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