Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(tan(x/2))/2-atan(1+sqrt(2)*tan(x/2))/4-1/(16*tan(x/2)^2)+atan(-1+sqrt(2)*tan(x/2))/4+tan(x/2)^2/16
¿Cómo vas a descomponer esta log(tan(x/2))/2-atan(1+sqrt(2)*tan(x/2))/4-1/(16*tan(x/2)^2)+atan(-1+sqrt(2)*tan(x/2))/4+tan(x/2)^2/16 expresión en fracciones?
Solución
Ha introducido
[src]
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-|
\ \2// \ \2// 1 \ \2// \2/
----------- - ---------------------- - ---------- + ----------------------- + -------
2 4 2/x\ 4 16
16*tan |-|
\2/
$$\left(\left(\left(\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4}\right) - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}\right) + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4}\right) + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16}$$
log(tan(x/2))/2 - atan(1 + sqrt(2)*tan(x/2))/4 - 1/(16*tan(x/2)^2) + atan(-1 + sqrt(2)*tan(x/2))/4 + tan(x/2)^2/16
Simplificación general
[src]
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-|
\ \2// \ \2// 1 \ \2// \2/
----------- - ---------------------- - ---------- + ----------------------- + -------
2 4 2/x\ 4 16
16*tan |-|
\2/
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
log(tan(x/2))/2 - atan(1 + sqrt(2)*tan(x/2))/4 - 1/(16*tan(x/2)^2) + atan(-1 + sqrt(2)*tan(x/2))/4 + tan(x/2)^2/16
Respuesta numérica
[src]
0.0625*tan(x/2)^2 + 0.25*atan(-1 + sqrt(2)*tan(x/2)) + 0.5*log(tan(x/2)) - 0.0625/tan(x/2)^2 - 0.25*atan(1 + sqrt(2)*tan(x/2))
0.0625*tan(x/2)^2 + 0.25*atan(-1 + sqrt(2)*tan(x/2)) + 0.5*log(tan(x/2)) - 0.0625/tan(x/2)^2 - 0.25*atan(1 + sqrt(2)*tan(x/2))
Denominador racional
[src]
4/x\ 2/x\ / / ___ /x\\ / /x\\\ 2/x\ / ___ /x\\
-512 + 512*tan |-| + 1024*tan |-|*|- 2*atan|1 + \/ 2 *tan|-|| + 4*log|tan|-||| + 2048*tan |-|*atan|-1 + \/ 2 *tan|-||
\2/ \2/ \ \ \2// \ \2/// \2/ \ \2//
---------------------------------------------------------------------------------------------------------------------
2/x\
8192*tan |-|
\2/
$$\frac{1024 \left(4 \log{\left(\tan{\left(\frac{x}{2} \right)} \right)} - 2 \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{x}{2} \right)} + 512 \tan^{4}{\left(\frac{x}{2} \right)} + 2048 \tan^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} - 512}{8192 \tan^{2}{\left(\frac{x}{2} \right)}}$$
(-512 + 512*tan(x/2)^4 + 1024*tan(x/2)^2*(-2*atan(1 + sqrt(2)*tan(x/2)) + 4*log(tan(x/2))) + 2048*tan(x/2)^2*atan(-1 + sqrt(2)*tan(x/2)))/(8192*tan(x/2)^2)
Denominador común
[src]
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-|
\ \2// \ \2// 1 \ \2// \2/
----------- - ---------------------- - ---------- + ----------------------- + -------
2 4 2/x\ 4 16
16*tan |-|
\2/
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
log(tan(x/2))/2 - atan(1 + sqrt(2)*tan(x/2))/4 - 1/(16*tan(x/2)^2) + atan(-1 + sqrt(2)*tan(x/2))/4 + tan(x/2)^2/16
Potencias
[src]
/ / I*x -I*x \\ / / I*x -I*x \\ / / I*x -I*x \\
| | --- -----|| | | --- -----|| | | --- -----||
| | 2 2 || | ___ | 2 2 || | ___ | 2 2 ||
|I*\- e + e /| | I*\/ 2 *\- e + e /| | I*\/ 2 *\- e + e /|
log|-------------------| atan|1 + -------------------------| atan|-1 + -------------------------| 2 2
| I*x -I*x | | I*x -I*x | | I*x -I*x | / I*x -I*x \ / I*x -I*x \
| --- ----- | | --- ----- | | --- ----- | | --- -----| | --- -----|
| 2 2 | | 2 2 | | 2 2 | | 2 2 | | 2 2 |
\ e + e / \ e + e / \ e + e / \- e + e / \e + e /
------------------------ - ----------------------------------- + ------------------------------------ - ------------------- + ---------------------
2 4 4 2 2
/ I*x -I*x \ / I*x -I*x \
| --- -----| | --- -----|
| 2 2 | | 2 2 |
16*\e + e / 16*\- e + e /
$$- \frac{\left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{16 \left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}} + \frac{\log{\left(\frac{i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{2} i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{\sqrt{2} i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} + 1 \right)}}{4} + \frac{\left(e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}{16 \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)^{2}}$$
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-|
\ \2// \ \2// 1 \ \2// \2/
----------- - ---------------------- - ---------- + ----------------------- + -------
2 4 2/x\ 4 16
16*tan |-|
\2/
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
log(tan(x/2))/2 - atan(1 + sqrt(2)*tan(x/2))/4 - 1/(16*tan(x/2)^2) + atan(-1 + sqrt(2)*tan(x/2))/4 + tan(x/2)^2/16
Combinatoria
[src]
/ 4/x\ 2/x\ / /x\\ 2/x\ / ___ /x\\ 2/x\ / ___ /x\\\
-|1 - tan |-| - 8*tan |-|*log|tan|-|| - 4*tan |-|*atan|-1 + \/ 2 *tan|-|| + 4*tan |-|*atan|1 + \/ 2 *tan|-|||
\ \2/ \2/ \ \2// \2/ \ \2// \2/ \ \2///
--------------------------------------------------------------------------------------------------------------
2/x\
16*tan |-|
\2/
$$- \frac{- 8 \log{\left(\tan{\left(\frac{x}{2} \right)} \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \tan^{4}{\left(\frac{x}{2} \right)} - 4 \tan^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)} + 1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
-(1 - tan(x/2)^4 - 8*tan(x/2)^2*log(tan(x/2)) - 4*tan(x/2)^2*atan(-1 + sqrt(2)*tan(x/2)) + 4*tan(x/2)^2*atan(1 + sqrt(2)*tan(x/2)))/(16*tan(x/2)^2)
Compilar la expresión
[src]
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-|
\ \2// \ \2// 1 \ \2// \2/
----------- - ---------------------- - ---------- + ----------------------- + -------
2 4 2/x\ 4 16
16*tan |-|
\2/
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
log(tan(x/2))/2 - atan(1 + sqrt(2)*tan(x/2))/4 - 1/(16*tan(x/2)^2) + atan(-1 + sqrt(2)*tan(x/2))/4 + tan(x/2)^2/16
Unión de expresiones racionales
[src]
4/x\ 2/x\ / / ___ /x\\ / /x\\\ 2/x\ / ___ /x\\
-1 + tan |-| + 4*tan |-|*|- atan|1 + \/ 2 *tan|-|| + 2*log|tan|-||| + 4*tan |-|*atan|-1 + \/ 2 *tan|-||
\2/ \2/ \ \ \2// \ \2/// \2/ \ \2//
-------------------------------------------------------------------------------------------------------
2/x\
16*tan |-|
\2/
$$\frac{4 \left(2 \log{\left(\tan{\left(\frac{x}{2} \right)} \right)} - \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \tan^{2}{\left(\frac{x}{2} \right)} + \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)} - 1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
(-1 + tan(x/2)^4 + 4*tan(x/2)^2*(-atan(1 + sqrt(2)*tan(x/2)) + 2*log(tan(x/2))) + 4*tan(x/2)^2*atan(-1 + sqrt(2)*tan(x/2)))/(16*tan(x/2)^2)
Parte trigonométrica
[src]
/ 2/x\\ / ___ 2/x\\ / ___ 2/x\\
|2*sin |-|| | 2*\/ 2 *sin |-|| | 2*\/ 2 *sin |-||
| \2/| | \2/| | \2/| 2/x\ 2/x\
log|---------| atan|1 + ---------------| atan|-1 + ---------------| cos |-| sin |-|
\ sin(x) / \ sin(x) / \ sin(x) / \2/ \2/
-------------- - ------------------------- + -------------------------- - -------------- + --------------
2 4 4 8*(1 - cos(x)) 8*(1 + cos(x))
$$\frac{\log{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1 \right)}}{4} + \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{8 \left(\cos{\left(x \right)} + 1\right)} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ /x pi\\ / ___ /x pi\\ / ___ /x pi\\
|cos|- - --|| | \/ 2 *cos|- - --|| | \/ 2 *cos|- - --||
| \2 2 /| | \2 2 /| | \2 2 /|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ | 2/x\ 2/x pi\
| cos|-| | | cos|-| | | cos|-| | cos |-| cos |- - --|
\ \2/ / \ \2/ / \ \2/ / \2/ \2 2 /
---------------- - --------------------------- - --------------------------- - -------------- + --------------
2 4 4 8*(1 - cos(x)) 8*(1 + cos(x))
$$\frac{\log{\left(\frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{8 \left(\cos{\left(x \right)} + 1\right)} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ / pi\\ / ___ / pi\\ / ___ / pi\\
|2*sec|x - --|| | 2*\/ 2 *sec|x - --|| | 2*\/ 2 *sec|x - --||
| \ 2 /| | \ 2 /| | \ 2 /|
log|-------------| atan|1 - -------------------| atan|1 + -------------------|
| 2/x pi\| | 2/x pi\ | | 2/x pi\ | 4/x pi\ 2/ pi\
| sec |- - --|| | sec |- - --| | | sec |- - --| | sec |- - --| sec |x - --|
\ \2 2 // \ \2 2 / / \ \2 2 / / \2 2 / \ 2 /
------------------ - ----------------------------- - ----------------------------- - --------------- + --------------
2 4 4 2/ pi\ 4/x pi\
64*sec |x - --| 4*sec |- - --|
\ 2 / \2 2 /
$$\frac{\log{\left(\frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{2 \sqrt{2} \sec{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{2 \sqrt{2} \sec{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{4} - \frac{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{64 \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{4 \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ /x\ \ / ___ /x\\ / ___ /x\\
| sec|-| | | \/ 2 *sec|-|| | \/ 2 *sec|-||
| \2/ | | \2/| | \2/|
log|-----------| atan|1 + ------------| atan|-1 + ------------|
| /x pi\| | /x pi\ | | /x pi\ |
|sec|- - --|| | sec|- - --| | | sec|- - --| |
\ \2 2 // \ \2 2 / / \ \2 2 / / 1 1
---------------- - ---------------------- + ----------------------- - ---------------------- + ---------------------------
2 4 4 / 1 \ 2/x\ / 1 \ 2/x pi\
8*|1 - ------|*sec |-| 8*|1 + ------|*sec |- - --|
\ sec(x)/ \2/ \ sec(x)/ \2 2 /
$$\frac{\log{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 \right)}}{4} + \frac{1}{8 \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{1}{8 \left(1 - \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} \right)}}$$
/ /x\\ / ___ /x\\ / ___ /x\\
|sec|-|| | \/ 2 *sec|-|| | \/ 2 *sec|-||
| \2/| | \2/| | \2/|
log|------| atan|1 + ------------| atan|1 - ------------|
| /x\| | /x\ | | /x\ | 2/x\ 2/x\
|csc|-|| | csc|-| | | csc|-| | csc |-| sec |-|
\ \2// \ \2/ / \ \2/ / \2/ \2/
----------- - ---------------------- - ---------------------- - ---------- + ----------
2 4 4 2/x\ 2/x\
16*sec |-| 16*csc |-|
\2/ \2/
$$\frac{\log{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\csc^{2}{\left(\frac{x}{2} \right)}}{16 \sec^{2}{\left(\frac{x}{2} \right)}} + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{16 \csc^{2}{\left(\frac{x}{2} \right)}}$$
/ /x pi\\ / ___ /x pi\\ / ___ /x pi\\
|cos|- - --|| | \/ 2 *cos|- - --|| | \/ 2 *cos|- - --||
| \2 2 /| | \2 2 /| | \2 2 /|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ | 2/x\ 2/x pi\
| cos|-| | | cos|-| | | cos|-| | cos |-| cos |- - --|
\ \2/ / \ \2/ / \ \2/ / \2/ \2 2 /
---------------- - --------------------------- - --------------------------- - -------------- + ------------
2 4 4 8*(1 - cos(x)) 2/x\
16*cos |-|
\2/
$$\frac{\log{\left(\frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{16 \cos^{2}{\left(\frac{x}{2} \right)}} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-| cos |-|
\ \2// \ \2// \ \2// \2/ \2/
----------- - ---------------------- + ----------------------- + ------- - --------------
2 4 4 16 8*(1 - cos(x))
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ /pi x\\ / ___ /pi x\\ / ___ /pi x\\
|csc|-- - -|| | \/ 2 *csc|-- - -|| | \/ 2 *csc|-- - -||
| \2 2/| | \2 2/| | \2 2/|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ | 2/x\ 2/pi x\
| csc|-| | | csc|-| | | csc|-| | csc |-| csc |-- - -|
\ \2/ / \ \2/ / \ \2/ / \2/ \2 2/
---------------- - --------------------------- - --------------------------- - --------------- + ------------
2 4 4 2/pi x\ 2/x\
16*csc |-- - -| 16*csc |-|
\2 2/ \2/
$$\frac{\log{\left(\frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\csc^{2}{\left(\frac{x}{2} \right)}}{16 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{16 \csc^{2}{\left(\frac{x}{2} \right)}}$$
2
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\ / 2/x\\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-| |1 - tan |-||
\ \2// \ \2// \ \2// \2/ \ \4//
----------- - ---------------------- + ----------------------- + ------- - ----------------------------------
2 4 4 16 / 2/x\\
2 | 1 - tan |-||
/ 2/x\\ | \2/|
8*|1 + tan |-|| *|1 - -----------|
\ \4// | 2/x\|
| 1 + tan |-||
\ \2//
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2}}{8 \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4}$$
/ /x\ \ / ___ /x\\ / ___ /x\\
| sec|-| | | \/ 2 *sec|-|| | \/ 2 *sec|-||
| \2/ | | \2/| | \2/|
log|-----------| atan|1 + ------------| atan|-1 + ------------|
| /x pi\| | /x pi\ | | /x pi\ | 2/x pi\
|sec|- - --|| | sec|- - --| | | sec|- - --| | sec |- - --|
\ \2 2 // \ \2 2 / / \ \2 2 / / \2 2 / 1
---------------- - ---------------------- + ----------------------- - ------------ + ---------------------------
2 4 4 2/x\ / 1 \ 2/x pi\
16*sec |-| 8*|1 + ------|*sec |- - --|
\2/ \ sec(x)/ \2 2 /
$$\frac{\log{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 \right)}}{4} - \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{16 \sec^{2}{\left(\frac{x}{2} \right)}} + \frac{1}{8 \left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 2/x\ / 2/x\\\ / ___ 2/x\ / 2/x\\\ / ___ 2/x\ / 2/x\\\
|4*tan |-|*|1 + tan |-||| | 4*\/ 2 *tan |-|*|1 + tan |-||| | 4*\/ 2 *tan |-|*|1 + tan |-|||
| \4/ \ \2//| | \4/ \ \2//| | \4/ \ \2//|
log|-----------------------| atan|1 - -----------------------------| atan|1 + -----------------------------|
| 2 | | 2 | | 2 | 2 4
| / 2/x\\ /x\ | | / 2/x\\ /x\ | | / 2/x\\ /x\ | / 2/x\\ 4/x\ / 2/x\\ 2/x\
| |1 + tan |-|| *tan|-| | | |1 + tan |-|| *tan|-| | | |1 + tan |-|| *tan|-| | |1 + tan |-|| *tan |-| |1 + tan |-|| *tan |-|
\ \ \4// \2/ / \ \ \4// \2/ / \ \ \4// \2/ / \ \2// \4/ \ \4// \2/
---------------------------- - --------------------------------------- - --------------------------------------- + ---------------------- - --------------------------
2 4 4 4 2
/ 2/x\\ 2/x\ / 2/x\\ 4/x\
|1 + tan |-|| *tan |-| 256*|1 + tan |-|| *tan |-|
\ \4// \2/ \ \2// \4/
$$- \frac{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan^{2}{\left(\frac{x}{2} \right)}}{256 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{x}{4} \right)}} + \frac{\log{\left(\frac{4 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{4 \sqrt{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{4 \sqrt{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan^{2}{\left(\frac{x}{2} \right)}}$$
/ ___ \ / ___ \
/2*csc(x)\ | 2*\/ 2 *csc(x)| | 2*\/ 2 *csc(x)|
log|--------| atan|1 - --------------| atan|1 + --------------|
| 2/x\ | | 2/x\ | | 2/x\ | 4/x\
|csc |-| | | csc |-| | | csc |-| | csc |-| 2
\ \2/ / \ \2/ / \ \2/ / \2/ csc (x)
------------- - ------------------------ - ------------------------ - ---------- + ---------
2 4 4 2 4/x\
64*csc (x) 4*csc |-|
\2/
$$\frac{\log{\left(\frac{2 \csc{\left(x \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{2 \sqrt{2} \csc{\left(x \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{2 \sqrt{2} \csc{\left(x \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\csc^{4}{\left(\frac{x}{2} \right)}}{64 \csc^{2}{\left(x \right)}} + \frac{\csc^{2}{\left(x \right)}}{4 \csc^{4}{\left(\frac{x}{2} \right)}}$$
/ 2/x\\ / ___ 2/x\\ / ___ 2/x\\
|2*sin |-|| | 2*\/ 2 *sin |-|| | 2*\/ 2 *sin |-||
| \2/| | \2/| | \2/| 2/x\
log|---------| atan|1 + ---------------| atan|-1 + ---------------| 2 sin |-|
\ sin(x) / \ sin(x) / \ sin(x) / sin (x) \2/
-------------- - ------------------------- + -------------------------- - ---------- + --------------
2 4 4 4/x\ 8*(1 + cos(x))
64*sin |-|
\2/
$$\frac{\log{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1 \right)}}{4} - \frac{\sin^{2}{\left(x \right)}}{64 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{8 \left(\cos{\left(x \right)} + 1\right)}$$
/ ___ \ / ___ \
/ 1 \ | \/ 2 | | \/ 2 |
log|------| atan|1 + ------| atan|1 - ------|
| /x\| | /x\| | /x\| 2/x\
|cot|-|| | cot|-|| | cot|-|| cot |-|
\ \2// \ \2// \ \2// \2/ 1
----------- - ---------------- - ---------------- - ------- + ----------
2 4 4 16 2/x\
16*cot |-|
\2/
$$\frac{\log{\left(\frac{1}{\cot{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{16} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{1}{16 \cot^{2}{\left(\frac{x}{2} \right)}}$$
2
/ /x\\ / ___ /x\\ / ___ /x\\ / 2/x\\ 4/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| |1 - tan |-|| *cos |-|
\ \2// \ \2// \ \2// 1 - cos(x) \ \4// \4/
----------- - ---------------------- + ----------------------- + --------------- - ----------------------
2 4 4 16*(1 + cos(x)) 8*(1 - cos(x))
$$\frac{1 - \cos{\left(x \right)}}{16 \left(\cos{\left(x \right)} + 1\right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \cos^{4}{\left(\frac{x}{4} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ ___ \ / ___ \
/ 1 \ | \/ 2 | | \/ 2 |
log|------| atan|1 + ------| atan|1 - ------|
| /x\| | /x\| | /x\| 2/x\ 2/x\
|cot|-|| | cot|-|| | cot|-|| cot |-| cot |-|
\ \2// \ \2// \ \2// \2/ \4/
----------- - ---------------- - ---------------- - ------- + -----------------------------------
2 4 4 16 / 2/x\\
2 | -1 + cot |-||
/ 2/x\\ | \2/|
2*|1 + cot |-|| *|1 + ------------|
\ \4// | 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{\log{\left(\frac{1}{\cot{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{16} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\cot^{2}{\left(\frac{x}{4} \right)}}{2 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
2
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\ / 2/x\\ 4/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-| |-1 + cot |-|| *sin |-|
\ \2// \ \2// \ \2// \2/ \ \4// \4/
----------- - ---------------------- + ----------------------- + ------- - -----------------------
2 4 4 16 8*(1 - cos(x))
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
2
/ /x\\ / ___ /x\\ / ___ /x\\ / 2/x\\ 4/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| |-1 + cot |-|| *sin |-|
\ \2// \ \2// \ \2// 1 - cos(x) \ \4// \4/
----------- - ---------------------- + ----------------------- + --------------- - -----------------------
2 4 4 16*(1 + cos(x)) 8*(1 - cos(x))
$$\frac{1 - \cos{\left(x \right)}}{16 \left(\cos{\left(x \right)} + 1\right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ 2/x pi\\ / ___ 2/x pi\\ / ___ 2/x pi\\
|2*cos |- - --|| | 2*\/ 2 *cos |- - --|| | 2*\/ 2 *cos |- - --||
| \2 2 /| | \2 2 /| | \2 2 /|
log|--------------| atan|1 + --------------------| atan|-1 + --------------------|
| / pi\ | | / pi\ | | / pi\ | 2/ pi\ 4/x pi\
| cos|x - --| | | cos|x - --| | | cos|x - --| | cos |x - --| cos |- - --|
\ \ 2 / / \ \ 2 / / \ \ 2 / / \ 2 / \2 2 /
------------------- - ------------------------------ + ------------------------------- - --------------- + --------------
2 4 4 4/x pi\ 2/ pi\
64*cos |- - --| 4*cos |x - --|
\2 2 / \ 2 /
$$\frac{\log{\left(\frac{2 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} \right)}}{2} + \frac{\cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{4 \cos^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + 1 \right)}}{4} - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{64 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 2/x\ / 2/x\\\ / ___ 2/x\ / 2/x\\\ / ___ 2/x\ / 2/x\\\
|4*cot |-|*|1 + cot |-||| | 4*\/ 2 *cot |-|*|1 + cot |-||| | 4*\/ 2 *cot |-|*|1 + cot |-|||
| \4/ \ \2//| | \4/ \ \2//| | \4/ \ \2//|
log|-----------------------| atan|1 - -----------------------------| atan|1 + -----------------------------|
| 2 | | 2 | | 2 | 2 4
| / 2/x\\ /x\ | | / 2/x\\ /x\ | | / 2/x\\ /x\ | / 2/x\\ 4/x\ / 2/x\\ 2/x\
| |1 + cot |-|| *cot|-| | | |1 + cot |-|| *cot|-| | | |1 + cot |-|| *cot|-| | |1 + cot |-|| *cot |-| |1 + cot |-|| *cot |-|
\ \ \4// \2/ / \ \ \4// \2/ / \ \ \4// \2/ / \ \2// \4/ \ \4// \2/
---------------------------- - --------------------------------------- - --------------------------------------- + ---------------------- - --------------------------
2 4 4 4 2
/ 2/x\\ 2/x\ / 2/x\\ 4/x\
|1 + cot |-|| *cot |-| 256*|1 + cot |-|| *cot |-|
\ \4// \2/ \ \2// \4/
$$- \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\frac{x}{2} \right)}}{256 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot^{4}{\left(\frac{x}{4} \right)}} + \frac{\log{\left(\frac{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{4 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{4 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \cot^{4}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\frac{x}{2} \right)}}$$
/ 2/x\\ / ___ 2/x\\ / ___ 2/x\\
|2*sin |-|| | 2*\/ 2 *sin |-|| | 2*\/ 2 *sin |-||
| \2/| | \2/| | \2/| 2/x\ 4/x\
log|---------| atan|1 + ---------------| atan|-1 + ---------------| cos |-| sin |-|
\ sin(x) / \ sin(x) / \ sin(x) / \2/ \2/
-------------- - ------------------------- + -------------------------- - -------------- + ---------
2 4 4 8*(1 - cos(x)) 2
4*sin (x)
$$\frac{\log{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} \right)}}{2} + \frac{\sin^{4}{\left(\frac{x}{2} \right)}}{4 \sin^{2}{\left(x \right)}} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1 \right)}}{4} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ /x\\ / ___ /x\\ / ___ /x\\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-||
\ \2// \ \2// \ \2// 1 - cos(x) 1 - cos(2*x)
----------- - ---------------------- + ----------------------- + --------------- - -------------------------------
2 4 4 16*(1 + cos(x)) 2 2
/ /x\\ / /x\\
128*|1 - cos|-|| *|1 + cos|-||
\ \2// \ \2//
$$\frac{1 - \cos{\left(x \right)}}{16 \left(\cos{\left(x \right)} + 1\right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1 - \cos{\left(2 x \right)}}{128 \left(1 - \cos{\left(\frac{x}{2} \right)}\right)^{2} \left(\cos{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
/ /x\\ / ___ /x\\ / ___ /x\\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-||
\ \2// \ \2// 1 \ \2// 1 - cos(x)
----------- - ---------------------- - ---------- + ----------------------- + ---------------
2 4 2/x\ 4 16*(1 + cos(x))
16*tan |-|
\2/
$$\frac{1 - \cos{\left(x \right)}}{16 \left(\cos{\left(x \right)} + 1\right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
/ /x\ \ / ___ /x\\ / ___ /x\\
| sec|-| | | \/ 2 *sec|-|| | \/ 2 *sec|-||
| \2/ | | \2/| | \2/|
log|-----------| atan|1 + ------------| atan|-1 + ------------|
| /x pi\| | /x pi\ | | /x pi\ | 2/x\
|sec|- - --|| | sec|- - --| | | sec|- - --| | sec |-|
\ \2 2 // \ \2 2 / / \ \2 2 / / 1 \2/
---------------- - ---------------------- + ----------------------- - ---------------------- + ---------------
2 4 4 / 1 \ 2/x\ 2/x pi\
8*|1 - ------|*sec |-| 16*sec |- - --|
\ sec(x)/ \2/ \2 2 /
$$\frac{\log{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 \right)}}{4} + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{16 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{1}{8 \left(1 - \frac{1}{\sec{\left(x \right)}}\right) \sec^{2}{\left(\frac{x}{2} \right)}}$$
/ /x pi\\ / ___ /x pi\\ / ___ /x pi\\
|cos|- - --|| | \/ 2 *cos|- - --|| | \/ 2 *cos|- - --||
| \2 2 /| | \2 2 /| | \2 2 /|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ | 2/x\ 2/x pi\
| cos|-| | | cos|-| | | cos|-| | cos |-| cos |- - --|
\ \2/ / \ \2/ / \ \2/ / \2/ \2 2 /
---------------- - --------------------------- - --------------------------- - --------------- + ------------
2 4 4 2/x pi\ 2/x\
16*cos |- - --| 16*cos |-|
\2 2 / \2/
$$\frac{\log{\left(\frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{16 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{16 \cos^{2}{\left(\frac{x}{2} \right)}}$$
/ 2/x\\ / ___ 2/x\\ / ___ 2/x\\
|2*sin |-|| | 2*\/ 2 *sin |-|| | 2*\/ 2 *sin |-||
| \2/| | \2/| | \2/| 4/x\
log|---------| atan|1 + ---------------| atan|-1 + ---------------| 2 sin |-|
\ sin(x) / \ sin(x) / \ sin(x) / sin (x) \2/
-------------- - ------------------------- + -------------------------- - ---------- + ---------
2 4 4 4/x\ 2
64*sin |-| 4*sin (x)
\2/
$$\frac{\log{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} \right)}}{2} + \frac{\sin^{4}{\left(\frac{x}{2} \right)}}{4 \sin^{2}{\left(x \right)}} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1 \right)}}{4} - \frac{\sin^{2}{\left(x \right)}}{64 \sin^{4}{\left(\frac{x}{2} \right)}}$$
/ /x\\ / ___ /x\\ / ___ /x\\
|sin|-|| | \/ 2 *sin|-|| | \/ 2 *sin|-||
| \2/| | \2/| | \2/|
log|------| atan|1 + ------------| atan|-1 + ------------|
| /x\| | /x\ | | /x\ | 2/x\ 2/x\
|cos|-|| | cos|-| | | cos|-| | cos |-| sin |-|
\ \2// \ \2/ / \ \2/ / \2/ \2/
----------- - ---------------------- + ----------------------- - ---------- + ----------
2 4 4 2/x\ 2/x\
16*sin |-| 16*cos |-|
\2/ \2/
$$\frac{\log{\left(\frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{2} + \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{16 \cos^{2}{\left(\frac{x}{2} \right)}} + \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + 1 \right)}}{4} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{16 \sin^{2}{\left(\frac{x}{2} \right)}}$$
/ /x\ \ / ___ /x\\ / ___ /x\\
| sec|-| | | \/ 2 *sec|-|| | \/ 2 *sec|-||
| \2/ | | \2/| | \2/|
log|-----------| atan|1 + ------------| atan|-1 + ------------|
| /x pi\| | /x pi\ | | /x pi\ | 2/x pi\ 2/x\
|sec|- - --|| | sec|- - --| | | sec|- - --| | sec |- - --| sec |-|
\ \2 2 // \ \2 2 / / \ \2 2 / / \2 2 / \2/
---------------- - ---------------------- + ----------------------- - ------------ + ---------------
2 4 4 2/x\ 2/x pi\
16*sec |-| 16*sec |- - --|
\2/ \2 2 /
$$\frac{\log{\left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{\sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1 \right)}}{4} + \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{16 \sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{16 \sec^{2}{\left(\frac{x}{2} \right)}}$$
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-|
\ \2// \ \2// 1 \ \2// \2/
----------- - ---------------------- - ---------- + ----------------------- + -------
2 4 2/x\ 4 16
16*tan |-|
\2/
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}}$$
/ /pi x\\ / ___ /pi x\\ / ___ /pi x\\
|csc|-- - -|| | \/ 2 *csc|-- - -|| | \/ 2 *csc|-- - -||
| \2 2/| | \2 2/| | \2 2/|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ | 2/x\
| csc|-| | | csc|-| | | csc|-| | csc |-|
\ \2/ / \ \2/ / \ \2/ / \2/ 1
---------------- - --------------------------- - --------------------------- - --------------- + ---------------------------
2 4 4 2/pi x\ / 1 \ 2/x\
16*csc |-- - -| 8*|1 + -----------|*csc |-|
\2 2/ | /pi \| \2/
| csc|-- - x||
\ \2 //
$$\frac{\log{\left(\frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\csc^{2}{\left(\frac{x}{2} \right)}}{16 \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{1}{8 \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(\frac{x}{2} \right)}}$$
/ ___ \ / ___ \
/ 1 \ | \/ 2 | | \/ 2 |
log|------| atan|1 + ------| atan|1 - ------| 2
| /x\| | /x\| | /x\| 2/x\ / 2/x\\
|cot|-|| | cot|-|| | cot|-|| cot |-| |-1 + cot |-||
\ \2// \ \2// \ \2// \4/ \ \4//
----------- - ---------------- - ---------------- + ----------------------------------- - -----------------------------------
2 4 4 / 2/x\\ / 2/x\\
2 | -1 + cot |-|| 2 | -1 + cot |-||
/ 2/x\\ | \2/| / 2/x\\ | \2/|
2*|1 + cot |-|| *|1 + ------------| 8*|1 + cot |-|| *|1 - ------------|
\ \4// | 2/x\ | \ \4// | 2/x\ |
| 1 + cot |-| | | 1 + cot |-| |
\ \2/ / \ \2/ /
$$\frac{\log{\left(\frac{1}{\cot{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\cot^{2}{\left(\frac{x}{4} \right)}}{2 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} - \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{8 \left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
/ /x\\ / ___ /x\\ 2/x\ / ___ /x\\
log|tan|-|| atan|1 + \/ 2 *tan|-|| cot |-| atan|-1 + \/ 2 *tan|-||
\ \2// \ \2// \2/ \ \2// 1 - cos(x)
----------- - ---------------------- - ------- + ----------------------- + ---------------
2 4 16 4 16*(1 + cos(x))
$$\frac{1 - \cos{\left(x \right)}}{16 \left(\cos{\left(x \right)} + 1\right)} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} - \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4}$$
/ /pi x\\ / ___ /pi x\\ / ___ /pi x\\
|csc|-- - -|| | \/ 2 *csc|-- - -|| | \/ 2 *csc|-- - -||
| \2 2/| | \2 2/| | \2 2/|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ | 2/pi x\
| csc|-| | | csc|-| | | csc|-| | csc |-- - -|
\ \2/ / \ \2/ / \ \2/ / 1 \2 2/
---------------- - --------------------------- - --------------------------- - -------------------------------- + ------------
2 4 4 / 1 \ 2/pi x\ 2/x\
8*|1 - -----------|*csc |-- - -| 16*csc |-|
| /pi \| \2 2/ \2/
| csc|-- - x||
\ \2 //
$$\frac{\log{\left(\frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{16 \csc^{2}{\left(\frac{x}{2} \right)}} - \frac{1}{8 \left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
/ ___ \ / ___ \
/ 1 \ | \/ 2 | | \/ 2 |
log|------| atan|1 + ------| atan|1 - ------| 2
| /x\| | /x\| | /x\| / 2/x\\
|cot|-|| | cot|-|| | cot|-|| |-1 + cot |-||
\ \2// \ \2// \ \2// 1 \ \4//
----------- - ---------------- - ---------------- + ---------- - -----------------------------------
2 4 4 2/x\ / 2/x\\
16*cot |-| 2 | -1 + cot |-||
\2/ / 2/x\\ | \2/|
8*|1 + cot |-|| *|1 - ------------|
\ \4// | 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{\log{\left(\frac{1}{\cot{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2}}{\cot{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{1}{16 \cot^{2}{\left(\frac{x}{2} \right)}} - \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{8 \left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
2
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\ / 2/x\\ 4/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-| |1 - tan |-|| *cos |-|
\ \2// \ \2// \ \2// \2/ \ \4// \4/
----------- - ---------------------- + ----------------------- + ------- - ----------------------
2 4 4 16 8*(1 - cos(x))
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{16} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \cos^{4}{\left(\frac{x}{4} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
2
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\ / 2/x\\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-| |1 - tan |-||
\ \2// \ \2// \ \2// \4/ \ \4//
----------- - ---------------------- + ----------------------- + ---------------------------------- - ----------------------------------
2 4 4 / 2/x\\ / 2/x\\
2 | 1 - tan |-|| 2 | 1 - tan |-||
/ 2/x\\ | \2/| / 2/x\\ | \2/|
2*|1 + tan |-|| *|1 + -----------| 8*|1 + tan |-|| *|1 - -----------|
\ \4// | 2/x\| \ \4// | 2/x\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2}}{8 \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} + \frac{\tan^{2}{\left(\frac{x}{4} \right)}}{2 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| sin |-|
\ \2// \ \2// 1 \ \2// \2/
----------- - ---------------------- - ---------- + ----------------------- + --------------
2 4 2/x\ 4 8*(1 + cos(x))
16*tan |-|
\2/
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{8 \left(\cos{\left(x \right)} + 1\right)}$$
/ /pi x\\ / ___ /pi x\\ / ___ /pi x\\
|csc|-- - -|| | \/ 2 *csc|-- - -|| | \/ 2 *csc|-- - -||
| \2 2/| | \2 2/| | \2 2/|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ |
| csc|-| | | csc|-| | | csc|-| |
\ \2/ / \ \2/ / \ \2/ / 1 1
---------------- - --------------------------- - --------------------------- - -------------------------------- + ---------------------------
2 4 4 / 1 \ 2/pi x\ / 1 \ 2/x\
8*|1 - -----------|*csc |-- - -| 8*|1 + -----------|*csc |-|
| /pi \| \2 2/ | /pi \| \2/
| csc|-- - x|| | csc|-- - x||
\ \2 // \ \2 //
$$\frac{\log{\left(\frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{1}{8 \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(\frac{x}{2} \right)}} - \frac{1}{8 \left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}$$
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| cos |-| sin |-|
\ \2// \ \2// \ \2// \2/ \2/
----------- - ---------------------- + ----------------------- - -------------- + --------------
2 4 4 8*(1 - cos(x)) 8*(1 + cos(x))
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} + \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{8 \left(\cos{\left(x \right)} + 1\right)} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{8 \left(1 - \cos{\left(x \right)}\right)}$$
/ /x pi\\ / ___ /x pi\\ / ___ /x pi\\
|cos|- - --|| | \/ 2 *cos|- - --|| | \/ 2 *cos|- - --||
| \2 2 /| | \2 2 /| | \2 2 /|
log|-----------| atan|1 + -----------------| atan|1 - -----------------|
| /x\ | | /x\ | | /x\ | 2/x\ 2/x pi\
| cos|-| | | cos|-| | | cos|-| | cos |-| cos |- - --|
\ \2/ / \ \2/ / \ \2/ / \2/ \2 2 /
---------------- - --------------------------- - --------------------------- - --------------- + --------------
2 4 4 2/x pi\ 8*(1 + cos(x))
16*cos |- - --|
\2 2 /
$$\frac{\log{\left(\frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{2} - \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{16 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\operatorname{atan}{\left(1 - \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} - \frac{\operatorname{atan}{\left(1 + \frac{\sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} \right)}}{4} + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{8 \left(\cos{\left(x \right)} + 1\right)}$$
/ 2/x\\ / ___ 2/x\\ / ___ 2/x\\
|2*sin |-|| | 2*\/ 2 *sin |-|| | 2*\/ 2 *sin |-||
| \2/| | \2/| | \2/| 2/x\
log|---------| atan|1 + ---------------| atan|-1 + ---------------| 2 sin |-|
\ sin(x) / \ sin(x) / \ sin(x) / sin (x) \2/
-------------- - ------------------------- + -------------------------- - ---------- + -------------------
2 4 4 4/x\ / / pi\\
64*sin |-| 8*|1 + sin|x + --||
\2/ \ \ 2 //
$$\frac{\log{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1 \right)}}{4} - \frac{\sin^{2}{\left(x \right)}}{64 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{8 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)}$$
/ 2/x\ \ / ___ 2/x\ \ / ___ 2/x\ \ 4/x\ 2 2 4/x\
log|2*sin |-|*csc(x)| atan|1 + 2*\/ 2 *sin |-|*csc(x)| atan|-1 + 2*\/ 2 *sin |-|*csc(x)| csc |-|*sin (x) csc (x)*sin |-|
\ \2/ / \ \2/ / \ \2/ / \2/ \2/
--------------------- - -------------------------------- + --------------------------------- - --------------- + ---------------
2 4 4 64 4
$$\frac{\log{\left(2 \sin^{2}{\left(\frac{x}{2} \right)} \csc{\left(x \right)} \right)}}{2} + \frac{\sin^{4}{\left(\frac{x}{2} \right)} \csc^{2}{\left(x \right)}}{4} - \frac{\sin^{2}{\left(x \right)} \csc^{4}{\left(\frac{x}{2} \right)}}{64} + \frac{\operatorname{atan}{\left(2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)} \csc{\left(x \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)} \csc{\left(x \right)} + 1 \right)}}{4}$$
/ 2/x\\ / ___ 2/x\\ / ___ 2/x\\
|2*sin |-|| | 2*\/ 2 *sin |-|| | 2*\/ 2 *sin |-||
| \2/| | \2/| | \2/| 2/pi x\ 4/x\
log|---------| atan|1 + ---------------| atan|-1 + ---------------| sin |-- + -| sin |-|
\ sin(x) / \ sin(x) / \ sin(x) / \2 2/ \2/
-------------- - ------------------------- + -------------------------- - ------------------- + ---------
2 4 4 / / pi\\ 2
8*|1 - sin|x + --|| 4*sin (x)
\ \ 2 //
$$\frac{\log{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} \right)}}{2} + \frac{\sin^{4}{\left(\frac{x}{2} \right)}}{4 \sin^{2}{\left(x \right)}} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1 \right)}}{4} - \frac{\sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{8 \left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right)}$$
/ 2/x\\ / ___ 2/x\\ / ___ 2/x\\
|2*sin |-|| | 2*\/ 2 *sin |-|| | 2*\/ 2 *sin |-||
| \2/| | \2/| | \2/| 2/pi x\ 2/x\
log|---------| atan|1 + ---------------| atan|-1 + ---------------| sin |-- + -| sin |-|
\ sin(x) / \ sin(x) / \ sin(x) / \2 2/ \2/
-------------- - ------------------------- + -------------------------- - ------------------- + -------------------
2 4 4 / / pi\\ / / pi\\
8*|1 - sin|x + --|| 8*|1 + sin|x + --||
\ \ 2 // \ \ 2 //
$$\frac{\log{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1 \right)}}{4} + \frac{\sin^{2}{\left(\frac{x}{2} \right)}}{8 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} - \frac{\sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{8 \left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right)}$$
/ /x\\ / ___ /x\\ / ___ /x\\ 2/x\
log|tan|-|| atan|1 + \/ 2 *tan|-|| atan|-1 + \/ 2 *tan|-|| tan |-|
\ \2// \ \2// 1 \ \2// \4/
----------- - ---------------------- - ---------- + ----------------------- + ----------------------------------
2 4 2/x\ 4 / 2/x\\
16*tan |-| 2 | 1 - tan |-||
\2/ / 2/x\\ | \2/|
2*|1 + tan |-|| *|1 + -----------|
\ \4// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{\log{\left(\tan{\left(\frac{x}{2} \right)} \right)}}{2} + \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} - 1 \right)}}{4} - \frac{\operatorname{atan}{\left(\sqrt{2} \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4} - \frac{1}{16 \tan^{2}{\left(\frac{x}{2} \right)}} + \frac{\tan^{2}{\left(\frac{x}{4} \right)}}{2 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}$$
log(tan(x/2))/2 - atan(1 + sqrt(2)*tan(x/2))/4 - 1/(16*tan(x/2)^2) + atan(-1 + sqrt(2)*tan(x/2))/4 + tan(x/4)^2/(2*(1 + tan(x/4)^2)^2*(1 + (1 - tan(x/2)^2)/(1 + tan(x/2)^2)))