Mister Exam
Least common denominator (-cos(x)^5)/5+2*cos(x)^3/3-cos(x)
The solution
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5 3
-cos (x) 2*cos (x)
--------- + --------- - cos(x)
5 3
$$\left(\frac{2 \cos^{3}{\left(x \right)}}{3} + \frac{\left(-1\right) \cos^{5}{\left(x \right)}}{5}\right) - \cos{\left(x \right)}$$
(-cos(x)^5)/5 + (2*cos(x)^3)/3 - cos(x)
General simplification
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5 3
cos (x) 2*cos (x)
-cos(x) - ------- + ---------
5 3
$$- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$
-cos(x) - cos(x)^5/5 + 2*cos(x)^3/3
Fraction decomposition
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-cos(x) - cos(x)^5/5 + 2*cos(x)^3/3
$$- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$
5 3
cos (x) 2*cos (x)
-cos(x) - ------- + ---------
5 3
Assemble expression
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5 3
cos (x) 2*cos (x)
-cos(x) - ------- + ---------
5 3
$$- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$
-cos(x) - cos(x)^5/5 + 2*cos(x)^3/3
Combining rational expressions
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/ 2 / 2 \\
\-15 + cos (x)*\10 - 3*cos (x)//*cos(x)
---------------------------------------
15
$$\frac{\left(\left(10 - 3 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} - 15\right) \cos{\left(x \right)}}{15}$$
(-15 + cos(x)^2*(10 - 3*cos(x)^2))*cos(x)/15
Rational denominator
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5 3
-15*cos(x) - 3*cos (x) + 10*cos (x)
-----------------------------------
15
$$\frac{- 3 \cos^{5}{\left(x \right)} + 10 \cos^{3}{\left(x \right)} - 15 \cos{\left(x \right)}}{15}$$
(-15*cos(x) - 3*cos(x)^5 + 10*cos(x)^3)/15
Common denominator
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5 3
cos (x) 2*cos (x)
-cos(x) - ------- + ---------
5 3
$$- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$
-cos(x) - cos(x)^5/5 + 2*cos(x)^3/3
Numerical answer
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-cos(x) + 0.666666666666667*cos(x)^3 - 0.2*cos(x)^5
-cos(x) + 0.666666666666667*cos(x)^3 - 0.2*cos(x)^5
Trigonometric part
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5 3
/ 2/x pi\\ / 2/x pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| 10/x\ | \2 2 /| 6/x\
|1 - ------------| *cos |-| 2*|1 - ------------| *cos |-|
| 2/x\ | \2/ | 2/x\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
-cos(x) - ---------------------------- + -----------------------------
5 3
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{5} \cos^{10}{\left(\frac{x}{2} \right)}}{5} + \frac{2 \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \cos^{6}{\left(\frac{x}{2} \right)}}{3} - \cos{\left(x \right)}$$
5 3
/ 2/pi x\\ / 2/pi x\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
|1 - ------------| 2*|1 - ------------|
| 2/x\ | | 2/x\ |
| csc |-| | | csc |-| |
1 \ \2/ / \ \2/ /
- ----------- - ------------------- + ---------------------
/pi \ 10/pi x\ 6/pi x\
csc|-- - x| 5*csc |-- - -| 3*csc |-- - -|
\2 / \2 2/ \2 2/
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{5}}{5 \csc^{10}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{2 \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)^{3}}{3 \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}$$
5 3
/ 2 \ / 2 \
| sin (x) | 10/x\ | sin (x) | 6/x\
|-1 + ---------| *sin |-| 2*|-1 + ---------| *sin |-|
| 4/x\| \2/ | 4/x\| \2/
| 4*sin |-|| | 4*sin |-||
/ pi\ \ \2// \ \2//
- sin|x + --| - -------------------------- + ---------------------------
\ 2 / 5 3
$$- \frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{5} \sin^{10}{\left(\frac{x}{2} \right)}}{5} + \frac{2 \left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)}}{3} - \sin{\left(x + \frac{\pi}{2} \right)}$$
5 10 3 6
2/x\ / 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\
1 - tan |-| |1 - tan |-|| *|1 - tan |-|| 2*|1 - tan |-|| *|1 - tan |-||
\2/ \ \2// \ \4// \ \2// \ \4//
- ----------- - ------------------------------ + -------------------------------
2/x\ 10 6
1 + tan |-| / 2/x\\ / 2/x\\
\2/ 5*|1 + tan |-|| 3*|1 + tan |-||
\ \4// \ \4//
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{10} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{5}}{5 \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{10}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3}}{3 \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} - \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
5 3
/ 2/x\ \ / 2/x\ \
| cos |-| | | cos |-| |
| \2/ | 10/x pi\ | \2/ | 6/x pi\
|-1 + ------------| *cos |- - --| 2*|-1 + ------------| *cos |- - --|
| 2/x pi\| \2 2 / | 2/x pi\| \2 2 /
| cos |- - --|| | cos |- - --||
\ \2 2 // \ \2 2 //
-cos(x) - ---------------------------------- + -----------------------------------
5 3
$$- \frac{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{5} \cos^{10}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{5} + \frac{2 \left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{3} \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{3} - \cos{\left(x \right)}$$
5 3
2/x\ / 2/x\\ 10/x\ / 2/x\\ 6/x\
-1 + cot |-| 1024*|-1 + cot |-|| *cot |-| 128*|-1 + cot |-|| *cot |-|
\2/ \ \2// \4/ \ \2// \4/
- ------------ - ----------------------------- + ---------------------------
2/x\ 10 6
1 + cot |-| / 2/x\\ / 2/x\\
\2/ 5*|1 + cot |-|| 3*|1 + cot |-||
\ \4// \ \4//
$$- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{128 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \cot^{6}{\left(\frac{x}{4} \right)}}{3 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} - \frac{1024 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{5} \cot^{10}{\left(\frac{x}{4} \right)}}{5 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{10}}$$
5 3
cos (x) 2*cos (x)
-cos(x) - ------- + ---------
5 3
$$- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$
5 3
2/x\ / 2/x\\ / 2/x\\
-1 + cot |-| |-1 + cot |-|| 2*|-1 + cot |-||
\2/ \ \2// \ \2//
- ------------ - ---------------- + -----------------
2/x\ 5 3
1 + cot |-| / 2/x\\ / 2/x\\
\2/ 5*|1 + cot |-|| 3*|1 + cot |-||
\ \2// \ \2//
$$- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{5}}{5 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{5}} + \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3}}{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} - \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$
5 3
/ 2/x\\ 10/x\ 20/x\ / 2/x\\ 6/x\ 12/x\
1024*|-1 + cot |-|| *cot |-|*sin |-| 128*|-1 + cot |-|| *cot |-|*sin |-|
\ \2// \4/ \4/ \ \2// \4/ \4/
-cos(x) - -------------------------------------- + ------------------------------------
5 3
$$- \frac{1024 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{5} \sin^{20}{\left(\frac{x}{4} \right)} \cot^{10}{\left(\frac{x}{4} \right)}}{5} + \frac{128 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)}}{3} - \cos{\left(x \right)}$$
5 10 3 6
/ 2/x\\ / 2/x\\ 20/x\ / 2/x\\ / 2/x\\ 12/x\
|1 - tan |-|| *|-1 + cot |-|| *sin |-| 2*|1 - tan |-|| *|-1 + cot |-|| *sin |-|
\ \2// \ \4// \4/ \ \2// \ \4// \4/
-cos(x) - ---------------------------------------- + -----------------------------------------
5 3
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{5} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{10} \sin^{20}{\left(\frac{x}{4} \right)}}{5} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)}}{3} - \cos{\left(x \right)}$$
5 3
/ 4/x\\ / 4/x\\
| 4*sin |-|| | 4*sin |-||
| \2/| 10/pi x\ | \2/| 6/pi x\
|1 - ---------| *sin |-- + -| 2*|1 - ---------| *sin |-- + -|
| 2 | \2 2/ | 2 | \2 2/
/ pi\ \ sin (x) / \ sin (x) /
- sin|x + --| - ------------------------------ + -------------------------------
\ 2 / 5 3
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{5} \sin^{10}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{5} + \frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)^{3} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{3} - \sin{\left(x + \frac{\pi}{2} \right)}$$
5 10/x\ 3 6/x\
32*cos (x)*cos |-| 16*cos (x)*cos |-|
\2/ \2/
-cos(x) - ------------------- + ------------------
5 3
5*(1 + cos(x)) 3*(1 + cos(x))
$$- \cos{\left(x \right)} + \frac{16 \cos^{6}{\left(\frac{x}{2} \right)} \cos^{3}{\left(x \right)}}{3 \left(\cos{\left(x \right)} + 1\right)^{3}} - \frac{32 \cos^{10}{\left(\frac{x}{2} \right)} \cos^{5}{\left(x \right)}}{5 \left(\cos{\left(x \right)} + 1\right)^{5}}$$
5 3
/ 1 \ 20/x\ 10/x\ / 1 \ 12/x\ 6/x\
1024*|-1 + -------| *cos |-|*tan |-| 128*|-1 + -------| *cos |-|*tan |-|
| 2/x\| \4/ \4/ | 2/x\| \4/ \4/
| tan |-|| | tan |-||
\ \2// \ \2//
-cos(x) - -------------------------------------- + ------------------------------------
5 3
$$- \frac{1024 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{5} \cos^{20}{\left(\frac{x}{4} \right)} \tan^{10}{\left(\frac{x}{4} \right)}}{5} + \frac{128 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \cos^{12}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)}}{3} - \cos{\left(x \right)}$$
5/ pi\ 3/ pi\
sin |x + --| 2*sin |x + --|
/ pi\ \ 2 / \ 2 /
- sin|x + --| - ------------ + --------------
\ 2 / 5 3
$$- \frac{\sin^{5}{\left(x + \frac{\pi}{2} \right)}}{5} + \frac{2 \sin^{3}{\left(x + \frac{\pi}{2} \right)}}{3} - \sin{\left(x + \frac{\pi}{2} \right)}$$
5/x pi\ 5/x pi\
32*cos |- + --|*sin |- + --|
3*cos(x) + cos(3*x) \2 4 / \2 4 /
-cos(x) + ------------------- - ----------------------------
6 5
$$\frac{3 \cos{\left(x \right)} + \cos{\left(3 x \right)}}{6} - \frac{32 \sin^{5}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cos^{5}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{5} - \cos{\left(x \right)}$$
5 3
/ 2/x\\ 10/x\ / 2/x\\ 6/x\
|-1 + cot |-|| *sin |-| 2*|-1 + cot |-|| *sin |-|
\ \2// \2/ \ \2// \2/
-cos(x) - ------------------------ + -------------------------
5 3
$$- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{5} \sin^{10}{\left(\frac{x}{2} \right)}}{5} + \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)}}{3} - \cos{\left(x \right)}$$
5 3
/ 2 \ / 2 \
| sin (x) | 10/x\ | sin (x) | 6/x\
|-1 + ---------| *sin |-| 2*|-1 + ---------| *sin |-|
| 4/x\| \2/ | 4/x\| \2/
| 4*sin |-|| | 4*sin |-||
\ \2// \ \2//
-cos(x) - -------------------------- + ---------------------------
5 3
$$- \frac{\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{5} \sin^{10}{\left(\frac{x}{2} \right)}}{5} + \frac{2 \left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)}}{3} - \cos{\left(x \right)}$$
5 3
/ 2/x\\ 10/x\ / 2/x\\ 6/x\
|1 - tan |-|| *cos |-| 2*|1 - tan |-|| *cos |-|
\ \2// \2/ \ \2// \2/
-cos(x) - ----------------------- + ------------------------
5 3
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{5} \cos^{10}{\left(\frac{x}{2} \right)}}{5} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3} \cos^{6}{\left(\frac{x}{2} \right)}}{3} - \cos{\left(x \right)}$$
5 3
/ 2/x pi\\ / 2/x pi\\
| sec |- - --|| | sec |- - --||
| \2 2 /| | \2 2 /|
|-1 + ------------| 2*|-1 + ------------|
| 2/x\ | | 2/x\ |
| sec |-| | | sec |-| |
1 \ \2/ / \ \2/ /
- ------ - -------------------- + ----------------------
sec(x) 10/x pi\ 6/x pi\
5*sec |- - --| 3*sec |- - --|
\2 2 / \2 2 /
$$- \frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{5}}{5 \sec^{10}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{2 \left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{3}}{3 \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(x \right)}}$$
5 10 3 6
/ 1 \ / 2/x\\ / 1 \ / 2/x\\
|1 - -------| *|-1 + cot |-|| 2*|1 - -------| *|-1 + cot |-||
2/x\ | 2/x\| \ \4// | 2/x\| \ \4//
-1 + cot |-| | cot |-|| | cot |-||
\2/ \ \2// \ \2//
- ------------ - ------------------------------- + --------------------------------
2/x\ 10 6
1 + cot |-| / 2/x\\ / 2/x\\
\2/ 5*|1 + cot |-|| 3*|1 + cot |-||
\ \4// \ \4//
$$- \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{5} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{10}}{5 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{10}} + \frac{2 \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{3 \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} - \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$
1 1 2
- ------ - --------- + ---------
sec(x) 5 3
5*sec (x) 3*sec (x)
$$- \frac{1}{\sec{\left(x \right)}} + \frac{2}{3 \sec^{3}{\left(x \right)}} - \frac{1}{5 \sec^{5}{\left(x \right)}}$$
1 1 2
- ----------- - -------------- + --------------
/pi \ 5/pi \ 3/pi \
csc|-- - x| 5*csc |-- - x| 3*csc |-- - x|
\2 / \2 / \2 /
$$- \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{2}{3 \csc^{3}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{5 \csc^{5}{\left(- x + \frac{\pi}{2} \right)}}$$
5 3
/ 2/x\ \ / 2/x\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
|1 - ------------| 2*|1 - ------------|
| 2/x pi\| | 2/x pi\|
| sec |- - --|| | sec |- - --||
1 \ \2 2 // \ \2 2 //
- ------ - ------------------- + ---------------------
sec(x) 10/x\ 6/x\
5*sec |-| 3*sec |-|
\2/ \2/
$$- \frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{5}}{5 \sec^{10}{\left(\frac{x}{2} \right)}} + \frac{2 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{3}}{3 \sec^{6}{\left(\frac{x}{2} \right)}} - \frac{1}{\sec{\left(x \right)}}$$
5 3
/ 1 \ 10/x\ / 1 \ 6/x\
1024*|-1 + -------| *tan |-| 128*|-1 + -------| *tan |-|
2/x\ | 2/x\| \4/ | 2/x\| \4/
1 - tan |-| | tan |-|| | tan |-||
\2/ \ \2// \ \2//
- ----------- - ----------------------------- + ---------------------------
2/x\ 10 6
1 + tan |-| / 2/x\\ / 2/x\\
\2/ 5*|1 + tan |-|| 3*|1 + tan |-||
\ \4// \ \4//
$$- \frac{1024 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{5} \tan^{10}{\left(\frac{x}{4} \right)}}{5 \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{10}} + \frac{128 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \tan^{6}{\left(\frac{x}{4} \right)}}{3 \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} - \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
5 3
/ 2/x\ \ / 2/x\ \
| csc |-| | | csc |-| |
| \2/ | | \2/ |
|-1 + ------------| 2*|-1 + ------------|
| 2/pi x\| | 2/pi x\|
| csc |-- - -|| | csc |-- - -||
1 \ \2 2// \ \2 2//
- ----------- - -------------------- + ----------------------
/pi \ 10/x\ 6/x\
csc|-- - x| 5*csc |-| 3*csc |-|
\2 / \2/ \2/
$$- \frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{5}}{5 \csc^{10}{\left(\frac{x}{2} \right)}} + \frac{2 \left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{3}}{3 \csc^{6}{\left(\frac{x}{2} \right)}} - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}$$
5 3
/ 1 \ 10/x\ / 1 \ 6/x\
|-1 + -------| *sin |-| 2*|-1 + -------| *sin |-|
| 2/x\| \2/ | 2/x\| \2/
| tan |-|| | tan |-||
\ \2// \ \2//
-cos(x) - ------------------------ + -------------------------
5 3
$$- \frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{5} \sin^{10}{\left(\frac{x}{2} \right)}}{5} + \frac{2 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{3} \sin^{6}{\left(\frac{x}{2} \right)}}{3} - \cos{\left(x \right)}$$
5 10 3 6
/ 2/x\\ / 2/x\\ 20/x\ / 2/x\\ / 2/x\\ 12/x\
|1 - tan |-|| *|1 - tan |-|| *cos |-| 2*|1 - tan |-|| *|1 - tan |-|| *cos |-|
\ \2// \ \4// \4/ \ \2// \ \4// \4/
-cos(x) - --------------------------------------- + ----------------------------------------
5 3
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{10} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{5} \cos^{20}{\left(\frac{x}{4} \right)}}{5} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3} \cos^{12}{\left(\frac{x}{4} \right)}}{3} - \cos{\left(x \right)}$$
5 3
2/x\ / 2/x\\ / 2/x\\
1 - tan |-| |1 - tan |-|| 2*|1 - tan |-||
\2/ \ \2// \ \2//
- ----------- - ---------------- + ----------------
2/x\ 5 3
1 + tan |-| / 2/x\\ / 2/x\\
\2/ 5*|1 + tan |-|| 3*|1 + tan |-||
\ \2// \ \2//
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{5}}{5 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{5}} + \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3}}{3 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} - \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
3 6/x\ 5 10/x\
16*cos (x)*sin |-| 32*cos (x)*sin |-|
\2/ \2/
-cos(x) - ------------------ + -------------------
3 5
3*(-1 + cos(x)) 5*(-1 + cos(x))
$$- \cos{\left(x \right)} - \frac{16 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{3}{\left(x \right)}}{3 \left(\cos{\left(x \right)} - 1\right)^{3}} + \frac{32 \sin^{10}{\left(\frac{x}{2} \right)} \cos^{5}{\left(x \right)}}{5 \left(\cos{\left(x \right)} - 1\right)^{5}}$$
-cos(x) - 16*cos(x)^3*sin(x/2)^6/(3*(-1 + cos(x))^3) + 32*cos(x)^5*sin(x/2)^10/(5*(-1 + cos(x))^5)
Combinatorics
[src]
/ 2 4 \
-\15 - 10*cos (x) + 3*cos (x)/*cos(x)
--------------------------------------
15
$$- \frac{\left(3 \cos^{4}{\left(x \right)} - 10 \cos^{2}{\left(x \right)} + 15\right) \cos{\left(x \right)}}{15}$$
-(15 - 10*cos(x)^2 + 3*cos(x)^4)*cos(x)/15
Powers
[src]
5 3
cos (x) 2*cos (x)
-cos(x) - ------- + ---------
5 3
$$- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$
5 3
/ I*x -I*x\ / I*x -I*x\
|e e | |e e |
I*x -I*x |---- + -----| 2*|---- + -----|
e e \ 2 2 / \ 2 2 /
- ---- - ----- - --------------- + -----------------
2 2 5 3
$$- \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{5}}{5} + \frac{2 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{3}}{3} - \frac{e^{i x}}{2} - \frac{e^{- i x}}{2}$$
-exp(i*x)/2 - exp(-i*x)/2 - (exp(i*x)/2 + exp(-i*x)/2)^5/5 + 2*(exp(i*x)/2 + exp(-i*x)/2)^3/3