Descomposición de una fracción
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tan(x/4 + pi/8)/2 - 1/(2*tan(x/4 + pi/8))
$$\frac{\tan{\left(\frac{x}{4} + \frac{\pi}{8} \right)}}{2} - \frac{1}{2 \tan{\left(\frac{x}{4} + \frac{\pi}{8} \right)}}$$
/x pi\
tan|- + --|
\4 8 / 1
----------- - -------------
2 /x pi\
2*tan|- + --|
\4 8 /
Denominador racional
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2/pi x\
-2 + 2*tan |-- + -|
\8 4/
-------------------
/pi x\
4*tan|-- + -|
\8 4/
$$\frac{2 \tan^{2}{\left(\frac{x}{4} + \frac{\pi}{8} \right)} - 2}{4 \tan{\left(\frac{x}{4} + \frac{\pi}{8} \right)}}$$
(-2 + 2*tan(pi/8 + x/4)^2)/(4*tan(pi/8 + x/4))
/ / x pi\ /x pi\\ / /x pi\ / x pi\\
| I*|- - - --| I*|- + --|| | I*|- + --| I*|- - - --||
| \ 4 8 / \4 8 /| | \4 8 / \ 4 8 /|
I*\e + e / I*\- e + e /
--------------------------------- + ---------------------------------
/ /x pi\ / x pi\\ / / x pi\ /x pi\\
| I*|- + --| I*|- - - --|| | I*|- - - --| I*|- + --||
| \4 8 / \ 4 8 /| | \ 4 8 / \4 8 /|
2*\- e + e / 2*\e + e /
$$\frac{i \left(e^{i \left(- \frac{x}{4} - \frac{\pi}{8}\right)} - e^{i \left(\frac{x}{4} + \frac{\pi}{8}\right)}\right)}{2 \left(e^{i \left(- \frac{x}{4} - \frac{\pi}{8}\right)} + e^{i \left(\frac{x}{4} + \frac{\pi}{8}\right)}\right)} + \frac{i \left(e^{i \left(- \frac{x}{4} - \frac{\pi}{8}\right)} + e^{i \left(\frac{x}{4} + \frac{\pi}{8}\right)}\right)}{2 \left(e^{i \left(- \frac{x}{4} - \frac{\pi}{8}\right)} - e^{i \left(\frac{x}{4} + \frac{\pi}{8}\right)}\right)}$$
i*(exp(i*(-x/4 - pi/8)) + exp(i*(x/4 + pi/8)))/(2*(-exp(i*(x/4 + pi/8)) + exp(i*(-x/4 - pi/8)))) + i*(-exp(i*(x/4 + pi/8)) + exp(i*(-x/4 - pi/8)))/(2*(exp(i*(-x/4 - pi/8)) + exp(i*(x/4 + pi/8))))
Abrimos la expresión
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/x\ /x\ ___ /x\
___ tan|-| tan|-| \/ 2 *tan|-|
1 1 \/ 2 \4/ \4/ \4/
- ----------------------------- - ----------------------- + ----------------------------- + ----------------------------- - ----------------------- + -----------------------
/ ___ /x\ /x\\ / ___ /x\\ / ___ /x\ /x\\ / ___ /x\ /x\\ / ___ /x\\ / ___ /x\\
2*|1 - \/ 2 *tan|-| + tan|-|| 2*|-1 + \/ 2 + tan|-|| 2*|1 - \/ 2 *tan|-| + tan|-|| 2*|1 - \/ 2 *tan|-| + tan|-|| 2*|-1 + \/ 2 + tan|-|| 2*|-1 + \/ 2 + tan|-||
\ \4/ \4// \ \4// \ \4/ \4// \ \4/ \4// \ \4// \ \4//
$$- \frac{\tan{\left(\frac{x}{4} \right)}}{2 \left(\tan{\left(\frac{x}{4} \right)} - 1 + \sqrt{2}\right)} + \frac{\sqrt{2} \tan{\left(\frac{x}{4} \right)}}{2 \left(\tan{\left(\frac{x}{4} \right)} - 1 + \sqrt{2}\right)} - \frac{1}{2 \left(\tan{\left(\frac{x}{4} \right)} - 1 + \sqrt{2}\right)} + \frac{\tan{\left(\frac{x}{4} \right)}}{2 \left(- \sqrt{2} \tan{\left(\frac{x}{4} \right)} + \tan{\left(\frac{x}{4} \right)} + 1\right)} - \frac{1}{2 \left(- \sqrt{2} \tan{\left(\frac{x}{4} \right)} + \tan{\left(\frac{x}{4} \right)} + 1\right)} + \frac{\sqrt{2}}{2 \left(- \sqrt{2} \tan{\left(\frac{x}{4} \right)} + \tan{\left(\frac{x}{4} \right)} + 1\right)}$$
-1/(2*(1 - sqrt(2)*tan(x/4) + tan(x/4))) - 1/(2*(-1 + sqrt(2) + tan(x/4))) + sqrt(2)/(2*(1 - sqrt(2)*tan(x/4) + tan(x/4))) + tan(x/4)/(2*(1 - sqrt(2)*tan(x/4) + tan(x/4))) - tan(x/4)/(2*(-1 + sqrt(2) + tan(x/4))) + sqrt(2)*tan(x/4)/(2*(-1 + sqrt(2) + tan(x/4)))
Unión de expresiones racionales
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2/pi + 2*x\
-1 + tan |--------|
\ 8 /
-------------------
/pi + 2*x\
2*tan|--------|
\ 8 /
$$\frac{\tan^{2}{\left(\frac{2 x + \pi}{8} \right)} - 1}{2 \tan{\left(\frac{2 x + \pi}{8} \right)}}$$
(-1 + tan((pi + 2*x)/8)^2)/(2*tan((pi + 2*x)/8))
0.5*tan(x/4 + pi/8) - 0.5/tan(x/4 + pi/8)
0.5*tan(x/4 + pi/8) - 0.5/tan(x/4 + pi/8)
/ /x pi\\ / /x pi\\
|1 + tan|- + --||*|-1 + tan|- + --||
\ \4 8 // \ \4 8 //
------------------------------------
/x pi\
2*tan|- + --|
\4 8 /
$$\frac{\left(\tan{\left(\frac{x}{4} + \frac{\pi}{8} \right)} - 1\right) \left(\tan{\left(\frac{x}{4} + \frac{\pi}{8} \right)} + 1\right)}{2 \tan{\left(\frac{x}{4} + \frac{\pi}{8} \right)}}$$
(1 + tan(x/4 + pi/8))*(-1 + tan(x/4 + pi/8))/(2*tan(x/4 + pi/8))