Mister Exam
Expression simplification/
Least common denominator/
sin(x)^cos(x/2)*(-log(sin(x))*sin(x/2)/2+cos(x)*cos(x/2)/sin(x))
Least common denominator sin(x)^cos(x/2)*(-log(sin(x))*sin(x/2)/2+cos(x)*cos(x/2)/sin(x))
The solution
You have entered
[src]
/x\ / /x\ /x\\
cos|-| |-log(sin(x))*sin|-| cos(x)*cos|-||
\2/ | \2/ \2/|
(sin(x)) *|------------------- + -------------|
\ 2 sin(x) /
$$\left(\frac{- \log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(x \right)} \cos{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
sin(x)^cos(x/2)*(((-log(sin(x)))*sin(x/2))/2 + (cos(x)*cos(x/2))/sin(x))
General simplification
[src]
/x\ / /x\\
-1 + cos|-| | log(sin(x))*sin(x)*sin|-||
\2/ | /x\ \2/|
(sin(x)) *|cos(x)*cos|-| - -------------------------|
\ \2/ 2 /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{2} + \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}\right) \sin^{\cos{\left(\frac{x}{2} \right)} - 1}{\left(x \right)}$$
sin(x)^(-1 + cos(x/2))*(cos(x)*cos(x/2) - log(sin(x))*sin(x)*sin(x/2)/2)
Numerical answer
[src]
sin(x)^cos(x/2)*(-0.5*log(sin(x))*sin(x/2) + cos(x)*cos(x/2)/sin(x))
sin(x)^cos(x/2)*(-0.5*log(sin(x))*sin(x/2) + cos(x)*cos(x/2)/sin(x))
Rational denominator
[src]
/x\
cos|-|
\2/ / /x\ /x\\
(sin(x)) *|2*cos(x)*cos|-| - log(sin(x))*sin(x)*sin|-||
\ \2/ \2//
------------------------------------------------------------
2*sin(x)
$$\frac{\left(- \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \sin{\left(\frac{x}{2} \right)} + 2 \cos{\left(x \right)} \cos{\left(\frac{x}{2} \right)}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{2 \sin{\left(x \right)}}$$
sin(x)^cos(x/2)*(2*cos(x)*cos(x/2) - log(sin(x))*sin(x)*sin(x/2))/(2*sin(x))
Combinatorics
[src]
/x\
cos|-|
\2/ / /x\ /x\\
(sin(x)) *|2*cos(x)*cos|-| - log(sin(x))*sin(x)*sin|-||
\ \2/ \2//
------------------------------------------------------------
2*sin(x)
$$\frac{\left(- \log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)} \sin{\left(x \right)} + 2 \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{2 \sin{\left(x \right)}}$$
sin(x)^cos(x/2)*(2*cos(x)*cos(x/2) - log(sin(x))*sin(x)*sin(x/2))/(2*sin(x))
Combining rational expressions
[src]
/x\
cos|-|
\2/ / /x\ /x\\
(sin(x)) *|2*cos(x)*cos|-| - log(sin(x))*sin(x)*sin|-||
\ \2/ \2//
------------------------------------------------------------
2*sin(x)
$$\frac{\left(- \log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)} \sin{\left(x \right)} + 2 \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{2 \sin{\left(x \right)}}$$
sin(x)^cos(x/2)*(2*cos(x)*cos(x/2) - log(sin(x))*sin(x)*sin(x/2))/(2*sin(x))
Common denominator
[src]
/x\ /x\
cos|-| cos|-|
\2/ /x\ \2/ /x\
(sin(x)) *log(sin(x))*sin|-| (sin(x)) *cos(x)*cos|-|
\2/ \2/
- --------------------------------- + ----------------------------
2 sin(x)
$$- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)} \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{2} + \frac{\sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)} \cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$$
-sin(x)^cos(x/2)*log(sin(x))*sin(x/2)/2 + sin(x)^cos(x/2)*cos(x)*cos(x/2)/sin(x)
Powers
[src]
I*x -I*x
--- -----
2 2 / / I*x -I*x \\
e e | / -I*x I*x\ | --- -----||
---- + ------ | | ----- ---| / / -I*x I*x\ \ / I*x -I*x\ | 2 2 ||
2 2 | | 2 2 | |-I*\- e + e / | |e e | |e e ||
/ / -I*x I*x\ \ |I*\- e + e /*log|--------------------| 2*I*|---- + -----|*|---- + ------||
|-I*\- e + e / | | \ 2 / \ 2 2 / \ 2 2 /|
|--------------------| *|--------------------------------------------- + ----------------------------------|
\ 2 / | 4 -I*x I*x |
\ - e + e /
$$\left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2}\right)^{\frac{e^{\frac{i x}{2}}}{2} + \frac{e^{- \frac{i x}{2}}}{2}} \left(\frac{2 i \left(\frac{e^{\frac{i x}{2}}}{2} + \frac{e^{- \frac{i x}{2}}}{2}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}{e^{i x} - e^{- i x}} + \frac{i \left(e^{\frac{i x}{2}} - e^{- \frac{i x}{2}}\right) \log{\left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} \right)}}{4}\right)$$
/x\ / /x\ /x\\
cos|-| | log(sin(x))*sin|-| cos(x)*cos|-||
\2/ | \2/ \2/|
(sin(x)) *|- ------------------ + -------------|
\ 2 sin(x) /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
sin(x)^cos(x/2)*(-log(sin(x))*sin(x/2)/2 + cos(x)*cos(x/2)/sin(x))
Assemble expression
[src]
/x\ / /x\ /x\\
cos|-| | log(sin(x))*sin|-| cos(x)*cos|-||
\2/ | \2/ \2/|
(sin(x)) *|- ------------------ + -------------|
\ 2 sin(x) /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(x \right)} \cos{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
sin(x)^cos(x/2)*(-log(sin(x))*sin(x/2)/2 + cos(x)*cos(x/2)/sin(x))
Trigonometric part
[src]
/x\
cos|-|
\2/ / log(sin(x)) cos(x)*log(sin(x)) \
(sin(x)) *|- ----------- + ------------------ + cos(x)|
\ 2 2 /
------------------------------------------------------------
/x\
2*sin|-|
\2/
$$\frac{\left(\frac{\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} \right)}}{2} + \cos{\left(x \right)}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{2 \sin{\left(\frac{x}{2} \right)}}$$
1
------
/x\
sec|-| / / 1 \ \
\2/ | log|------| |
/ 1 \ | \csc(x)/ csc(x) |
|------| *|- ----------- + -------------|
\csc(x)/ | /x\ /x\|
| 2*csc|-| sec(x)*sec|-||
\ \2/ \2//
$$\left(- \frac{\log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{2 \csc{\left(\frac{x}{2} \right)}} + \frac{\csc{\left(x \right)}}{\sec{\left(\frac{x}{2} \right)} \sec{\left(x \right)}}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{\sec{\left(\frac{x}{2} \right)}}}$$
2/x\
-1 + cot |-|
\4/
------------ / / /x\ \\
2/x\ | | 2*cot|-| ||
1 + cot |-| | /x\ | \2/ ||
\4/ | 2 2*cot|-|*log|-----------||
/ /x\ \ | /x\ / 2/x\\ \4/ | 2/x\||
| 2*cot|-| | | 2*cot|-| |-1 + cot |-|| |1 + cot |-|||
| \2/ | | \4/ \ \4// \ \2//|
|-----------| *|- ----------- + ---------------------- - -------------------------|
| 2/x\| | 2/x\ / 2/x\\ /x\ 2/x\ |
|1 + cot |-|| | 1 + cot |-| 2*|1 + cot |-||*cot|-| 1 + cot |-| |
\ \2// \ \4/ \ \4// \4/ \4/ /
----------------------------------------------------------------------------------------------
2
$$\frac{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{2 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right) \cot{\left(\frac{x}{4} \right)}} - \frac{2 \log{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \cot{\left(\frac{x}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} - \frac{2 \cot{\left(\frac{x}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}\right)}{2}$$
/x\ / /x\ / 2/x\ 2/x\\ \
cos|-| | log(sin(x))*sin|-| |cos |-| - sin |-||*cos(x)|
\2/ | \2/ \ \4/ \4// |
(sin(x)) *|- ------------------ + --------------------------|
| 2 /x\ |
| (1 + cos(x))*tan|-| |
\ \2/ /
$$\left(\frac{\left(- \sin^{2}{\left(\frac{x}{4} \right)} + \cos^{2}{\left(\frac{x}{4} \right)}\right) \cos{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
2/x\
1 - tan |-|
\4/
----------- / / /x\ \ \
2/x\ | | 2*tan|-| | |
1 + tan |-| | | \2/ | /x\|
\4/ | 2 2*log|-----------|*tan|-||
/ /x\ \ | /x\ / 2/x\\ | 2/x\| \4/|
| 2*tan|-| | | 2*tan|-| |1 - tan |-|| |1 + tan |-|| |
| \2/ | | \4/ \ \4// \ \2// |
|-----------| *|- ----------- + ---------------------- - -------------------------|
| 2/x\| | 2/x\ / 2/x\\ /x\ 2/x\ |
|1 + tan |-|| | 1 + tan |-| 2*|1 + tan |-||*tan|-| 1 + tan |-| |
\ \2// \ \4/ \ \4// \4/ \4/ /
---------------------------------------------------------------------------------------------
2
$$\frac{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2}}{2 \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right) \tan{\left(\frac{x}{4} \right)}} - \frac{2 \log{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \tan{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1} - \frac{2 \tan{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}\right)}{2}$$
/ / 4/x\ \ / 4/x\\ \
| | 8*sin |-| | | 4*sin |-|| |
| 2/x\ | \4/ | | \2/| |
/x\ | /x\ cos |-|*|1 - ----------|*|1 - ---------|*sin(x)|
cos|-| | log(sin(x))*sin|-| \4/ \ 1 - cos(x)/ | 2 | |
\2/ | \2/ \ sin (x) / |
(sin(x)) *|- ------------------ + -----------------------------------------------|
\ 2 2*(1 - cos(x)) /
$$\left(\frac{\left(1 - \frac{8 \sin^{4}{\left(\frac{x}{4} \right)}}{1 - \cos{\left(x \right)}}\right) \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)} \cos^{2}{\left(\frac{x}{4} \right)}}{2 \left(1 - \cos{\left(x \right)}\right)} - \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
1
-----------
/pi x\
csc|-- - -| / /x\ / 1 \\
\2 2/ | csc|-| log|------||
/ 1 \ | 1 \2/ \csc(x)/|
|------| *|- ------ + ------------ - -----------|
\csc(x)/ | /x\ 2/pi x\ /x\ |
| csc|-| csc |-- - -| csc|-| |
\ \2/ \2 2/ \2/ /
-----------------------------------------------------------
2
$$\frac{\left(- \frac{\log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{\csc{\left(\frac{x}{2} \right)}} + \frac{\csc{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(\frac{x}{2} \right)}}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}}}{2}$$
/x\ / 2/x\ \
cos|-| | cos |-| |
\2/ | /x\ \2/ /x\|
(sin(x)) *|- sin|-| + ------- - log(sin(x))*sin|-||
| \2/ /x\ \2/|
| sin|-| |
\ \2/ /
--------------------------------------------------------
2
$$\frac{\left(- \log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)} - \sin{\left(\frac{x}{2} \right)} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{2}$$
1
-----------
/pi x\
csc|-- - -| / / 1 \ \
\2 2/ | log|------| |
/ 1 \ | \csc(x)/ csc(x) |
|------| *|- ----------- + -----------------------|
\csc(x)/ | /x\ /pi \ /pi x\|
| 2*csc|-| csc|-- - x|*csc|-- - -||
\ \2/ \2 / \2 2//
$$\left(- \frac{\log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{2 \csc{\left(\frac{x}{2} \right)}} + \frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}}$$
/ / 4/x\\ / 4/x\\ \
| | 4*sin |-|| | 4*sin |-|| |
| 2/pi x\ | \2/| | \4/| |
| sin |-- + -|*|1 - ---------|*|1 - ---------|*sin(x)|
/pi x\ | /x\ \2 4/ | 2 | | 2/x\ | |
sin|-- + -| | log(sin(x))*sin|-| \ sin (x) / | sin |-| | |
\2 2/ | \2/ \ \2/ / |
(sin(x)) *|- ------------------ + ---------------------------------------------------|
| 2 2/x\ |
| 4*sin |-| |
\ \2/ /
$$\left(\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{4} \right)}}{\sin^{2}{\left(\frac{x}{2} \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)} \sin^{2}{\left(\frac{x}{4} + \frac{\pi}{2} \right)}}{4 \sin^{2}{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2}\right) \sin^{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(x \right)}$$
/x\ / /x\\
cos|-| | log(sin(x))*sin|-||
\2/ | 1 /x\ \2/|
(sin(x)) *|-------- - sin|-| - ------------------|
| /x\ \2/ 2 |
|2*sin|-| |
\ \2/ /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} - \sin{\left(\frac{x}{2} \right)} + \frac{1}{2 \sin{\left(\frac{x}{2} \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
2/x\
-1 + cot |-|
\4/
------------ / / /x\ \\
2/x\ | | 2*cot|-| ||
1 + cot |-| | /x\ | \2/ ||
\4/ | cot|-|*log|-----------||
/ /x\ \ |/ 2/x\\ \4/ | 2/x\||
| 2*cot|-| | ||-1 + cot |-||*cot(x) |1 + cot |-|||
| \2/ | |\ \4// \ \2//|
|-----------| *|--------------------- - -----------------------|
| 2/x\| | 2/x\ 2/x\ |
|1 + cot |-|| | 1 + cot |-| 1 + cot |-| |
\ \2// \ \4/ \4/ /
$$\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \cot{\left(x \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1} - \frac{\log{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \cot{\left(\frac{x}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}\right)$$
2/x\
-1 + cot |-|
\4/
------------ / / /x\ \ \
2/x\ | | 2*cot|-| | |
1 + cot |-| | /x\ | \2/ | |
\4/ | cot|-|*log|-----------| |
/ /x\ \ | \4/ | 2/x\| / 2/x\\ / 2/x\\|
| 2*cot|-| | | |1 + cot |-|| |-1 + cot |-||*|-1 + cot |-|||
| \2/ | | \ \2// \ \2// \ \4//|
|-----------| *|- ----------------------- + -----------------------------|
| 2/x\| | 2/x\ / 2/x\\ /x\ |
|1 + cot |-|| | 1 + cot |-| 2*|1 + cot |-||*cot|-| |
\ \2// \ \4/ \ \4// \2/ /
$$\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \cot{\left(\frac{x}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}\right)$$
/x\ / /x\ /x\\
cos|-| |cos|-| log(sin(x))*sin|-||
\2/ | \2/ \2/|
(sin(x)) *|------ - ------------------|
\tan(x) 2 /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(\frac{x}{2} \right)}}{\tan{\left(x \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
2/x\
1 - tan |-|
\4/
----------- / / /x\ \ \
2/x\ | | 2*tan|-| | |
1 + tan |-| | | \2/ | /x\|
\4/ | log|-----------|*tan|-||
/ /x\ \ | 2/x\ | 2/x\| \4/|
| 2*tan|-| | | 1 - tan |-| |1 + tan |-|| |
| \2/ | | \4/ \ \2// |
|-----------| *|-------------------- - -----------------------|
| 2/x\| |/ 2/x\\ 2/x\ |
|1 + tan |-|| ||1 + tan |-||*tan(x) 1 + tan |-| |
\ \2// \\ \4// \4/ /
$$\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right) \tan{\left(x \right)}} - \frac{\log{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \tan{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}\right)$$
2/x\
1 - tan |-|
\4/
----------- / / /x\ \ \
2/x\ | | 2*tan|-| | |
1 + tan |-| | | \2/ | /x\ |
\4/ | log|-----------|*tan|-| |
/ /x\ \ | | 2/x\| \4/ / 2/x\\ / 2/x\\|
| 2*tan|-| | | |1 + tan |-|| |1 - tan |-||*|1 - tan |-|||
| \2/ | | \ \2// \ \2// \ \4//|
|-----------| *|- ----------------------- + ---------------------------|
| 2/x\| | 2/x\ / 2/x\\ /x\ |
|1 + tan |-|| | 1 + tan |-| 2*|1 + tan |-||*tan|-| |
\ \2// \ \4/ \ \4// \2/ /
$$\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{2 \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \tan{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}\right)$$
1
------ / / 1 \\
/x\ | log|-----------||
sec|-| | /x pi\ | / pi\||
\2/ | sec|- - --| |sec|x - --|||
/ 1 \ | 1 \2 2 / \ \ 2 //|
|-----------| *|- ----------- + ----------- - ----------------|
| / pi\| | /x pi\ 2/x\ /x pi\ |
|sec|x - --|| | sec|- - --| sec |-| sec|- - --| |
\ \ 2 // \ \2 2 / \2/ \2 2 / /
--------------------------------------------------------------------
2
$$\frac{\left(- \frac{\log{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{\sec{\left(\frac{x}{2} \right)}}}}{2}$$
/ / 2/x pi\\ / 2/ pi x\\ \
| | cos |- - --|| | cos |- -- + -|| |
| 2/x\ | \2 2 /| | \ 2 4/| /x\|
/x\ | cos |-|*|1 - ------------|*|1 - --------------|*cos|-||
cos|-| | /x pi\ / / pi\\ \4/ | 2/x\ | | 2/x\ | \2/|
\2/ | cos|- - --|*log|cos|x - --|| | cos |-| | | cos |-| | |
/ / pi\\ | \2 2 / \ \ 2 // \ \2/ / \ \4/ / |
|cos|x - --|| *|- ---------------------------- + ------------------------------------------------------|
\ \ 2 // | 2 /x pi\ |
| 2*cos|- - --| |
\ \2 2 / /
$$\left(\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{4} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{4} \right)}}\right) \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos^{2}{\left(\frac{x}{4} \right)} \cos{\left(\frac{x}{2} \right)}}{2 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\log{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2}\right) \cos^{\cos{\left(\frac{x}{2} \right)}}{\left(x - \frac{\pi}{2} \right)}$$
2/x\
-1 + cot |-|
\4/
------------ / / /x\ \ \
2/x\ | | 2*cot|-| | 2 |
1 + cot |-| | /x\ | \2/ | / 2/x\\ / 1 \ / 1 \ /x\|
\4/ | cot|-|*log|-----------| |-1 + cot |-|| *|1 - -------|*|1 - -------|*cot|-||
/ /x\ \ | \4/ | 2/x\| \ \8// | 2/x\| | 2/x\| \2/|
| 2*cot|-| | | |1 + cot |-|| | cot |-|| | cot |-|| |
| \2/ | | \ \2// \ \2// \ \4// |
|-----------| *|- ----------------------- + --------------------------------------------------|
| 2/x\| | 2/x\ 2 |
|1 + cot |-|| | 1 + cot |-| / 2/x\\ |
\ \2// | \4/ 2*|1 + cot |-|| |
\ \ \8// /
$$\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{\cot^{2}{\left(\frac{x}{4} \right)} - 1}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{4} \right)}}\right) \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right)^{2} \cot{\left(\frac{x}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2}} - \frac{\log{\left(\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \cot{\left(\frac{x}{4} \right)}}{\cot^{2}{\left(\frac{x}{4} \right)} + 1}\right)$$
/x\
cos|-| / /x pi\ / / pi\\ /x\\
\2/ | cos|- - --|*log|cos|x - --|| cos(x)*cos|-||
/ / pi\\ | \2 2 / \ \ 2 // \2/|
|cos|x - --|| *|- ---------------------------- + -------------|
\ \ 2 // | 2 / pi\ |
| cos|x - --| |
\ \ 2 / /
$$\left(- \frac{\log{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2} + \frac{\cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}}\right) \cos^{\cos{\left(\frac{x}{2} \right)}}{\left(x - \frac{\pi}{2} \right)}$$
/x\
cos|-| / 2/x\ \
\2/ | cos |-| |
/ / pi\\ | /x pi\ \2/ /x pi\ / / pi\\|
|cos|x - --|| *|- cos|- - --| + ----------- - cos|- - --|*log|cos|x - --|||
\ \ 2 // | \2 2 / /x pi\ \2 2 / \ \ 2 //|
| cos|- - --| |
\ \2 2 / /
--------------------------------------------------------------------------------
2
$$\frac{\left(- \log{\left(\cos{\left(x - \frac{\pi}{2} \right)} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}\right) \cos^{\cos{\left(\frac{x}{2} \right)}}{\left(x - \frac{\pi}{2} \right)}}{2}$$
2/x\
1 - tan |-|
\4/
----------- / / /x\ \ \
2/x\ | | 2*tan|-| | |
1 + tan |-| | | \2/ | /x\ |
\4/ | log|-----------|*tan|-| 2 |
/ /x\ \ | | 2/x\| \4/ / 2/x\\ / 2/x\\ / 2/x\\|
| 2*tan|-| | | |1 + tan |-|| |1 - tan |-|| *|1 - tan |-||*|1 - tan |-|||
| \2/ | | \ \2// \ \8// \ \2// \ \4//|
|-----------| *|- ----------------------- + ------------------------------------------|
| 2/x\| | 2/x\ 2 |
|1 + tan |-|| | 1 + tan |-| / 2/x\\ /x\ |
\ \2// | \4/ 2*|1 + tan |-|| *tan|-| |
\ \ \8// \2/ /
$$\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\frac{1 - \tan^{2}{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}} \left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{2 \left(\tan^{2}{\left(\frac{x}{8} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)} \tan{\left(\frac{x}{4} \right)}}{\tan^{2}{\left(\frac{x}{4} \right)} + 1}\right)$$
/x\ / /x\ 2/x\ / 2/x\\ / 2/x\\\
cos|-| | log(sin(x))*sin|-| cos |-|*|1 - tan |-||*|1 - tan |-|||
\2/ | \2/ \4/ \ \2// \ \4//|
(sin(x)) *|- ------------------ + -----------------------------------|
| 2 /x\ |
| 2*tan|-| |
\ \2/ /
$$\left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{2}{\left(\frac{x}{4} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
/x\
cos|-|
\2/ / /x\ /3*x\ /3*x\ /x\ \
(sin(x)) *|2*cos|-| + 2*cos|---| + cos|---|*log(sin(x)) - cos|-|*log(sin(x))|
\ \2/ \ 2 / \ 2 / \2/ /
----------------------------------------------------------------------------------
4*sin(x)
$$\frac{\left(- \log{\left(\sin{\left(x \right)} \right)} \cos{\left(\frac{x}{2} \right)} + \log{\left(\sin{\left(x \right)} \right)} \cos{\left(\frac{3 x}{2} \right)} + 2 \cos{\left(\frac{x}{2} \right)} + 2 \cos{\left(\frac{3 x}{2} \right)}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{4 \sin{\left(x \right)}}$$
/pi x\ / /x\ /pi x\\
sin|-- + -| | log(sin(x))*sin|-| sin(2*x)*sin|-- + -||
\2 2/ | \2/ \2 2/|
(sin(x)) *|- ------------------ + --------------------|
| 2 2 |
\ 2*sin (x) /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\sin{\left(2 x \right)} \sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{2 \sin^{2}{\left(x \right)}}\right) \sin^{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(x \right)}$$
/ / 2/x\ \ / 2/x\ \ \
1 | | sec |-| | | sec |-| | |
------ | / 1 \ | \2/ | | \4/ | /x pi\|
/x\ | log|-----------| |1 - ------------|*|1 - --------------|*sec|- - --||
sec|-| | | / pi\| | 2/x pi\| | 2/ pi x\| \2 2 /|
\2/ | |sec|x - --|| | sec |- - --|| | sec |- -- + -|| |
/ 1 \ | \ \ 2 // \ \2 2 // \ \ 2 4// |
|-----------| *|- ---------------- + ---------------------------------------------------|
| / pi\| | /x pi\ /x\ 2/x\ |
|sec|x - --|| | 2*sec|- - --| 2*sec|-|*sec |-| |
\ \ 2 // \ \2 2 / \2/ \4/ /
$$\left(\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{4} \right)}}{\sec^{2}{\left(\frac{x}{4} - \frac{\pi}{2} \right)}} + 1\right) \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{2 \sec^{2}{\left(\frac{x}{4} \right)} \sec{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{\sec{\left(\frac{x}{2} \right)}}}$$
1
------ / / 1 \ \
/x\ | log|-----------| |
sec|-| | | / pi\| / pi\ |
\2/ | |sec|x - --|| sec|x - --| |
/ 1 \ | \ \ 2 // \ 2 / |
|-----------| *|- ---------------- + -------------|
| / pi\| | /x pi\ /x\|
|sec|x - --|| | 2*sec|- - --| sec(x)*sec|-||
\ \ 2 // \ \2 2 / \2//
$$\left(- \frac{\log{\left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}}{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(\frac{x}{2} \right)} \sec{\left(x \right)}}\right) \left(\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{1}{\sec{\left(\frac{x}{2} \right)}}}$$
/ 2 \
/x\ | /x\ / 2/x\\ 4/x\ / 2/x\\ / 2/x\\ /x\|
cos|-| | log(sin(x))*sin|-| |-1 + cot |-|| *sin |-|*|1 - tan |-||*|1 - tan |-||*cot|-||
\2/ | \2/ \ \8// \8/ \ \2// \ \4// \2/|
(sin(x)) *|- ------------------ + ----------------------------------------------------------|
\ 2 2 /
$$\left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\cot^{2}{\left(\frac{x}{8} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{8} \right)} \cot{\left(\frac{x}{2} \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
/pi x\ / /x\ / pi\ /pi x\\
sin|-- + -| | log(sin(x))*sin|-| sin|x + --|*sin|-- + -||
\2 2/ | \2/ \ 2 / \2 2/|
(sin(x)) *|- ------------------ + -----------------------|
\ 2 sin(x) /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(x \right)}}\right) \sin^{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(x \right)}$$
/x\ / /x\ /x\\
cos|-| | log(sin(x))*sin|-| cos(x)*cos|-||
\2/ | \2/ \2/|
(sin(x)) *|- ------------------ + -------------|
\ 2 sin(x) /
$$\left(- \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(\frac{x}{2} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
/ / 2/pi x\\ / 2/pi x\\ \
1 | | csc |-- - -|| | csc |-- - -|| |
----------- | | \2 2/| | \2 4/| /x\|
/pi x\ | |1 - ------------|*|1 - ------------|*csc|-||
csc|-- - -| | / 1 \ | 2/x\ | | 2/x\ | \2/|
\2 2/ | log|------| | csc |-| | | csc |-| | |
/ 1 \ | \csc(x)/ \ \2/ / \ \4/ / |
|------| *|- ----------- + --------------------------------------------|
\csc(x)/ | /x\ /pi x\ 2/pi x\ |
| 2*csc|-| 2*csc|-- - -|*csc |-- - -| |
\ \2/ \2 2/ \2 4/ /
$$\left(\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{4} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{4} \right)}}\right) \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{2 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)} \csc^{2}{\left(- \frac{x}{4} + \frac{\pi}{2} \right)}} - \frac{\log{\left(\frac{1}{\csc{\left(x \right)}} \right)}}{2 \csc{\left(\frac{x}{2} \right)}}\right) \left(\frac{1}{\csc{\left(x \right)}}\right)^{\frac{1}{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}}$$
/ 2 \
/x\ | /x\ / 2/x\\ 4/x\ / 2/x\\ / 2/x\\|
cos|-| | log(sin(x))*sin|-| |1 - tan |-|| *cos |-|*|1 - tan |-||*|1 - tan |-|||
\2/ | \2/ \ \8// \8/ \ \2// \ \4//|
(sin(x)) *|- ------------------ + --------------------------------------------------|
| 2 /x\ |
| 2*tan|-| |
\ \2/ /
$$\left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{8} \right)}\right)^{2} \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right) \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cos^{4}{\left(\frac{x}{8} \right)}}{2 \tan{\left(\frac{x}{2} \right)}} - \frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)}}{2}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}$$
/pi x\ / 2/pi x\ \
sin|-- + -| | sin |-- + -| |
\2 2/ | /x\ \2 2/ /x\|
(sin(x)) *|- sin|-| + ------------ - log(sin(x))*sin|-||
| \2/ /x\ \2/|
| sin|-| |
\ \2/ /
------------------------------------------------------------------
2
$$\frac{\left(- \log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)} - \sin{\left(\frac{x}{2} \right)} + \frac{\sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}}\right) \sin^{\sin{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(x \right)}}{2}$$
/x\
cos|-|
\2/ / 1 /x\ /x\\
(sin(x)) *|------ - 2*sin|-| - log(sin(x))*sin|-||
| /x\ \2/ \2/|
|sin|-| |
\ \2/ /
-------------------------------------------------------
2
$$\frac{\left(- \log{\left(\sin{\left(x \right)} \right)} \sin{\left(\frac{x}{2} \right)} - 2 \sin{\left(\frac{x}{2} \right)} + \frac{1}{\sin{\left(\frac{x}{2} \right)}}\right) \sin^{\cos{\left(\frac{x}{2} \right)}}{\left(x \right)}}{2}$$
sin(x)^cos(x/2)*(1/sin(x/2) - 2*sin(x/2) - log(sin(x))*sin(x/2))/2