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