{2} \left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 4/x\\
| 4*sin |-||
___ 2/x\ 4/pi x\ | \2/|
2*\/ 2 *sin |-|*sin |-- + -|*|1 - ---------|
\2/ \2 2/ | 2 |
\ sin (x) /
--------------------------------------------
/ sin(2*x)\ / pi\
|1 - --------|*sin(x)*sin|x + --|
\ 2 / \ 4 /
$$\frac{2 \sqrt{2} \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 - \frac{\sin{\left(2 x \right)}}{2}\right) \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{4} \right)}}$$
___
\/ 2 *sin(2*x)
-----------------------------
/ pi\ / pi\
3*cos|x - --| + cos|3*x + --|
\ 4 / \ 4 /
$$\frac{\sqrt{2} \sin{\left(2 x \right)}}{3 \cos{\left(x - \frac{\pi}{4} \right)} + \cos{\left(3 x + \frac{\pi}{4} \right)}}$$
___ 4/x\ / 2/x pi\\ / 2/x\\ /x\
8*\/ 2 *cot |-|*|1 + cot |- + --||*|-1 + cot |-||*cot|-|
\4/ \ \2 8 // \ \2// \2/
--------------------------------------------------------
4
/ 2/x\\ / cot(x) \ /x pi\
|1 + cot |-|| *|1 - -----------|*cot|- + --|
\ \4// | 2 | \2 8 /
\ 1 + cot (x)/
$$\frac{8 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right) \cot^{4}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \frac{\cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
1
------------------------------------------------------------------
/ 1 \ / 1 1 \ /pi \
|1 - ------------------|*|------ + -----------|*csc(x)*csc|-- - x|
| /pi \| |csc(x) /pi \| \2 /
| csc(x)*csc|-- - x|| | csc|-- - x||
\ \2 // \ \2 //
$$\frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
___
\/ 2 *sin(2*x)
-------------------------------
/ pi\ / 3*pi\
3*sin|x + --| + sin|3*x + ----|
\ 4 / \ 4 /
$$\frac{\sqrt{2} \sin{\left(2 x \right)}}{3 \sin{\left(x + \frac{\pi}{4} \right)} + \sin{\left(3 x + \frac{3 \pi}{4} \right)}}$$
___
2*\/ 2 *cot(x)
------------------------------------------------------
/ 2/pi 3*x\ /x pi\ \
|-1 + cot |-- + ---| 6*cot|- + --| |
/ 2 \ | \8 2 / \2 8 / |
\1 + cot (x)/*|------------------- + ----------------|
| 2/pi 3*x\ 2/x pi\|
| 1 + cot |-- + ---| 1 + cot |- + --||
\ \8 2 / \2 8 //
$$\frac{2 \sqrt{2} \cot{\left(x \right)}}{\left(\frac{\cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} + \frac{\pi}{8} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
/ 2/pi x\\
| csc |-- - -||
___ | \2 2/| / pi\
\/ 2 *|1 - ------------|*csc|x + --|
| 2/x\ | \ 4 /
| csc |-| |
\ \2/ /
------------------------------------
/ 1 \ /x\ 3/pi x\
|1 - ----------|*csc|-|*csc |-- - -|
\ 2*csc(2*x)/ \2/ \2 2/
$$\frac{\sqrt{2} \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(x + \frac{\pi}{4} \right)}}{\left(1 - \frac{1}{2 \csc{\left(2 x \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
1
---------------------------------------------------
/ 1 \ / 1 1 \
|1 - -------------|*|------ + ------|*csc(x)*sec(x)
\ csc(x)*sec(x)/ \csc(x) sec(x)/
$$\frac{1}{\left(1 - \frac{1}{\csc{\left(x \right)} \sec{\left(x \right)}}\right) \left(\frac{1}{\sec{\left(x \right)}} + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)} \sec{\left(x \right)}}$$
/ 2/x\\ /x\
2*|-1 + cot |-||*cot|-|
\ \2// \2/
-------------------------------------------------------------------------
/ / 2/x\\ /x\\ / 2/x\ /x\ \
2 | 2*|-1 + cot |-||*cot|-|| |-1 + cot |-| 2*cot|-| |
/ 2/x\\ | \ \2// \2/| | \2/ \2/ |
|1 + cot |-|| *|1 - -----------------------|*|------------ + -----------|
\ \2// | 2 | | 2/x\ 2/x\|
| / 2/x\\ | |1 + cot |-| 1 + cot |-||
| |1 + cot |-|| | \ \2/ \2//
\ \ \2// /
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(- \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
4
___ / 2/x\\ / 2/x pi\\ / 1 \
\/ 2 *|-1 + cot |-|| *|1 + cot |- + --||*|1 - -------|
\ \4// \ \2 8 // | 2/x\|
| cot |-||
\ \2//
------------------------------------------------------
4
/ 2/x\\ / cot(x) \ /x\ /x pi\
2*|1 + cot |-|| *|1 - -----------|*cot|-|*cot|- + --|
\ \4// | 2 | \2/ \2 8 /
\ 1 + cot (x)/
$$\frac{\sqrt{2} \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}{2 \left(1 - \frac{\cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}$$
sqrt(2)*(-1 + cot(x/4)^2)^4*(1 + cot(x/2 + pi/8)^2)*(1 - 1/cot(x/2)^2)/(2*(1 + cot(x/4)^2)^4*(1 - cot(x)/(1 + cot(x)^2))*cot(x/2)*cot(x/2 + pi/8))