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