Parte trigonométrica
[src]
4
/ 2/x\\ /x\
|1 + cot |-|| *cot|-|
\ \4// \2/
-------------------------------
2 4
/ 1 \ / 2/x\\
2*|1 + ------| *|-1 + cot |-||
| /x\| \ \4//
| cot|-||
\ \2//
$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot{\left(\frac{x}{2} \right)}}{2 \left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}}$$
/x\
csc|-|
\2/
-----------------------
2
/ /x\\ 3/x\
2*|1 + tan|-|| *cos |-|
\ \2// \2/
$$\frac{\csc{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{3}{\left(\frac{x}{2} \right)}}$$
3/pi x\ /x\
csc |-- - -|*csc|-|
\2 2/ \2/
--------------------
2
/ /pi x\\
| csc|-- - -||
| \2 2/|
2*|1 + -----------|
| /x\ |
| csc|-| |
\ \2/ /
$$\frac{\csc{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{2 \left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2}}$$
/ 4/x pi\ 2/x pi\\
| cos |- - --| 4*cos |- - --||
| \2 2 / \2 2 /| /x\
|1 + ------------ + --------------|*cos|-|
| 4/x\ 1 + cos(x) | \2/
| cos |-| |
\ \2/ /
------------------------------------------------
/ 2/x pi\ /x pi\\
| 4*cos |- - --| 4*cos|- - --||
| \2 2 / \2 2 /| /x pi\
|2 + -------------- + -------------|*cos|- - --|
| 1 + cos(x) /x\ | \2 2 /
| cos|-| |
\ \2/ /
$$\frac{\left(1 + \frac{\cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}} + \frac{4 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + \frac{4 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 4/x\ 8/x\\
| 8*sin |-| 16*sin |-||
3 | \2/ \2/|
sin (x)*|1 + --------- + ----------|
| 2 4 |
\ sin (x) sin (x) /
-----------------------------------------------
2
/ / pi\\ / ___ / 3*pi\\
2*|1 - sin|x + --||*|-1 + \/ 2 *sin|x + ----||
\ \ 2 // \ \ 4 //
$$\frac{\left(\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x \right)}} + \frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{3}{\left(x \right)}}{2 \left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \left(\sqrt{2} \sin{\left(x + \frac{3 \pi}{4} \right)} - 1\right)^{2}}$$
2
/ 2/x\\
|1 - cos |-||
1 2 \ \2//
- ------ + ------------- + --------------
/x\ /x\ /x\ 4/x\ /x\
tan|-| cos|-|*sin|-| cos |-|*tan|-|
\2/ \2/ \2/ \2/ \2/
-----------------------------------------
2
/ /x\\
2*|1 + tan|-||
\ \2//
$$\frac{\frac{\left(1 - \cos^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\cos^{4}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}} - \frac{1}{\tan{\left(\frac{x}{2} \right)}} + \frac{2}{\sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
3/x\ 3 3/x\ cos(x)
tan |-| + ------ + tan |-|*cos(x) - ------
\2/ /x\ \2/ /x\
tan|-| tan|-|
\2/ \2/
------------------------------------------
4*(1 + sin(x))
$$\frac{\cos{\left(x \right)} \tan^{3}{\left(\frac{x}{2} \right)} - \frac{\cos{\left(x \right)}}{\tan{\left(\frac{x}{2} \right)}} + \tan^{3}{\left(\frac{x}{2} \right)} + \frac{3}{\tan{\left(\frac{x}{2} \right)}}}{4 \left(\sin{\left(x \right)} + 1\right)}$$
/ 2 \
| / 2/x\\ 4/x\|
| 32*|1 + tan |-|| *tan |-||
3/x\ | 4/x\ \ \2// \4/|
4*tan |-|*|1 + tan |-| + -------------------------|
\2/ | \2/ 4 |
| / 2/x\\ 2/x\ |
| |1 + tan |-|| *tan |-| |
\ \ \4// \2/ /
-----------------------------------------------------------------
2
/ 2/x\\ / ___ / 2/x pi\\\
3 | 1 - tan |-|| | \/ 2 *|1 - tan |- + --|||
/ 2/x\\ | \2/| | \ \2 8 //|
|1 + tan |-|| *|1 - -----------|*|-1 + ------------------------|
\ \2// | 2/x\| | 2/x pi\ |
| 1 + tan |-|| | 1 + tan |- + --| |
\ \2// \ \2 8 / /
$$\frac{4 \left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{32 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan^{3}{\left(\frac{x}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - 1\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
/ 4/x\ 2/x\\
| sec |-| 2*sec |-||
| \2/ \2/| /x\
|1 + ------- + ---------|*csc|-|
| 4/x\ 2/x\ | \2/
| csc |-| csc |-| |
\ \2/ \2/ /
---------------------------------
/ 2/x\ /x\\
| 2*sec |-| 4*sec|-||
| \2/ \2/| /x\
|2 + --------- + --------|*sec|-|
| 2/x\ /x\ | \2/
| csc |-| csc|-| |
\ \2/ \2/ /
$$\frac{\left(1 + \frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}} + \frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \sec{\left(\frac{x}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} + \frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}}$$
3/x\
sec |-|
\2/
----------------------
2
/ /x\\ /x\
2*|1 + tan|-|| *sin|-|
\ \2// \2/
$$\frac{\sec^{3}{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \sin{\left(\frac{x}{2} \right)}}$$
/ 2/x\ \
| 16*cot |-| |
/ 2/x\\ | 1 \4/ |
|-1 + cot |-||*|1 + ------- + ---------------------------------|
\ \4// | 4/x\ / 2/x\\|
| cot |-| 2 | -1 + cot |-|||
| \2/ / 2/x\\ | \2/||
| |1 + cot |-|| *|1 + ------------||
| \ \4// | 2/x\ ||
| | 1 + cot |-| ||
\ \ \2/ //
----------------------------------------------------------------
/ 2/x\ \
| 16*cot |-| |
| 4 \4/ | /x\
2*|2 + ------ + ---------------------------------|*cot|-|
| /x\ / 2/x\\| \4/
| cot|-| 2 | -1 + cot |-|||
| \2/ / 2/x\\ | \2/||
| |1 + cot |-|| *|1 + ------------||
| \ \4// | 2/x\ ||
| | 1 + cot |-| ||
\ \ \2/ //
$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \left(1 + \frac{1}{\cot^{4}{\left(\frac{x}{2} \right)}} + \frac{16 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right)}{2 \left(2 + \frac{4}{\cot{\left(\frac{x}{2} \right)}} + \frac{16 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right) \cot{\left(\frac{x}{4} \right)}}$$
/ 4/x\ 2/x\\
| sin |-| 2*sin |-||
| \2/ \2/| /x\
|1 + ------- + ---------|*cos|-|
| 4/x\ 2/x\ | \2/
| cos |-| cos |-| |
\ \2/ \2/ /
---------------------------------
/ 2/x\ /x\\
| 2*sin |-| 4*sin|-||
| \2/ \2/| /x\
|2 + --------- + --------|*sin|-|
| 2/x\ /x\ | \2/
| cos |-| cos|-| |
\ \2/ \2/ /
$$\frac{\left(\frac{\sin^{4}{\left(\frac{x}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} \right)}}{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + \frac{4 \sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + 2\right) \sin{\left(\frac{x}{2} \right)}}$$
/ 4/x\ \
| sec |-| |
| \2/ 4 | /x pi\
|1 + ------------ + -------------------------|*sec|- - --|
| 4/x pi\ / 1 \ 2/x pi\| \2 2 /
| sec |- - --| |1 + ------|*sec |- - --||
\ \2 2 / \ sec(x)/ \2 2 //
----------------------------------------------------------
/ /x\ \
| 4*sec|-| |
| 4 \2/ | /x\
|2 + ------------------------- + -----------|*sec|-|
| / 1 \ 2/x pi\ /x pi\| \2/
| |1 + ------|*sec |- - --| sec|- - --||
\ \ sec(x)/ \2 2 / \2 2 //
$$\frac{\left(\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(\frac{4 \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 2 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}}$$
4/x\ 2/x\
1 + tan |-| + 2*tan |-|
\2/ \2/
---------------------------------
/ 2/x\ /x\\ /x\
|2 + 2*tan |-| + 4*tan|-||*tan|-|
\ \2/ \2// \2/
$$\frac{\tan^{4}{\left(\frac{x}{2} \right)} + 2 \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\left(2 \tan^{2}{\left(\frac{x}{2} \right)} + 4 \tan{\left(\frac{x}{2} \right)} + 2\right) \tan{\left(\frac{x}{2} \right)}}$$
/ 4/pi x\ 2/pi x\\
| csc |-- - -| 2*csc |-- - -||
| \2 2/ \2 2/| /x\
|1 + ------------ + --------------|*csc|-|
| 4/x\ 2/x\ | \2/
| csc |-| csc |-| |
\ \2/ \2/ /
------------------------------------------------
/ 2/pi x\ /pi x\\
| 2*csc |-- - -| 4*csc|-- - -||
| \2 2/ \2 2/| /pi x\
|2 + -------------- + -------------|*csc|-- - -|
| 2/x\ /x\ | \2 2/
| csc |-| csc|-| |
\ \2/ \2/ /
$$\frac{\left(1 + \frac{2 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}} + \frac{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} + \frac{2 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
/ 2/x\ 8/x\\
| 4*sin |-| 16*sin |-||
| \2/ \2/| /pi x\
|1 + --------------- + ----------|*sin|-- + -|
| / pi\ 4 | \2 2/
| 1 + sin|x + --| sin (x) |
\ \ 2 / /
----------------------------------------------
/ 2/x\ 2/x\\
| 4*sin |-| 8*sin |-||
| \2/ \2/| /x\
|2 + --------------- + ---------|*sin|-|
| / pi\ sin(x) | \2/
| 1 + sin|x + --| |
\ \ 2 / /
$$\frac{\left(\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x \right)}} + 1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}\right) \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\frac{8 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 2 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}\right) \sin{\left(\frac{x}{2} \right)}}$$
4/pi x\
csc |-- - -| 2
\2 2/ 8*csc (x)
1 + ------------ + ---------
4/x\ 4/x\
csc |-| csc |-|
\2/ \2/
------------------------------------------------
2
/ ___ \
/ 1 \ | \/ 2 | 3
2*|1 - -----------|*|-1 + ------------| *csc (x)
| /pi \| | / pi\|
| csc|-- - x|| | csc|-x + --||
\ \2 // \ \ 4 //
$$\frac{1 + \frac{8 \csc^{2}{\left(x \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}} + \frac{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}}}{2 \left(-1 + \frac{\sqrt{2}}{\csc{\left(- x + \frac{\pi}{4} \right)}}\right)^{2} \left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(x \right)}}$$
1
---------------------------------------------
4 2
/ 2/x\\ / /x\\ 8/x\ /x\
2*|1 - tan |-|| *|1 + tan|-|| *cos |-|*tan|-|
\ \4// \ \2// \4/ \2/
$$\frac{1}{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}$$
4
------------------------------------------------
2
/ /x\ \
| sec|-| |
| \2/ | / 1 2 \
|1 + -----------| *|------------- + -----------|
| /x pi\| | / pi\ / pi\|
| sec|- - --|| |sec|2*x - --| sec|x - --||
\ \2 2 // \ \ 2 / \ 2 //
$$\frac{4}{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \left(\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)}$$
sin(x)
---------------------------------------------
2
/ 1 1 \ 4/x\
2*(1 - cos(x))*|1 + ------ - ------| *cos |-|
\ sin(x) tan(x)/ \2/
$$\frac{\sin{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}$$
4
--------------------------------------
2
/ /pi x\\
| csc|-- - -||
| \2 2/| / 1 2 \
|1 + -----------| *|-------- + ------|
| /x\ | \csc(2*x) csc(x)/
| csc|-| |
\ \2/ /
$$\frac{4}{\left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{1}{\csc{\left(2 x \right)}} + \frac{2}{\csc{\left(x \right)}}\right)}$$
1
----------------------------------------
2
/ /x pi\\
| cos|- - --||
| \2 2 /| 3/x\ /x pi\
2*|1 + -----------| *cos |-|*cos|- - --|
| /x\ | \2/ \2 2 /
| cos|-| |
\ \2/ /
$$\frac{1}{2 \left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
3 / 4/x\ 2 4/x\\
sin (x)*|1 + tan |-| + 8*csc (x)*sin |-||
\ \2/ \2//
-----------------------------------------
2
/ ___ / pi\\
2*(1 - cos(x))*|-1 + \/ 2 *cos|x + --||
\ \ 4 //
$$\frac{\left(8 \sin^{4}{\left(\frac{x}{2} \right)} \csc^{2}{\left(x \right)} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \sin^{3}{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}$$
/ 4/x pi\ 2/x pi\\
| cos |- - --| 2*cos |- - --||
| \2 2 / \2 2 /| /x\
|1 + ------------ + --------------|*cos|-|
| 4/x\ 2/x\ | \2/
| cos |-| cos |-| |
\ \2/ \2/ /
------------------------------------------------
/ 2/x pi\ /x pi\\
| 2*cos |- - --| 4*cos|- - --||
| \2 2 / \2 2 /| /x pi\
|2 + -------------- + -------------|*cos|- - --|
| 2/x\ /x\ | \2 2 /
| cos |-| cos|-| |
\ \2/ \2/ /
$$\frac{\left(1 + \frac{2 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + \frac{\cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + \frac{2 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/x\
cot|-|
\2/
-----------------------
2
/ /x\\ 4/x\
2*|1 + tan|-|| *cos |-|
\ \2// \2/
$$\frac{\cot{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{x}{2} \right)}}$$
3/x\ /x pi\
sec |-|*sec|- - --|
\2/ \2 2 /
--------------------
2
/ /x\ \
| sec|-| |
| \2/ |
2*|1 + -----------|
| /x pi\|
| sec|- - --||
\ \2 2 //
$$\frac{\sec^{3}{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}$$
/ 2/x\ \
| 4*sin |-| |
| 4/x\ \2/ | /x\
|1 + tan |-| + ----------|*cos|-|
\ \2/ 1 + cos(x)/ \2/
----------------------------------
/ 2/x\ \
| 4*sin |-| |
| /x\ \2/ | /x\
|2 + 4*tan|-| + ----------|*sin|-|
\ \2/ 1 + cos(x)/ \2/
$$\frac{\left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} \right)}}{\left(4 \tan{\left(\frac{x}{2} \right)} + 2 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \sin{\left(\frac{x}{2} \right)}}$$
4
-----------------------------------------
/ /x\ \
2 | 4*tan|-| |
/ /x\\ | 2*tan(x) \2/ |
|1 + tan|-|| *|----------- + -----------|
\ \2// | 2 2/x\|
|1 + tan (x) 1 + tan |-||
\ \2//
$$\frac{4}{\left(\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + \frac{4 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
4
-----------------------------------------
/ /x\ \
2 | 4*cot|-| |
/ 1 \ | 2*cot(x) \2/ |
|1 + ------| *|----------- + -----------|
| /x\| | 2 2/x\|
| cot|-|| |1 + cot (x) 1 + cot |-||
\ \2// \ \2//
$$\frac{4}{\left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right)^{2} \left(\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} + \frac{4 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)}$$
4
-----------------------------------
2
/ /x\\
|1 + tan|-|| *(2*sin(x) + sin(2*x))
\ \2//
$$\frac{4}{\left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
/x\
cot|-|
\2/
---------------------------------------
2 4
/ /x\\ / 2/x\\ 8/x\
2*|1 + tan|-|| *|-1 + cot |-|| *sin |-|
\ \2// \ \4// \4/
$$\frac{\cot{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)}}$$
/ 4/x pi\ 4/x pi\\
| cos |- - --| 8*cos |- - --||
3/ pi\ | \2 2 / \2 2 /|
cos |x - --|*|1 + ------------ + --------------|
\ 2 / | 4/x\ 2/ pi\ |
| cos |-| cos |x - --| |
\ \2/ \ 2 / /
------------------------------------------------
2
/ ___ / pi\\
2*(1 - cos(x))*|-1 + \/ 2 *cos|x + --||
\ \ 4 //
$$\frac{\left(\frac{8 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1 + \frac{\cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{4}{\left(\frac{x}{2} \right)}}\right) \cos^{3}{\left(x - \frac{\pi}{2} \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}$$
/ 2/x\ \
| 16*tan |-| |
/ 2/x\\ | 4/x\ \4/ |
|1 - tan |-||*|1 + tan |-| + --------------------------------|
\ \4// | \2/ / 2/x\\|
| 2 | 1 - tan |-|||
| / 2/x\\ | \2/||
| |1 + tan |-|| *|1 + -----------||
| \ \4// | 2/x\||
| | 1 + tan |-|||
\ \ \2///
--------------------------------------------------------------
/ 2/x\ \
| 16*tan |-| |
| /x\ \4/ | /x\
2*|2 + 4*tan|-| + --------------------------------|*tan|-|
| \2/ / 2/x\\| \4/
| 2 | 1 - tan |-|||
| / 2/x\\ | \2/||
| |1 + tan |-|| *|1 + -----------||
| \ \4// | 2/x\||
| | 1 + tan |-|||
\ \ \2///
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{16 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right)}{2 \left(4 \tan{\left(\frac{x}{2} \right)} + 2 + \frac{16 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right) \tan{\left(\frac{x}{4} \right)}}$$
3/x\ 6/x\ / 4/x\ 2*(1 - cos(x))\
4*cot |-|*sin |-|*|1 + tan |-| + --------------|
\2/ \2/ \ \2/ 1 + cos(x) /
-----------------------------------------------------------
2
/ ___ 2/x pi\ / 2/x pi\\\
(1 - cos(x))*|-1 + \/ 2 *sin |- + --|*|-1 + cot |- + --|||
\ \2 8 / \ \2 8 ///
$$\frac{4 \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \sin^{6}{\left(\frac{x}{2} \right)} \cot^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)^{2}}$$
/ 4/x\ 8/x\\
| 8*sin |-| 16*sin |-||
| \2/ \2/|
|1 + --------- + ----------|*sin(x)
| 2 4 |
\ sin (x) sin (x) /
-------------------------------------
/ 2/x\ 4/x\\
| 8*sin |-| 8*sin |-||
| \2/ \2/| 2/x\
2*|2 + --------- + ---------|*sin |-|
| sin(x) 2 | \2/
\ sin (x) /
$$\frac{\left(\frac{16 \sin^{8}{\left(\frac{x}{2} \right)}}{\sin^{4}{\left(x \right)}} + \frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{2 \left(\frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \frac{8 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 2\right) \sin^{2}{\left(\frac{x}{2} \right)}}$$
/ 1 2 \ /x\
|1 + ------- + -------|*cot|-|
| 4/x\ 2/x\| \2/
| cot |-| cot |-||
\ \2/ \2//
------------------------------
2 4
2 + ------- + ------
2/x\ /x\
cot |-| cot|-|
\2/ \2/
$$\frac{\left(1 + \frac{2}{\cot^{2}{\left(\frac{x}{2} \right)}} + \frac{1}{\cot^{4}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}{2 + \frac{4}{\cot{\left(\frac{x}{2} \right)}} + \frac{2}{\cot^{2}{\left(\frac{x}{2} \right)}}}$$
2*sin(x)
--------------------------------------
2
/ ___ / pi\\
(1 + cos(x))*|-1 + \/ 2 *cos|x + --||
\ \ 4 //
$$\frac{2 \sin{\left(x \right)}}{\left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2} \left(\cos{\left(x \right)} + 1\right)}$$
/ 4/pi x\ \
| csc |-- - -| |
| \2 2/ 4 | /x\
|1 + ------------ + -------------------------|*csc|-|
| 4/x\ / 1 \ 2/x\| \2/
| csc |-| |1 + -----------|*csc |-||
| \2/ | /pi \| \2/|
| | csc|-- - x|| |
\ \ \2 // /
-----------------------------------------------------------
/ /pi x\\
| 4*csc|-- - -||
| 4 \2 2/| /pi x\
|2 + ------------------------- + -------------|*csc|-- - -|
| / 1 \ 2/x\ /x\ | \2 2/
| |1 + -----------|*csc |-| csc|-| |
| | /pi \| \2/ \2/ |
| | csc|-- - x|| |
\ \ \2 // /
$$\frac{\left(1 + \frac{\csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{4}{\left(\frac{x}{2} \right)}} + \frac{4}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{\left(2 + \frac{4 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} + \frac{4}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
/ 4/x\ \
| sec |-| |
| \2/ 4 | /pi x\
|1 + ------------ + -------------------------|*sec|-- - -|
| 4/x pi\ / 1 \ 2/x pi\| \2 2/
| sec |- - --| |1 + ------|*sec |- - --||
\ \2 2 / \ sec(x)/ \2 2 //
----------------------------------------------------------
/ /x\ \
| 4*sec|-| |
| 4 \2/ | /x\
|2 + ------------------------- + -----------|*sec|-|
| / 1 \ 2/x pi\ /x pi\| \2/
| |1 + ------|*sec |- - --| sec|- - --||
\ \ sec(x)/ \2 2 / \2 2 //
$$\frac{\left(\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(\frac{4 \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 2 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}}$$
4/x\ 2/ pi\
sec |-| 8*sec |x - --|
\2/ \ 2 /
1 + ------------ + --------------
4/x pi\ 4/x pi\
sec |- - --| sec |- - --|
\2 2 / \2 2 /
-----------------------------------------------
2
/ ___ \
/ 1 \ | \/ 2 | 3/ pi\
2*|1 - ------|*|-1 + -----------| *sec |x - --|
\ sec(x)/ | / pi\| \ 2 /
| sec|x + --||
\ \ 4 //
$$\frac{\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 + \frac{8 \sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}{2 \left(-1 + \frac{\sqrt{2}}{\sec{\left(x + \frac{\pi}{4} \right)}}\right)^{2} \left(1 - \frac{1}{\sec{\left(x \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}}$$
2
---------------------------------
2
/ /x\\
(1 + cos(x))*|1 + tan|-|| *sin(x)
\ \2//
$$\frac{2}{\left(\cos{\left(x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \sin{\left(x \right)}}$$
/ 2/x\ \
| 4*sin |-| |
| 4/x\ \2/ | /x\ /x\
|1 + tan |-| + ----------|*cos|-|*csc|-|
\ \2/ 1 + cos(x)/ \2/ \2/
----------------------------------------
2/x\
4*sin |-|
/x\ \2/
2 + 4*tan|-| + ----------
\2/ 1 + cos(x)
$$\frac{\left(\tan^{4}{\left(\frac{x}{2} \right)} + 1 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}\right) \cos{\left(\frac{x}{2} \right)} \csc{\left(\frac{x}{2} \right)}}{4 \tan{\left(\frac{x}{2} \right)} + 2 + \frac{4 \sin^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(x \right)} + 1}}$$
/ 4/x\\
| 8*sin |-||
3 | 4/x\ \2/|
sin (x)*|1 + tan |-| + ---------|
| \2/ 2 |
\ sin (x) /
----------------------------------------
2
/ ___ / pi\\
2*(1 - cos(x))*|-1 + \/ 2 *cos|x + --||
\ \ 4 //
$$\frac{\left(\frac{8 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \sin^{3}{\left(x \right)}}{2 \left(1 - \cos{\left(x \right)}\right) \left(\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} - 1\right)^{2}}$$
/ 2/x\\ / 4/x\ 2*(1 - cos(x))\
|1 - tan |-||*|1 + tan |-| + --------------|
\ \4// \ \2/ 1 + cos(x) /
--------------------------------------------
/ /x\ 2*(1 - cos(x))\ /x\
2*|2 + 4*tan|-| + --------------|*tan|-|
\ \2/ 1 + cos(x) / \4/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right)}{2 \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + 4 \tan{\left(\frac{x}{2} \right)} + 2\right) \tan{\left(\frac{x}{4} \right)}}$$
4
--------------------------------------
2
/ 2/x\\
| 2*sin |-||
| \2/|
|1 + ---------| *(2*sin(x) + sin(2*x))
\ sin(x) /
$$\frac{4}{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right)^{2} \left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right)}$$
4
--------------------------------------------------
2
/ /x pi\\
| cos|- - --||
| \2 2 /| / / pi\ / pi\\
|1 + -----------| *|2*cos|x - --| + cos|2*x - --||
| /x\ | \ \ 2 / \ 2 //
| cos|-| |
\ \2/ /
$$\frac{4}{\left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right)^{2} \left(2 \cos{\left(x - \frac{\pi}{2} \right)} + \cos{\left(2 x - \frac{\pi}{2} \right)}\right)}$$
/ 2/x\\ / 4/x\ 2*(1 - cos(x))\ /x\
|-1 + cot |-||*|1 + tan |-| + --------------|*tan|-|
\ \4// \ \2/ 1 + cos(x) / \4/
----------------------------------------------------
/ /x\ 2*(1 - cos(x))\
2*|2 + 4*tan|-| + --------------|
\ \2/ 1 + cos(x) /
$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + \tan^{4}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{4} \right)}}{2 \left(\frac{2 \left(1 - \cos{\left(x \right)}\right)}{\cos{\left(x \right)} + 1} + 4 \tan{\left(\frac{x}{2} \right)} + 2\right)}$$
/ 2 \
| / 2/x\\ 4/x\|
| 32*|1 + cot |-|| *cot |-||
3/x\ | 1 \ \2// \4/|
4*cot |-|*|1 + ------- + -------------------------|
\2/ | 4/x\ 4 |
| cot |-| / 2/x\\ 2/x\ |
| \2/ |1 + cot |-|| *cot |-| |
\ \ \4// \2/ /
-------------------------------------------------------------------
2
/ 2/x\\ / ___ / 2/x pi\\\
3 | -1 + cot |-|| | \/ 2 *|-1 + cot |- + --|||
/ 2/x\\ | \2/| | \ \2 8 //|
|1 + cot |-|| *|1 - ------------|*|-1 + -------------------------|
\ \2// | 2/x\ | | 2/x pi\ |
| 1 + cot |-| | | 1 + cot |- + --| |
\ \2/ / \ \2 8 / /
$$\frac{4 \left(1 + \frac{1}{\cot^{4}{\left(\frac{x}{2} \right)}} + \frac{32 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot^{4}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot^{3}{\left(\frac{x}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
4
/ 2/x\\
|1 + tan |-||
\ \4//
-------------------------------------
4 2
/ 2/x\\ / /x\\ /x\
2*|1 - tan |-|| *|1 + tan|-|| *tan|-|
\ \4// \ \2// \2/
$$\frac{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}}$$
sin(x)
---------------------------------------
2
/ 2/x\\
| 2*sin |-||
| \2/| 2/x\ 4/pi x\
4*|1 + ---------| *sin |-|*sin |-- + -|
\ sin(x) / \2/ \2 2/
$$\frac{\sin{\left(x \right)}}{4 \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}$$
/ 4/x\ 2/x\ \
| sec |-| 2*sec |-| |
| \2/ \2/ | /x pi\
|1 + ------------ + ------------|*sec|- - --|
| 4/x pi\ 2/x pi\| \2 2 /
| sec |- - --| sec |- - --||
\ \2 2 / \2 2 //
---------------------------------------------
/ 2/x\ /x\ \
| 2*sec |-| 4*sec|-| |
| \2/ \2/ | /x\
|2 + ------------ + -----------|*sec|-|
| 2/x pi\ /x pi\| \2/
| sec |- - --| sec|- - --||
\ \2 2 / \2 2 //
$$\frac{\left(\frac{\sec^{4}{\left(\frac{x}{2} \right)}}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(\frac{2 \sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{4 \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(\frac{x}{2} \right)}}$$
(1 + sec(x/2)^4/sec(x/2 - pi/2)^4 + 2*sec(x/2)^2/sec(x/2 - pi/2)^2)*sec(x/2 - pi/2)/((2 + 2*sec(x/2)^2/sec(x/2 - pi/2)^2 + 4*sec(x/2)/sec(x/2 - pi/2))*sec(x/2))