Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(-sin(x)/cos(2*x)^(1/4)+cos(x)*sin(2*x)/(2*cos(2*x)^(5/4)))/(1+cos(x)^2/sqrt(cos(2*x)))
¿Cómo vas a descomponer esta (-sin(x)/cos(2*x)^(1/4)+cos(x)*sin(2*x)/(2*cos(2*x)^(5/4)))/(1+cos(x)^2/sqrt(cos(2*x))) expresión en fracciones?
Solución
Ha introducido
[src]
-sin(x) cos(x)*sin(2*x)
------------ + ---------------
4 __________ 5/4
\/ cos(2*x) 2*cos (2*x)
------------------------------
2
cos (x)
1 + ------------
__________
\/ cos(2*x)
$$\frac{\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \cos^{\frac{5}{4}}{\left(2 x \right)}} + \frac{\left(-1\right) \sin{\left(x \right)}}{\sqrt[4]{\cos{\left(2 x \right)}}}}{\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1}$$
((-sin(x))/cos(2*x)^(1/4) + (cos(x)*sin(2*x))/((2*cos(2*x)^(5/4))))/(1 + cos(x)^2/sqrt(cos(2*x)))
Simplificación general
[src]
3
sin (x)
------------------------------------
/ 2 __________\ 3/4
\cos (x) + \/ cos(2*x) /*cos (2*x)
$$\frac{\sin^{3}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + \sqrt{\cos{\left(2 x \right)}}\right) \cos^{\frac{3}{4}}{\left(2 x \right)}}$$
sin(x)^3/((cos(x)^2 + sqrt(cos(2*x)))*cos(2*x)^(3/4))
Potencias
[src]
/ I*x -I*x\
|e e | / -2*I*x 2*I*x\
/ -I*x I*x\ I*|---- + -----|*\- e + e /
I*\- e + e / \ 2 2 /
------------------------- - -------------------------------------
__________________ 5/4
/ -2*I*x 2*I*x / -2*I*x 2*I*x\
/ e e |e e |
2*4 / ------- + ------ 4*|------- + ------|
\/ 2 2 \ 2 2 /
-----------------------------------------------------------------
2
/ I*x -I*x\
|e e |
|---- + -----|
\ 2 2 /
1 + -----------------------
__________________
/ -2*I*x 2*I*x
/ e e
/ ------- + ------
\/ 2 2
$$\frac{- \frac{i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{2 i x} - e^{- 2 i x}\right)}{4 \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)^{\frac{5}{4}}} + \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \sqrt[4]{\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}}}}{\frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}}{\sqrt{\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}}} + 1}$$
sin(x) cos(x)*sin(2*x)
- ------------ + ---------------
4 __________ 5/4
\/ cos(2*x) 2*cos (2*x)
--------------------------------
2
cos (x)
1 + ------------
__________
\/ cos(2*x)
$$\frac{- \frac{\sin{\left(x \right)}}{\sqrt[4]{\cos{\left(2 x \right)}}} + \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \cos^{\frac{5}{4}}{\left(2 x \right)}}}{\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1}$$
(-sin(x)/cos(2*x)^(1/4) + cos(x)*sin(2*x)/(2*cos(2*x)^(5/4)))/(1 + cos(x)^2/sqrt(cos(2*x)))
Combinatoria
[src]
-(-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x)) ---------------------------------------- / 2 __________\ 3/4 2*\cos (x) + \/ cos(2*x) /*cos (2*x)
$$- \frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)} - \sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} + \sqrt{\cos{\left(2 x \right)}}\right) \cos^{\frac{3}{4}}{\left(2 x \right)}}$$
-(-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x))/(2*(cos(x)^2 + sqrt(cos(2*x)))*cos(2*x)^(3/4))
Denominador común
[src]
-(-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x))
----------------------------------------
5/4 2 3/4
2*cos (2*x) + 2*cos (x)*cos (2*x)
$$- \frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)} - \sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \cos^{\frac{3}{4}}{\left(2 x \right)} + 2 \cos^{\frac{5}{4}}{\left(2 x \right)}}$$
-(-cos(x)*sin(2*x) + 2*cos(2*x)*sin(x))/(2*cos(2*x)^(5/4) + 2*cos(x)^2*cos(2*x)^(3/4))
Denominador racional
[src]
7/4 3 4 __________ 3/4 2 5/4
2*cos (2*x)*sin(x) + cos (x)*\/ cos(2*x) *sin(2*x) - cos (2*x)*cos(x)*sin(2*x) - 2*cos (x)*cos (2*x)*sin(x)
-----------------------------------------------------------------------------------------------------------------
/ 4 \
2*\cos (x) - cos(2*x)/*cos(2*x)
$$\frac{- 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos^{\frac{5}{4}}{\left(2 x \right)} + 2 \sin{\left(x \right)} \cos^{\frac{7}{4}}{\left(2 x \right)} + \sin{\left(2 x \right)} \cos^{3}{\left(x \right)} \sqrt[4]{\cos{\left(2 x \right)}} - \sin{\left(2 x \right)} \cos{\left(x \right)} \cos^{\frac{3}{4}}{\left(2 x \right)}}{2 \left(\cos^{4}{\left(x \right)} - \cos{\left(2 x \right)}\right) \cos{\left(2 x \right)}}$$
(2*cos(2*x)^(7/4)*sin(x) + cos(x)^3*cos(2*x)^(1/4)*sin(2*x) - cos(2*x)^(3/4)*cos(x)*sin(2*x) - 2*cos(x)^2*cos(2*x)^(5/4)*sin(x))/(2*(cos(x)^4 - cos(2*x))*cos(2*x))
Unión de expresiones racionales
[src]
cos(x)*sin(2*x) - 2*cos(2*x)*sin(x) -------------------------------------- / 2 __________\ 3/4 2*\cos (x) + \/ cos(2*x) /*cos (2*x)
$$\frac{- 2 \sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} + \sqrt{\cos{\left(2 x \right)}}\right) \cos^{\frac{3}{4}}{\left(2 x \right)}}$$
(cos(x)*sin(2*x) - 2*cos(2*x)*sin(x))/(2*(cos(x)^2 + sqrt(cos(2*x)))*cos(2*x)^(3/4))
Compilar la expresión
[src]
sin(x) cos(x)*sin(2*x)
- ------------ + ---------------
4 __________ 5/4
\/ cos(2*x) 2*cos (2*x)
--------------------------------
2
cos (x)
1 + ------------
__________
\/ cos(2*x)
$$\frac{- \frac{\sin{\left(x \right)}}{\sqrt[4]{\cos{\left(2 x \right)}}} + \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \cos^{\frac{5}{4}}{\left(2 x \right)}}}{\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1}$$
(-sin(x)/cos(2*x)^(1/4) + cos(x)*sin(2*x)/(2*cos(2*x)^(5/4)))/(1 + cos(x)^2/sqrt(cos(2*x)))
Respuesta numérica
[src]
(-cos(2*x)^(-0.25)*sin(x) + 0.5*cos(2*x)^(-1.25)*cos(x)*sin(2*x))/(1.0 + cos(x)^2*cos(2*x)^(-0.5))
(-cos(2*x)^(-0.25)*sin(x) + 0.5*cos(2*x)^(-1.25)*cos(x)*sin(2*x))/(1.0 + cos(x)^2*cos(2*x)^(-0.5))
Parte trigonométrica
[src]
3
sin (x)
-------------------------------------
5/4 / 2 \
/ 2 \ | cos (x) |
\1 - 2*sin (x)/ *|1 + ------------|
| __________|
\ \/ cos(2*x) /
$$\frac{\sin^{3}{\left(x \right)}}{\left(1 - 2 \sin^{2}{\left(x \right)}\right)^{\frac{5}{4}} \left(\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1\right)}$$
3
sin (x)
------------------------------
/ 2 \
| cos (x) | 5/4
|1 + ------------|*cos (2*x)
| __________|
\ \/ cos(2*x) /
$$\frac{\sin^{3}{\left(x \right)}}{\left(\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1\right) \cos^{\frac{5}{4}}{\left(2 x \right)}}$$
3/x\
8*cot |-|
\2/
--------------------------------------------------------------------------------
5/4 / 2 \
/ 2/x\ \ | / 2/x\\ |
3 | 8*cot |-| | | |-1 + cot |-|| |
/ 2/x\\ | \2/ | | \ \2// |
|1 + cot |-|| *|1 - --------------| *|1 + -----------------------------------|
\ \2// | 2| | ______________ |
| / 2/x\\ | | / 2 2|
| |1 + cot |-|| | | / -1 + cot (x) / 2/x\\ |
\ \ \2// / | / ------------ *|1 + cot |-|| |
| / 2 \ \2// |
\ \/ 1 + cot (x) /
$$\frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{8 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right)^{\frac{5}{4}} \left(1 + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\sqrt{\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
/3*x\ /x\
2*cot|---| 6*cot|-|
\ 2 / \2/
- ------------- + -----------
2/3*x\ 2/x\
1 + cot |---| 1 + cot |-|
\ 2 / \2/
----------------------------------------------------------------------------------------
/ 2 \
3/4 | ______________ / 2/x\\ / 2/x\\|
/ 2 \ | / 2 | -1 + cot |-|| 2*|-1 + cot |-|||
|-1 + cot (x)| | / -1 + cot (x) | \2/| \ \2//|
4*|------------| *|-1 + / ------------ + |1 - ------------| + ----------------|
| 2 | | / 2 | 2/x\ | 2/x\ |
\1 + cot (x) / | \/ 1 + cot (x) | 1 + cot |-| | 1 + cot |-| |
\ \ \2/ / \2/ /
$$\frac{- \frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}\right)^{\frac{3}{4}} \left(\sqrt{\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}} + \left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} + \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - 1\right)}$$
1 1
- --------------------- + -------------------------------
__________ 5/4
/ 1 / 1 \
4 / -------- *csc(x) 2*|--------| *csc(2*x)*sec(x)
\/ sec(2*x) \sec(2*x)/
---------------------------------------------------------
1
1 + ----------------------
__________
/ 1 2
/ -------- *sec (x)
\/ sec(2*x)
$$\frac{- \frac{1}{\sqrt[4]{\frac{1}{\sec{\left(2 x \right)}}} \csc{\left(x \right)}} + \frac{1}{2 \left(\frac{1}{\sec{\left(2 x \right)}}\right)^{\frac{5}{4}} \csc{\left(2 x \right)} \sec{\left(x \right)}}}{1 + \frac{1}{\sqrt{\frac{1}{\sec{\left(2 x \right)}}} \sec^{2}{\left(x \right)}}}$$
1 3
- -------- + ------
csc(3*x) csc(x)
------------------------------------------------------------------------------------
3/4 / _______________ 2 \
/ 1 \ | / 1 / 1 \ 2 |
4*|-------------| *|-1 + / ------------- + |1 - -----------| + -----------|
| /pi \| | / /pi \ | /pi \| /pi \|
|csc|-- - 2*x|| | / csc|-- - 2*x| | csc|-- - x|| csc|-- - x||
\ \2 // \ \/ \2 / \ \2 // \2 //
$$\frac{- \frac{1}{\csc{\left(3 x \right)}} + \frac{3}{\csc{\left(x \right)}}}{4 \left(\left(1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{2} + \sqrt{\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}} - 1 + \frac{2}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right)^{\frac{3}{4}}}$$
/x\ / 2/x\\
2*tan|-| |1 - tan |-||*tan(x)
\2/ \ \2//
- --------------------------------- + --------------------------------------------
_____________ 5/4
/ 2 / 2 \
/ 1 - tan (x) / 2/x\\ |1 - tan (x)| / 2 \ / 2/x\\
/ ----------- *|1 + tan |-|| |-----------| *\1 + tan (x)/*|1 + tan |-||
4 / 2 \ \2// | 2 | \ \2//
\/ 1 + tan (x) \1 + tan (x)/
----------------------------------------------------------------------------------
2
/ 2/x\\
|1 - tan |-||
\ \2//
1 + ----------------------------------
_____________
/ 2 2
/ 1 - tan (x) / 2/x\\
/ ----------- *|1 + tan |-||
/ 2 \ \2//
\/ 1 + tan (x)
$$\frac{- \frac{2 \tan{\left(\frac{x}{2} \right)}}{\sqrt[4]{\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(x \right)}}{\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{\frac{5}{4}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}}{1 + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\sqrt{\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}}$$
-sin(3*x) + 3*sin(x) ---------------------------------------------------------------------------------- / 2 _______________ \ | / / pi\\ / /pi \ / pi\| 3/4/pi \ 4*|-1 + |1 - sin|x + --|| + / sin|-- + 2*x| + 2*sin|x + --||*sin |-- + 2*x| \ \ \ 2 // \/ \2 / \ 2 // \2 /
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right)^{2} + 2 \sin{\left(x + \frac{\pi}{2} \right)} + \sqrt{\sin{\left(2 x + \frac{\pi}{2} \right)}} - 1\right) \sin^{\frac{3}{4}}{\left(2 x + \frac{\pi}{2} \right)}}$$
/x\ / 2/x\\
2*cot|-| |-1 + cot |-||*cot(x)
\2/ \ \2//
- ---------------------------------- + ---------------------------------------------
______________ 5/4
/ 2 / 2 \
/ -1 + cot (x) / 2/x\\ |-1 + cot (x)| / 2 \ / 2/x\\
/ ------------ *|1 + cot |-|| |------------| *\1 + cot (x)/*|1 + cot |-||
4 / 2 \ \2// | 2 | \ \2//
\/ 1 + cot (x) \1 + cot (x) /
------------------------------------------------------------------------------------
2
/ 2/x\\
|-1 + cot |-||
\ \2//
1 + -----------------------------------
______________
/ 2 2
/ -1 + cot (x) / 2/x\\
/ ------------ *|1 + cot |-||
/ 2 \ \2//
\/ 1 + cot (x)
$$\frac{- \frac{2 \cot{\left(\frac{x}{2} \right)}}{\sqrt[4]{\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \cot{\left(x \right)}}{\left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}\right)^{\frac{5}{4}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}}{1 + \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\sqrt{\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}}$$
3/ pi\
cos |x - --|
\ 2 /
------------------------------------------
5/4 / 2 \
/ 2/ pi\\ | cos (x) |
|1 - 2*cos |x - --|| *|1 + ------------|
\ \ 2 // | __________|
\ \/ cos(2*x) /
$$\frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\left(1 - 2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}\right)^{\frac{5}{4}} \left(\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1\right)}$$
/ pi\
sin(2*x)*sin|x + --|
sin(x) \ 2 /
- ------------------- + --------------------
_______________ 5/4/pi \
/ /pi \ 2*sin |-- + 2*x|
4 / sin|-- + 2*x| \2 /
\/ \2 /
--------------------------------------------
2/ pi\
sin |x + --|
\ 2 /
1 + -------------------
_______________
/ /pi \
/ sin|-- + 2*x|
\/ \2 /
$$\frac{- \frac{\sin{\left(x \right)}}{\sqrt[4]{\sin{\left(2 x + \frac{\pi}{2} \right)}}} + \frac{\sin{\left(2 x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{2 \sin^{\frac{5}{4}}{\left(2 x + \frac{\pi}{2} \right)}}}{\frac{\sin^{2}{\left(x + \frac{\pi}{2} \right)}}{\sqrt{\sin{\left(2 x + \frac{\pi}{2} \right)}}} + 1}$$
/3*x\ /x\
2*tan|---| 6*tan|-|
\ 2 / \2/
- ------------- + -----------
2/3*x\ 2/x\
1 + tan |---| 1 + tan |-|
\ 2 / \2/
------------------------------------------------------------------------------------
/ 2 \
3/4 | _____________ / 2/x\\ / 2/x\\|
/ 2 \ | / 2 | 1 - tan |-|| 2*|1 - tan |-|||
|1 - tan (x)| | / 1 - tan (x) | \2/| \ \2//|
4*|-----------| *|-1 + / ----------- + |1 - -----------| + ---------------|
| 2 | | / 2 | 2/x\| 2/x\ |
\1 + tan (x)/ | \/ 1 + tan (x) | 1 + tan |-|| 1 + tan |-| |
\ \ \2// \2/ /
$$\frac{- \frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)^{\frac{3}{4}} \left(\sqrt{\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)^{2} - 1\right)}$$
1 1
- ---------------------------- + -----------------------------------------
_______________ 5/4
/ 1 / 1 \ /pi \
/ ------------- *csc(x) 2*|-------------| *csc(2*x)*csc|-- - x|
/ /pi \ | /pi \| \2 /
4 / csc|-- - 2*x| |csc|-- - 2*x||
\/ \2 / \ \2 //
--------------------------------------------------------------------------
1
1 + ----------------------------------
_______________
/ 1 2/pi \
/ ------------- *csc |-- - x|
/ /pi \ \2 /
/ csc|-- - 2*x|
\/ \2 /
$$\frac{- \frac{1}{\sqrt[4]{\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}} \csc{\left(x \right)}} + \frac{1}{2 \left(\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right)^{\frac{5}{4}} \csc{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}}{1 + \frac{1}{\sqrt{\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}}$$
-sin(3*x) + 3*sin(x) ------------------------------------------------------------ / 2 __________ \ 3/4 4*\-1 + (1 - cos(x)) + \/ cos(2*x) + 2*cos(x)/*cos (2*x)
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\left(1 - \cos{\left(x \right)}\right)^{2} + 2 \cos{\left(x \right)} + \sqrt{\cos{\left(2 x \right)}} - 1\right) \cos^{\frac{3}{4}}{\left(2 x \right)}}$$
sin(x) cos(x)*sin(2*x)
- ------------ + ---------------
4 __________ 5/4
\/ cos(2*x) 2*cos (2*x)
--------------------------------
2
cos (x)
1 + ------------
__________
\/ cos(2*x)
$$\frac{- \frac{\sin{\left(x \right)}}{\sqrt[4]{\cos{\left(2 x \right)}}} + \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{2 \cos^{\frac{5}{4}}{\left(2 x \right)}}}{\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1}$$
1 3
- ------------- + -----------
/ pi\ / pi\
sec|3*x - --| sec|x - --|
\ 2 / \ 2 /
--------------------------------------------------------------
3/4 / __________ 2 \
/ 1 \ | / 1 / 1 \ 2 |
4*|--------| *|-1 + / -------- + |1 - ------| + ------|
\sec(2*x)/ \ \/ sec(2*x) \ sec(x)/ sec(x)/
$$\frac{- \frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(x - \frac{\pi}{2} \right)}}}{4 \left(\left(1 - \frac{1}{\sec{\left(x \right)}}\right)^{2} + \sqrt{\frac{1}{\sec{\left(2 x \right)}}} - 1 + \frac{2}{\sec{\left(x \right)}}\right) \left(\frac{1}{\sec{\left(2 x \right)}}\right)^{\frac{3}{4}}}$$
/ pi\ / pi\
cos|x - --| cos(x)*cos|2*x - --|
\ 2 / \ 2 /
- ------------ + --------------------
4 __________ 5/4
\/ cos(2*x) 2*cos (2*x)
-------------------------------------
2
cos (x)
1 + ------------
__________
\/ cos(2*x)
$$\frac{\frac{\cos{\left(x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)}}{2 \cos^{\frac{5}{4}}{\left(2 x \right)}} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\sqrt[4]{\cos{\left(2 x \right)}}}}{\frac{\cos^{2}{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + 1}$$
1
-----------------------------------------------------------------
5/4
/ 2 \ / 1 \ 3
|1 - -------| *|1 + ----------------------------------|*csc (x)
| 2 | | _______________ |
\ csc (x)/ | / 1 2/pi \|
| / ------------- *csc |-- - x||
| / /pi \ \2 /|
| / csc|-- - 2*x| |
\ \/ \2 / /
$$\frac{1}{\left(1 + \frac{1}{\sqrt{\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}\right) \left(1 - \frac{2}{\csc^{2}{\left(x \right)}}\right)^{\frac{5}{4}} \csc^{3}{\left(x \right)}}$$
/ pi\ / pi\
- cos|3*x - --| + 3*cos|x - --|
\ 2 / \ 2 /
------------------------------------------------------------
/ 2 __________ \ 3/4
4*\-1 + (1 - cos(x)) + \/ cos(2*x) + 2*cos(x)/*cos (2*x)
$$\frac{3 \cos{\left(x - \frac{\pi}{2} \right)} - \cos{\left(3 x - \frac{\pi}{2} \right)}}{4 \left(\left(1 - \cos{\left(x \right)}\right)^{2} + 2 \cos{\left(x \right)} + \sqrt{\cos{\left(2 x \right)}} - 1\right) \cos^{\frac{3}{4}}{\left(2 x \right)}}$$
3/x\
8*tan |-|
\2/
-------------------------------------------------------------------------------
5/4 / 2 \
/ 2/x\ \ | / 2/x\\ |
3 | 8*tan |-| | | |1 - tan |-|| |
/ 2/x\\ | \2/ | | \ \2// |
|1 + tan |-|| *|1 - --------------| *|1 + ----------------------------------|
\ \2// | 2| | _____________ |
| / 2/x\\ | | / 2 2|
| |1 + tan |-|| | | / 1 - tan (x) / 2/x\\ |
\ \ \2// / | / ----------- *|1 + tan |-|| |
| / 2 \ \2// |
\ \/ 1 + tan (x) /
$$\frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - \frac{8 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right)^{\frac{5}{4}} \left(1 + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\sqrt{\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
3
sin (x)
--------------------------------------------
/ 2/ pi\ \
5/4 | sin |x + --| |
/ 2 \ | \ 2 / |
\1 - 2*sin (x)/ *|1 + -------------------|
| _______________|
| / /pi \ |
| / sin|-- + 2*x| |
\ \/ \2 / /
$$\frac{\sin^{3}{\left(x \right)}}{\left(1 - 2 \sin^{2}{\left(x \right)}\right)^{\frac{5}{4}} \left(\frac{\sin^{2}{\left(x + \frac{\pi}{2} \right)}}{\sqrt{\sin{\left(2 x + \frac{\pi}{2} \right)}}} + 1\right)}$$
1
---------------------------------------------------------------
5/4
/ 2 \ / 1 \ 3/ pi\
|1 - ------------| *|1 + ----------------------|*sec |x - --|
| 2/ pi\| | __________ | \ 2 /
| sec |x - --|| | / 1 2 |
\ \ 2 // | / -------- *sec (x)|
\ \/ sec(2*x) /
$$\frac{1}{\left(1 + \frac{1}{\sqrt{\frac{1}{\sec{\left(2 x \right)}}} \sec^{2}{\left(x \right)}}\right) \left(1 - \frac{2}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}\right)^{\frac{5}{4}} \sec^{3}{\left(x - \frac{\pi}{2} \right)}}$$
-sin(3*x) + 3*sin(x) ------------------------------------------- /1 __________ cos(2*x)\ 3/4 4*|- + \/ cos(2*x) + --------|*cos (2*x) \2 2 /
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\sqrt{\cos{\left(2 x \right)}} + \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right) \cos^{\frac{3}{4}}{\left(2 x \right)}}$$
1 1
- -------------------------- + ------------------------------------
__________ 5/4
/ 1 / pi\ / 1 \ / pi\
4 / -------- *sec|x - --| 2*|--------| *sec(x)*sec|2*x - --|
\/ sec(2*x) \ 2 / \sec(2*x)/ \ 2 /
-------------------------------------------------------------------
1
1 + ----------------------
__________
/ 1 2
/ -------- *sec (x)
\/ sec(2*x)
$$\frac{- \frac{1}{\sqrt[4]{\frac{1}{\sec{\left(2 x \right)}}} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{2 \left(\frac{1}{\sec{\left(2 x \right)}}\right)^{\frac{5}{4}} \sec{\left(x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}}{1 + \frac{1}{\sqrt{\frac{1}{\sec{\left(2 x \right)}}} \sec^{2}{\left(x \right)}}}$$
(-1/((1/sec(2*x))^(1/4)*sec(x - pi/2)) + 1/(2*(1/sec(2*x))^(5/4)*sec(x)*sec(2*x - pi/2)))/(1 + 1/(sqrt(1/sec(2*x))*sec(x)^2))