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