Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(tan(x))/(1+x^2)+(1+tan(x)^2)*atan(x)*cos(tan(x))
¿Cómo vas a descomponer esta sin(tan(x))/(1+x^2)+(1+tan(x)^2)*atan(x)*cos(tan(x)) expresión en fracciones?
Solución
Ha introducido
[src]
sin(tan(x)) / 2 \
----------- + \1 + tan (x)/*atan(x)*cos(tan(x))
2
1 + x
$$\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)} \cos{\left(\tan{\left(x \right)} \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
sin(tan(x))/(1 + x^2) + ((1 + tan(x)^2)*atan(x))*cos(tan(x))
Simplificación general
[src]
/ 2\
\1 + x /*atan(x)*cos(tan(x))
---------------------------- + sin(tan(x))
2
cos (x)
------------------------------------------
2
1 + x
$$\frac{\frac{\left(x^{2} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
((1 + x^2)*atan(x)*cos(tan(x))/cos(x)^2 + sin(tan(x)))/(1 + x^2)
Respuesta numérica
[src]
sin(tan(x))/(1.0 + x^2) + (1.0 + tan(x)^2)*atan(x)*cos(tan(x))
sin(tan(x))/(1.0 + x^2) + (1.0 + tan(x)^2)*atan(x)*cos(tan(x))
Denominador común
[src]
sin(tan(x)) 2
----------- + atan(x)*cos(tan(x)) + tan (x)*atan(x)*cos(tan(x))
2
1 + x
$$\cos{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
sin(tan(x))/(1 + x^2) + atan(x)*cos(tan(x)) + tan(x)^2*atan(x)*cos(tan(x))
Denominador racional
[src]
/ 2\ / 2 \
\1 + x /*\1 + tan (x)/*atan(x)*cos(tan(x)) + sin(tan(x))
--------------------------------------------------------
2
1 + x
$$\frac{\left(x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
((1 + x^2)*(1 + tan(x)^2)*atan(x)*cos(tan(x)) + sin(tan(x)))/(1 + x^2)
Unión de expresiones racionales
[src]
/ 2\ / 2 \
\1 + x /*\1 + tan (x)/*atan(x)*cos(tan(x)) + sin(tan(x))
--------------------------------------------------------
2
1 + x
$$\frac{\left(x^{2} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
((1 + x^2)*(1 + tan(x)^2)*atan(x)*cos(tan(x)) + sin(tan(x)))/(1 + x^2)
Combinatoria
[src]
2 2 2 2
atan(x)*cos(tan(x)) + x *atan(x)*cos(tan(x)) + tan (x)*atan(x)*cos(tan(x)) + x *tan (x)*atan(x)*cos(tan(x)) + sin(tan(x))
-------------------------------------------------------------------------------------------------------------------------
2
1 + x
$$\frac{x^{2} \cos{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + x^{2} \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \sin{\left(\tan{\left(x \right)} \right)} + \cos{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)} \operatorname{atan}{\left(x \right)} + \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{x^{2} + 1}$$
(atan(x)*cos(tan(x)) + x^2*atan(x)*cos(tan(x)) + tan(x)^2*atan(x)*cos(tan(x)) + x^2*tan(x)^2*atan(x)*cos(tan(x)) + sin(tan(x)))/(1 + x^2)
Potencias
[src]
sin(tan(x)) / 2 \
----------- + \1 + tan (x)/*atan(x)*cos(tan(x))
2
1 + x
$$\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
/ I*x -I*x / I*x -I*x\ \ / I*x -I*x / I*x -I*x\ \
| - e + e -\- e + e / | | - e + e -\- e + e / |
| -------------- ------------------| | -------------- ------------------|
/ 2\ | I*x -I*x I*x -I*x | | I*x -I*x I*x -I*x |
| / I*x -I*x\ | | e + e e + e | | e + e e + e |
| \- e + e / | |e e | I*\- e + e /
|1 - -----------------|*|--------------- + -------------------|*atan(x) - -------------------------------------------
| 2 | \ 2 2 / / 2\
| / I*x -I*x\ | 2*\1 + x /
\ \e + e / /
$$\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1\right) \left(\frac{e^{\frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}}}{2} + \frac{e^{- \frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}}}{2}\right) \operatorname{atan}{\left(x \right)} - \frac{i \left(- e^{\frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}} + e^{- \frac{- e^{i x} + e^{- i x}}{e^{i x} + e^{- i x}}}\right)}{2 \left(x^{2} + 1\right)}$$
(1 - (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp((-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/2 + exp(-(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))/2)*atan(x) - i*(-exp((-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x))) + exp(-(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x))))/(2*(1 + x^2))
Compilar la expresión
[src]
sin(tan(x)) / 2 \
----------- + \1 + tan (x)/*atan(x)*cos(tan(x))
2
1 + x
$$\left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
sin(tan(x))/(1 + x^2) + (1 + tan(x)^2)*atan(x)*cos(tan(x))
Parte trigonométrica
[src]
/ / pi\\ / / / pi\\\
| cos|x - --|| | | cos|x - --|||
| pi \ 2 /| | 2| pi \ 2 /||
cos|- -- + -----------| |1 - 2*cos |- -- + -----------||*atan(x)
\ 2 cos(x) / \ \ 2 2*cos(x) //
----------------------- + ----------------------------------------
2 2
1 + x cos (x)
$$\frac{\left(1 - 2 \cos^{2}{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}} \right)}\right) \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1}$$
/ 2/pi \\
| csc |-- - x||
| \2 /|
|1 + ------------|*atan(x)
| 2 |
1 \ csc (x) /
------------------------- + --------------------------
/ /pi \\ /pi sec(x)\
|csc|-- - x|| csc|-- - ------|
/ 2\ | \2 /| \2 csc(x)/
\1 + x /*csc|-----------|
\ csc(x) /
$$\frac{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)}}{\csc{\left(\frac{\pi}{2} - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}$$
/ 2/ 1 \ \
2 | 8*cot |--------| |
/ 2/x\\ | \4*cot(x)/ |
|1 + cot |-|| *|1 - ---------------------|*atan(x)
\ \2// | 2|
/ 1 \ | / 2/ 1 \\ |
2*cot|--------| | |1 + cot |--------|| |
\2*cot(x)/ \ \ \4*cot(x)// /
----------------------------- + --------------------------------------------------
/ 2\ / 2/ 1 \\ 2
\1 + x /*|1 + cot |--------|| / 2/x\\
\ \2*cot(x)// |-1 + cot |-||
\ \2//
$$\frac{\left(1 - \frac{8 \cot^{2}{\left(\frac{1}{4 \cot{\left(x \right)}} \right)}}{\left(\cot^{2}{\left(\frac{1}{4 \cot{\left(x \right)}} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}} + \frac{2 \cot{\left(\frac{1}{2 \cot{\left(x \right)}} \right)}}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)}$$
/ 2 \ / / 2 \\
|2*sin (x)| | 2|sin (x) ||
sin|---------| |1 - 2*sin |--------||*atan(x)
\ sin(2*x)/ \ \sin(2*x)//
-------------- + ------------------------------
2 2/ pi\
1 + x sin |x + --|
\ 2 /
$$\frac{\left(1 - 2 \sin^{2}{\left(\frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}\right) \operatorname{atan}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1}$$
/ / pi\\ / / pi\\
| cos|x - --|| |cos|x - --||
| pi \ 2 /| | \ 2 /|
cos|- -- + -----------| atan(x)*cos|-----------|
\ 2 cos(x) / \ cos(x) /
----------------------- + ------------------------
2 2
1 + x cos (x)
$$\frac{\cos{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1}$$
/ 2/tan(x)\ \
2 | 8*tan |------| |
/ 2/x\\ | \ 4 / |
|1 + tan |-|| *|1 - -------------------|*atan(x)
\ \2// | 2|
/tan(x)\ | / 2/tan(x)\\ |
2*tan|------| | |1 + tan |------|| |
\ 2 / \ \ \ 4 // /
--------------------------- + ------------------------------------------------
/ 2\ / 2/tan(x)\\ 2
\1 + x /*|1 + tan |------|| / 2/x\\
\ \ 2 // |1 - tan |-||
\ \2//
$$\frac{\left(1 - \frac{8 \tan^{2}{\left(\frac{\tan{\left(x \right)}}{4} \right)}}{\left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{4} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}} + \frac{2 \tan{\left(\frac{\tan{\left(x \right)}}{2} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)}$$
/ 2 \
| sec (x)|
|1 + -------|*atan(x)
| 2 |
1 \ csc (x)/
-------------------- + ---------------------
/ 2\ /sec(x)\ / sec(x) \
\1 + x /*csc|------| sec|-----------|
\csc(x)/ | /pi \|
|sec|-- - x||
\ \2 //
$$\frac{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(- x + \frac{\pi}{2} \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}}$$
/ 1 \ / 2/ 1 \\
/ 1 \ |1 + -------|*|-1 + cot |--------||*atan(x)
2*cot|--------| | 2 | \ \2*cot(x)//
\2*cot(x)/ \ cot (x)/
----------------------------- + -------------------------------------------
/ 2\ / 2/ 1 \\ 2/ 1 \
\1 + x /*|1 + cot |--------|| 1 + cot |--------|
\ \2*cot(x)// \2*cot(x)/
$$\frac{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} - 1\right) \operatorname{atan}{\left(x \right)}}{\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1} + \frac{2 \cot{\left(\frac{1}{2 \cot{\left(x \right)}} \right)}}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)}$$
2
1 sec (x)*atan(x)
-------------------------------- + ----------------
/ 2\ / pi sec(x) \ / sec(x) \
\1 + x /*sec|- -- + -----------| sec|-----------|
| 2 / pi\| | / pi\|
| sec|x - --|| |sec|x - --||
\ \ 2 // \ \ 2 //
$$\frac{\operatorname{atan}{\left(x \right)} \sec^{2}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}}$$
/sin(x)\
sin|------| / 2 \
\cos(x)/ | sin (x)| /sin(x)\
----------- + |1 + -------|*atan(x)*cos|------|
2 | 2 | \cos(x)/
1 + x \ cos (x)/
$$\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1}$$
sin(tan(x)) 2
----------- + sec (x)*atan(x)*cos(tan(x))
2
1 + x
$$\cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} \sec^{2}{\left(x \right)} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
/ 2 \
| sec (x) |
|1 + ------------|*atan(x)
| 2/ pi\|
| sec |x - --||
1 \ \ 2 //
-------------------------------- + --------------------------
/ 2\ / pi sec(x) \ / sec(x) \
\1 + x /*sec|- -- + -----------| sec|-----------|
| 2 / pi\| | / pi\|
| sec|x - --|| |sec|x - --||
\ \ 2 // \ \ 2 //
$$\frac{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \operatorname{atan}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}}$$
2
/tan(x)\ / 2/x\\ / 2/tan(x)\\
2*tan|------| |1 + tan |-|| *|1 - tan |------||*atan(x)
\ 2 / \ \2// \ \ 2 //
--------------------------- + -----------------------------------------
/ 2\ / 2/tan(x)\\ 2
\1 + x /*|1 + tan |------|| / 2/tan(x)\\ / 2/x\\
\ \ 2 // |1 + tan |------||*|1 - tan |-||
\ \ 2 // \ \2//
$$\frac{2 \tan{\left(\frac{\tan{\left(x \right)}}{2} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)} + \frac{\left(1 - \tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)}$$
2/pi \
csc |-- - x|*atan(x)
1 \2 /
------------------------- + --------------------
/ /pi \\ /pi sec(x)\
|csc|-- - x|| csc|-- - ------|
/ 2\ | \2 /| \2 csc(x)/
\1 + x /*csc|-----------|
\ csc(x) /
$$\frac{\operatorname{atan}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(\frac{\pi}{2} - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}$$
/ 2 \
|2*sin (x)|
sin|---------| / 4 \ / 2 \
\ sin(2*x)/ | 4*sin (x)| |2*sin (x)|
-------------- + |1 + ---------|*atan(x)*cos|---------|
2 | 2 | \ sin(2*x)/
1 + x \ sin (2*x)/
$$\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \cos{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1}$$
1 2 / 2 \
-------------------------------- + sec (x)*|1 - --------------------------|*atan(x)
/ 2\ / pi sec(x) \ | 2/ pi sec(x) \|
\1 + x /*sec|- -- + -----------| | sec |- -- + -------------||
| 2 / pi\| | | 2 / pi\||
| sec|x - --|| | | 2*sec|x - --|||
\ \ 2 // \ \ \ 2 ///
$$\left(1 - \frac{2}{\sec^{2}{\left(\frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}}\right) \operatorname{atan}{\left(x \right)} \sec^{2}{\left(x \right)} + \frac{1}{\left(x^{2} + 1\right) \sec{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\pi}{2} \right)}}$$
1 2/pi \ / 2 \
------------------------- + csc |-- - x|*|1 - -----------------|*atan(x)
/ /pi \\ \2 / | / /pi \\|
|csc|-- - x|| | |csc|-- - x|||
/ 2\ | \2 /| | 2| \2 /||
\1 + x /*csc|-----------| | csc |-----------||
\ csc(x) / \ \ 2*csc(x) //
$$\left(1 - \frac{2}{\csc^{2}{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}} \right)}}\right) \operatorname{atan}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}$$
sin(tan(x)) 2*atan(x)*cos(tan(x))
----------- + ---------------------
2 1 + cos(2*x)
1 + x
$$\frac{2 \cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{\cos{\left(2 x \right)} + 1} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
/ 2/tan(x)\\
|1 - 2*sin |------||*atan(x)
sin(tan(x)) \ \ 2 //
----------- + ----------------------------
2 2
1 + x cos (x)
$$\frac{\left(1 - 2 \sin^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)}\right) \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
2
/ 1 \ / 2/x\\ / 2/ 1 \\
2*cot|--------| |1 + cot |-|| *|-1 + cot |--------||*atan(x)
\2*cot(x)/ \ \2// \ \2*cot(x)//
----------------------------- + --------------------------------------------
/ 2\ / 2/ 1 \\ 2
\1 + x /*|1 + cot |--------|| / 2/ 1 \\ / 2/x\\
\ \2*cot(x)// |1 + cot |--------||*|-1 + cot |-||
\ \2*cot(x)// \ \2//
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} - 1\right) \operatorname{atan}{\left(x \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{1}{2 \cot{\left(x \right)}} \right)}}{\left(x^{2} + 1\right) \left(\cot^{2}{\left(\frac{1}{2 \cot{\left(x \right)}} \right)} + 1\right)}$$
/tan(x)\ / 2 \ / 2/tan(x)\\
2*tan|------| \1 + tan (x)/*|1 - tan |------||*atan(x)
\ 2 / \ \ 2 //
--------------------------- + ----------------------------------------
/ 2\ / 2/tan(x)\\ 2/tan(x)\
\1 + x /*|1 + tan |------|| 1 + tan |------|
\ \ 2 // \ 2 /
$$\frac{\left(1 - \tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(x \right)}}{\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{\tan{\left(x \right)}}{2} \right)}}{\left(x^{2} + 1\right) \left(\tan^{2}{\left(\frac{\tan{\left(x \right)}}{2} \right)} + 1\right)}$$
/ / pi\\
| cos|x - --||
| pi \ 2 /| / 2/ pi\\ / / pi\\
cos|- -- + -----------| | cos |x - --|| |cos|x - --||
\ 2 cos(x) / | \ 2 /| | \ 2 /|
----------------------- + |1 + ------------|*atan(x)*cos|-----------|
2 | 2 | \ cos(x) /
1 + x \ cos (x) /
$$\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)} + \frac{\cos{\left(- \frac{\pi}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{x^{2} + 1}$$
sin(tan(x)) atan(x)*cos(tan(x))
----------- + -------------------
2 2
1 + x cos (x)
$$\frac{\cos{\left(\tan{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin{\left(\tan{\left(x \right)} \right)}}{x^{2} + 1}$$
/ 2 \ / 2 \
|2*sin (x)| |pi 2*sin (x)|
sin|---------| atan(x)*sin|-- + ---------|
\ sin(2*x)/ \2 sin(2*x)/
-------------- + ---------------------------
2 2/ pi\
1 + x sin |x + --|
\ 2 /
$$\frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\pi}{2} \right)} \operatorname{atan}{\left(x \right)}}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1}$$
/ 2 \
| sec (x)|
|1 + -------|*atan(x)
| 2 |
1 \ csc (x)/
------------------------- + ---------------------
/ /pi \\ /sec(x)\
|csc|-- - x|| sec|------|
/ 2\ | \2 /| \csc(x)/
\1 + x /*csc|-----------|
\ csc(x) /
$$\frac{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \operatorname{atan}{\left(x \right)}}{\sec{\left(\frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}} + \frac{1}{\left(x^{2} + 1\right) \csc{\left(\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}$$
/ 2 \
|2*sin (x)|
sin|---------| / 4 \ / 2 \
\ sin(2*x)/ | 4*sin (x)| |pi 2*sin (x)|
-------------- + |1 + ---------|*atan(x)*sin|-- + ---------|
2 | 2 | \2 sin(2*x)/
1 + x \ sin (2*x)/
$$\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{\pi}{2} \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} \right)}}{x^{2} + 1}$$
sin(2*sin(x)^2/sin(2*x))/(1 + x^2) + (1 + 4*sin(x)^4/sin(2*x)^2)*atan(x)*sin(pi/2 + 2*sin(x)^2/sin(2*x))