Sr Examen
¿Cómo vas a descomponer esta sina/(sin^3a+3cos^3a) expresión en fracciones?
Solución
Ha introducido
[src]
sin(a) ------------------- 3 3 sin (a) + 3*cos (a)
$$\frac{\sin{\left(a \right)}}{\sin^{3}{\left(a \right)} + 3 \cos^{3}{\left(a \right)}}$$
sin(a)/(sin(a)^3 + 3*cos(a)^3)
Potencias
[src]
/ -I*a I*a\
-I*\- e + e /
-------------------------------------------
/ 3 3\
| / I*a -I*a\ / -I*a I*a\ |
| |e e | I*\- e + e / |
2*|3*|---- + -----| + -------------------|
\ \ 2 2 / 8 /
$$- \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(3 \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{3} + \frac{i \left(e^{i a} - e^{- i a}\right)^{3}}{8}\right)}$$
-i*(-exp(-i*a) + exp(i*a))/(2*(3*(exp(i*a)/2 + exp(-i*a)/2)^3 + i*(-exp(-i*a) + exp(i*a))^3/8))
Parte trigonométrica
[src]
2
/ 2/a\\ 4/a\ /a\
2*|1 - tan |-|| *cos |-|*tan|-|
\ \4// \4/ \2/
----------------------------------------------------------------
3 6
sin(3*a) 3*sin(a) / 2/a\\ / 2/a\\ 12/a\
- -------- + -------- + 3*|1 - tan |-|| *|1 - tan |-|| *cos |-|
4 4 \ \2// \ \4// \4/
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \cos^{4}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \cos^{12}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}$$
2 -------------------------------------------------------------------- / 3\ | / 2/pi a\\ | | | csc |-- - -|| | | | \2 2/| | | 3*|1 - ------------| | | | 2/a\ | | | | csc |-| | | | 1 3 \ \2/ / | /a\ /pi a\ |- ---------- + -------- + ---------------------|*csc|-|*csc|-- - -| | 4*csc(3*a) 4*csc(a) 6/pi a\ | \2/ \2 2/ | csc |-- - -| | \ \2 2/ /
$$\frac{2}{\left(\frac{3 \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{3}}{\csc^{6}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{1}{4 \csc{\left(3 a \right)}} + \frac{3}{4 \csc{\left(a \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
/ pi\
cos|a - --|
\ 2 /
------------------------
3/ pi\ 3
cos |a - --| + 3*cos (a)
\ 2 /
$$\frac{\cos{\left(a - \frac{\pi}{2} \right)}}{3 \cos^{3}{\left(a \right)} + \cos^{3}{\left(a - \frac{\pi}{2} \right)}}$$
/a\ /a\
2*cos|-|*sin|-|
\2/ \2/
------------------------------------------
3 6/a\
24*cos (a)*cos |-|
sin(3*a) 3*sin(a) \2/
- -------- + -------- + ------------------
4 4 3
(1 + cos(a))
$$\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{\frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4} + \frac{24 \cos^{6}{\left(\frac{a}{2} \right)} \cos^{3}{\left(a \right)}}{\left(\cos{\left(a \right)} + 1\right)^{3}}}$$
2
/ 2/a\\
2*|-1 + cot |-||
\ \4//
------------------------------------------------------------------------------------------------
/ 3 6\
| / 1 \ / 2/a\\ |
| 3*|1 - -------| *|-1 + cot |-|| |
| /3*a\ /a\ | 2/a\| \ \4// |
2 | cot|---| 3*cot|-| | cot |-|| |
/ 2/a\\ | \ 2 / \2/ \ \2// | /a\
|1 + cot |-|| *|- ----------------- + --------------- + --------------------------------|*cot|-|
\ \4// | / 2/3*a\\ / 2/a\\ 6 | \2/
| 2*|1 + cot |---|| 2*|1 + cot |-|| / 2/a\\ |
| \ \ 2 // \ \2// |1 + cot |-|| |
\ \ \4// /
$$\frac{2 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\frac{3 \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}} - \frac{\cot{\left(\frac{3 a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \cot{\left(\frac{a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}\right) \cot{\left(\frac{a}{2} \right)}}$$
4*sin(a) -------------------------------------------- -sin(3*a) + 3*cos(3*a) + 3*sin(a) + 9*cos(a)
$$\frac{4 \sin{\left(a \right)}}{3 \sin{\left(a \right)} - \sin{\left(3 a \right)} + 9 \cos{\left(a \right)} + 3 \cos{\left(3 a \right)}}$$
/a\ /a pi\
2*cos|-|*cos|- - --|
\2/ \2 2 /
---------------------------------------------------------------
3
/ pi\ / pi\ / 2/a pi\\
cos|3*a - --| 3*cos|a - --| | cos |- - --||
\ 2 / \ 2 / | \2 2 /| 6/a\
- ------------- + ------------- + 3*|1 - ------------| *cos |-|
4 4 | 2/a\ | \2/
| cos |-| |
\ \2/ /
$$\frac{2 \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{3 \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \cos^{6}{\left(\frac{a}{2} \right)} + \frac{3 \cos{\left(a - \frac{\pi}{2} \right)}}{4} - \frac{\cos{\left(3 a - \frac{\pi}{2} \right)}}{4}}$$
/a\
2*cot|-|
\2/
--------------------------------------------------
/ 3 \
| / 2/a\\ 3/a\ |
|3*|-1 + cot |-|| 8*cot |-| |
/ 2/a\\ | \ \2// \2/ |
|1 + cot |-||*|----------------- + --------------|
\ \2// | 3 3|
| / 2/a\\ / 2/a\\ |
| |1 + cot |-|| |1 + cot |-|| |
\ \ \2// \ \2// /
$$\frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(\frac{3 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{3}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}} + \frac{8 \cot^{3}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
2 ------------------------------------------------------------------------------- / 3\ | / 2/a pi\\ | | | sec |- - --|| | | | \2 2 /| | | 3*|-1 + ------------| | | | 2/a\ | | | | sec |-| | | | 1 3 \ \2/ / | /a\ /a pi\ |- --------------- + ------------- + ----------------------|*sec|-|*sec|- - --| | / pi\ / pi\ 6/a pi\ | \2/ \2 2 / | 4*sec|3*a - --| 4*sec|a - --| sec |- - --| | \ \ 2 / \ 2 / \2 2 / /
$$\frac{2}{\left(\frac{3 \left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{3}}{\sec^{6}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{1}{4 \sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{3}{4 \sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
/a\ /a\
2*cos|-|*sin|-|
\2/ \2/
-------------------------------------------------
3
sin(3*a) 3*sin(a) / 1 \ 6/a\
- -------- + -------- + 3*|-1 + -------| *sin |-|
4 4 | 2/a\| \2/
| tan |-||
\ \2//
$$\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{3 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}$$
2 ------------------------------------------------------------------------------ / 3\ | / 2/a\ \ | | | sec |-| | | | | \2/ | | | 3*|1 - ------------| | | | 2/a pi\| | | | sec |- - --|| | | 1 3 \ \2 2 // | /a\ /a pi\ |- --------------- + ------------- + ---------------------|*sec|-|*sec|- - --| | / pi\ / pi\ 6/a\ | \2/ \2 2 / | 4*sec|3*a - --| 4*sec|a - --| sec |-| | \ \ 2 / \ 2 / \2/ /
$$\frac{2}{\left(\frac{3 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{3}}{\sec^{6}{\left(\frac{a}{2} \right)}} - \frac{1}{4 \sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{3}{4 \sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
sin(a)
------------------------
3 3/ pi\
sin (a) + 3*sin |a + --|
\ 2 /
$$\frac{\sin{\left(a \right)}}{\sin^{3}{\left(a \right)} + 3 \sin^{3}{\left(a + \frac{\pi}{2} \right)}}$$
2
/ 2/a\\ /a\
2*|1 - tan |-|| *tan|-|
\ \4// \2/
----------------------------------------------------------------------------------------
/ 3 6\
| /3*a\ /a\ / 2/a\\ / 2/a\\ |
2 | tan|---| 3*tan|-| 3*|1 - tan |-|| *|1 - tan |-|| |
/ 2/a\\ | \ 2 / \2/ \ \2// \ \4// |
|1 + tan |-|| *|- ----------------- + --------------- + -------------------------------|
\ \4// | / 2/3*a\\ / 2/a\\ 6 |
| 2*|1 + tan |---|| 2*|1 + tan |-|| / 2/a\\ |
| \ \ 2 // \ \2// |1 + tan |-|| |
\ \ \4// /
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\frac{3 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}} - \frac{\tan{\left(\frac{3 a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \tan{\left(\frac{a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}\right)}$$
2/a\ /a\
2*sin |-|*cot|-|
\2/ \2/
------------------------------------------------
3
-sin(3*a) + 3*sin(a) / 2/a\\ 6/a\
-------------------- + 3*|-1 + cot |-|| *sin |-|
4 \ \2// \2/
$$\frac{2 \sin^{2}{\left(\frac{a}{2} \right)} \cot{\left(\frac{a}{2} \right)}}{\frac{3 \sin{\left(a \right)} - \sin{\left(3 a \right)}}{4} + 3 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{3} \sin^{6}{\left(\frac{a}{2} \right)}}$$
2 --------------------------------------------------------------------- / 3\ | / 2/a\ \ | | | csc |-| | | | | \2/ | | | 3*|-1 + ------------| | | | 2/pi a\| | | | csc |-- - -|| | | 1 3 \ \2 2// | /a\ /pi a\ |- ---------- + -------- + ----------------------|*csc|-|*csc|-- - -| | 4*csc(3*a) 4*csc(a) 6/a\ | \2/ \2 2/ | csc |-| | \ \2/ /
$$\frac{2}{\left(\frac{3 \left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{3}}{\csc^{6}{\left(\frac{a}{2} \right)}} - \frac{1}{4 \csc{\left(3 a \right)}} + \frac{3}{4 \csc{\left(a \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
1 ------------------------------- / 1 3 \ |------- + ------------|*csc(a) | 3 3/pi \| |csc (a) csc |-- - a|| \ \2 //
$$\frac{1}{\left(\frac{3}{\csc^{3}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{3}{\left(a \right)}}\right) \csc{\left(a \right)}}$$
2/a\ /a\
2*cos |-|*tan|-|
\2/ \2/
-----------------------------------------------
3
-sin(3*a) + 3*sin(a) / 2/a\\ 6/a\
-------------------- + 3*|1 - tan |-|| *cos |-|
4 \ \2// \2/
$$\frac{2 \cos^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \cos^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)} - \sin{\left(3 a \right)}}{4}}$$
/a\ /a\
2*cos|-|*sin|-|
\2/ \2/
------------------------------------------
3 6/a\
24*cos (a)*sin |-|
sin(3*a) 3*sin(a) \2/
- -------- + -------- - ------------------
4 4 3
(-1 + cos(a))
$$\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{\frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4} - \frac{24 \sin^{6}{\left(\frac{a}{2} \right)} \cos^{3}{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right)^{3}}}$$
2/a\ 4/a\ /a\
8*cot |-|*sin |-|*cot|-|
\4/ \4/ \2/
------------------------------------------------------------
3
sin(3*a) 3*sin(a) / 2/a\\ 6/a\ 12/a\
- -------- + -------- + 192*|-1 + cot |-|| *cot |-|*sin |-|
4 4 \ \2// \4/ \4/
$$\frac{8 \sin^{4}{\left(\frac{a}{4} \right)} \cot^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{192 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{3} \sin^{12}{\left(\frac{a}{4} \right)} \cot^{6}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}$$
4/a\ 2/a\
8*cos |-|*tan |-|
\4/ \4/
---------------------------------------------------------------------
/ 3 \
| sin(3*a) 3*sin(a) / 1 \ 12/a\ 6/a\| /a\
|- -------- + -------- + 192*|-1 + -------| *cos |-|*tan |-||*tan|-|
| 4 4 | 2/a\| \4/ \4/| \2/
| | tan |-|| |
\ \ \2// /
$$\frac{8 \cos^{4}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)}}{\left(192 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \cos^{12}{\left(\frac{a}{4} \right)} \tan^{6}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}\right) \tan{\left(\frac{a}{2} \right)}}$$
/a\ /a pi\
2*cos|-|*cos|- - --|
\2/ \2 2 /
---------------------------------------------------------------------
3
/ pi\ / pi\ / 2/a\ \
cos|3*a - --| 3*cos|a - --| | cos |-| |
\ 2 / \ 2 / | \2/ | 6/a pi\
- ------------- + ------------- + 3*|-1 + ------------| *cos |- - --|
4 4 | 2/a pi\| \2 2 /
| cos |- - --||
\ \2 2 //
$$\frac{2 \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{3 \left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{3} \cos^{6}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} + \frac{3 \cos{\left(a - \frac{\pi}{2} \right)}}{4} - \frac{\cos{\left(3 a - \frac{\pi}{2} \right)}}{4}}$$
sin(a)
---------------------------------------------------
3
/ 2 \
sin(3*a) 3*sin(a) | sin (a) | 6/a\
- -------- + -------- + 3*|-1 + ---------| *sin |-|
4 4 | 4/a\| \2/
| 4*sin |-||
\ \2//
$$\frac{\sin{\left(a \right)}}{3 \left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}$$
2/a\ /a\
8*cot |-|*cot|-|
\4/ \2/
------------------------------------------------------------------------------------
/ 3 \
| /3*a\ /a\ / 2/a\\ 6/a\|
2 | cot|---| 3*cot|-| 192*|-1 + cot |-|| *cot |-||
/ 2/a\\ | \ 2 / \2/ \ \2// \4/|
|1 + cot |-|| *|- ----------------- + --------------- + ---------------------------|
\ \4// | / 2/3*a\\ / 2/a\\ 6 |
| 2*|1 + cot |---|| 2*|1 + cot |-|| / 2/a\\ |
| \ \ 2 // \ \2// |1 + cot |-|| |
\ \ \4// /
$$\frac{8 \cot^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(- \frac{\cot{\left(\frac{3 a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \cot{\left(\frac{a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} + \frac{192 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{3} \cot^{6}{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}}\right)}$$
/a\ /a\
2*cos|-|*sin|-|
\2/ \2/
------------------------------------------------
3
sin(3*a) 3*sin(a) / 2/a\\ 6/a\
- -------- + -------- + 3*|1 - tan |-|| *cos |-|
4 4 \ \2// \2/
$$\frac{2 \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \cos^{6}{\left(\frac{a}{2} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}$$
1 ------------------------------------ / 1 3 \ / pi\ |------------ + -------|*sec|a - --| | 3/ pi\ 3 | \ 2 / |sec |a - --| sec (a)| \ \ 2 / /
$$\frac{1}{\left(\frac{1}{\sec^{3}{\left(a - \frac{\pi}{2} \right)}} + \frac{3}{\sec^{3}{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
/a\
2*tan|-|
\2/
-------------------------------------------------
/ 3 \
| / 2/a\\ 3/a\ |
|3*|1 - tan |-|| 8*tan |-| |
/ 2/a\\ | \ \2// \2/ |
|1 + tan |-||*|---------------- + --------------|
\ \2// | 3 3|
| / 2/a\\ / 2/a\\ |
| |1 + tan |-|| |1 + tan |-|| |
\ \ \2// \ \2// /
$$\frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(\frac{3 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}} + \frac{8 \tan^{3}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{3}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
2/a\
8*tan |-|
\4/
-------------------------------------------------------------------------------------------
/ 3 \
| / 1 \ 6/a\|
| 192*|-1 + -------| *tan |-||
| /3*a\ /a\ | 2/a\| \4/|
2 | tan|---| 3*tan|-| | tan |-|| |
/ 2/a\\ | \ 2 / \2/ \ \2// | /a\
|1 + tan |-|| *|- ----------------- + --------------- + ---------------------------|*tan|-|
\ \4// | / 2/3*a\\ / 2/a\\ 6 | \2/
| 2*|1 + tan |---|| 2*|1 + tan |-|| / 2/a\\ |
| \ \ 2 // \ \2// |1 + tan |-|| |
\ \ \4// /
$$\frac{8 \tan^{2}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\frac{192 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{3} \tan^{6}{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{6}} - \frac{\tan{\left(\frac{3 a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)} + \frac{3 \tan{\left(\frac{a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}\right) \tan{\left(\frac{a}{2} \right)}}$$
1 -------------------------- / 1 3 \ |------- + -------|*csc(a) | 3 3 | \csc (a) sec (a)/
$$\frac{1}{\left(\frac{3}{\sec^{3}{\left(a \right)}} + \frac{1}{\csc^{3}{\left(a \right)}}\right) \csc{\left(a \right)}}$$
2/a\ 2/pi a\
4*sin |-|*sin |-- + -|
\2/ \2 2/
----------------------------------------------------------------
/ 3 \
| / 4/a\\ |
| | 4*sin |-|| |
| sin(3*a) 3*sin(a) | \2/| 6/pi a\|
|- -------- + -------- + 3*|1 - ---------| *sin |-- + -||*sin(a)
| 4 4 | 2 | \2 2/|
\ \ sin (a) / /
$$\frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{\left(3 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{3} \sin^{6}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}\right) \sin{\left(a \right)}}$$
2
/ 2/a\\ 4/a\ /a\
2*|-1 + cot |-|| *sin |-|*tan|-|
\ \4// \4/ \2/
-----------------------------------------------------------------
3 6
sin(3*a) 3*sin(a) / 2/a\\ / 2/a\\ 12/a\
- -------- + -------- + 3*|1 - tan |-|| *|-1 + cot |-|| *sin |-|
4 4 \ \2// \ \4// \4/
$$\frac{2 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} \right)}}{3 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)^{3} \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{a}{4} \right)} + \frac{3 \sin{\left(a \right)}}{4} - \frac{\sin{\left(3 a \right)}}{4}}$$
2*(-1 + cot(a/4)^2)^2*sin(a/4)^4*tan(a/2)/(-sin(3*a)/4 + 3*sin(a)/4 + 3*(1 - tan(a/2)^2)^3*(-1 + cot(a/4)^2)^6*sin(a/4)^12)