Mister Exam
Least common denominator 4*((sin(8*x)/2+4*x)/8-sin(4*x)/2+x)
The solution
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/sin(8*x) \ |-------- + 4*x | | 2 sin(4*x) | 4*|-------------- - -------- + x| \ 8 2 /
$$4 \left(x + \left(\frac{4 x + \frac{\sin{\left(8 x \right)}}{2}}{8} - \frac{\sin{\left(4 x \right)}}{2}\right)\right)$$
4*((sin(8*x)/2 + 4*x)/8 - sin(4*x)/2 + x)
General simplification
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sin(8*x)
-2*sin(4*x) + 6*x + --------
4
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(8 x \right)}}{4}$$
-2*sin(4*x) + 6*x + sin(8*x)/4
Common denominator
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sin(8*x)
-2*sin(4*x) + 6*x + --------
4
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(8 x \right)}}{4}$$
-2*sin(4*x) + 6*x + sin(8*x)/4
Rational denominator
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-64*sin(4*x) + 8*sin(8*x) + 192*x
---------------------------------
32
$$\frac{192 x - 64 \sin{\left(4 x \right)} + 8 \sin{\left(8 x \right)}}{32}$$
(-64*sin(4*x) + 8*sin(8*x) + 192*x)/32
Assemble expression
[src]
sin(8*x)
-2*sin(4*x) + 6*x + --------
4
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(8 x \right)}}{4}$$
-2*sin(4*x) + 6*x + sin(8*x)/4
Combinatorics
[src]
sin(8*x)
-2*sin(4*x) + 6*x + --------
4
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(8 x \right)}}{4}$$
-2*sin(4*x) + 6*x + sin(8*x)/4
Combining rational expressions
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-8*sin(4*x) + 24*x + sin(8*x)
-----------------------------
4
$$\frac{24 x - 8 \sin{\left(4 x \right)} + \sin{\left(8 x \right)}}{4}$$
(-8*sin(4*x) + 24*x + sin(8*x))/4
Powers
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/ -8*I*x 8*I*x\
/ -4*I*x 4*I*x\ I*\- e + e /
6*x + I*\- e + e / - ----------------------
8
$$6 x + i \left(e^{4 i x} - e^{- 4 i x}\right) - \frac{i \left(e^{8 i x} - e^{- 8 i x}\right)}{8}$$
sin(8*x)
-2*sin(4*x) + 6*x + --------
4
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(8 x \right)}}{4}$$
-2*sin(4*x) + 6*x + sin(8*x)/4
Trigonometric part
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2
4*cot(2*x) 2*cot (2*x)*cot(4*x)
6*x - ------------- + --------------------
2 2
1 + cot (2*x) / 2 \
\1 + cot (2*x)/
$$6 x - \frac{4 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1} + \frac{2 \cot^{2}{\left(2 x \right)} \cot{\left(4 x \right)}}{\left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}$$
sin(4*x)
-2*sin(4*x) + 6*x + ---------------
/pi \
2*csc|-- - 4*x|
\2 /
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(4 x \right)}}{2 \csc{\left(- 4 x + \frac{\pi}{2} \right)}}$$
2
/ 2 \ 4
\1 - tan (2*x)/ *cos (2*x)*tan(4*x)
-2*sin(4*x) + 6*x + -----------------------------------
2
$$6 x + \frac{\left(1 - \tan^{2}{\left(2 x \right)}\right)^{2} \cos^{4}{\left(2 x \right)} \tan{\left(4 x \right)}}{2} - 2 \sin{\left(4 x \right)}$$
2 1 - -------- + 6*x + ---------- csc(4*x) 4*csc(8*x)
$$6 x + \frac{1}{4 \csc{\left(8 x \right)}} - \frac{2}{\csc{\left(4 x \right)}}$$
2/ pi\ / 2/ pi\\
sin |4*x + --|*|1 - cot |4*x + --||
\ 4 / \ \ 4 //
-2*sin(4*x) + 6*x + -----------------------------------
4
$$6 x + \frac{\left(1 - \cot^{2}{\left(4 x + \frac{\pi}{4} \right)}\right) \sin^{2}{\left(4 x + \frac{\pi}{4} \right)}}{4} - 2 \sin{\left(4 x \right)}$$
2 1
- ------------- + 6*x + ---------------
/ pi\ /pi \
sec|4*x - --| 4*sec|-- - 8*x|
\ 2 / \2 /
$$6 x - \frac{2}{\sec{\left(4 x - \frac{\pi}{2} \right)}} + \frac{1}{4 \sec{\left(- 8 x + \frac{\pi}{2} \right)}}$$
cot(4*x) 4*cot(2*x)
6*x + ----------------- - -------------
/ 2 \ 2
2*\1 + cot (4*x)/ 1 + cot (2*x)
$$6 x + \frac{\cot{\left(4 x \right)}}{2 \left(\cot^{2}{\left(4 x \right)} + 1\right)} - \frac{4 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1}$$
cos(4*x)*sin(4*x)
-2*sin(4*x) + 6*x + -----------------
2
$$6 x + \frac{\sin{\left(4 x \right)} \cos{\left(4 x \right)}}{2} - 2 \sin{\left(4 x \right)}$$
2
cos (4*x)*tan(4*x)
-2*sin(4*x) + 6*x + ------------------
2
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\cos^{2}{\left(4 x \right)} \tan{\left(4 x \right)}}{2}$$
2 1 - -------- + 6*x + --------------- csc(4*x) 4*csc(pi - 8*x)
$$6 x + \frac{1}{4 \csc{\left(\pi - 8 x \right)}} - \frac{2}{\csc{\left(4 x \right)}}$$
/ pi\
cos|8*x - --|
\ 2 /
-2*sin(4*x) + 6*x + -------------
4
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\cos{\left(8 x - \frac{\pi}{2} \right)}}{4}$$
2
/ 2 \ 4
\-1 + cot (2*x)/ *sin (2*x)*tan(4*x)
-2*sin(4*x) + 6*x + ------------------------------------
2
$$6 x + \frac{\left(\cot^{2}{\left(2 x \right)} - 1\right)^{2} \sin^{4}{\left(2 x \right)} \tan{\left(4 x \right)}}{2} - 2 \sin{\left(4 x \right)}$$
2 1
- -------- + 6*x + ------------------------
csc(4*x) /pi \
2*csc(4*x)*csc|-- - 4*x|
\2 /
$$6 x - \frac{2}{\csc{\left(4 x \right)}} + \frac{1}{2 \csc{\left(4 x \right)} \csc{\left(- 4 x + \frac{\pi}{2} \right)}}$$
2
/ 2 \
4*cot(2*x) \-1 + cot (2*x)/
6*x - ------------- + ---------------------------
2 2
1 + cot (2*x) / 2 \
2*\1 + cot (2*x)/ *cot(4*x)
$$6 x + \frac{\left(\cot^{2}{\left(2 x \right)} - 1\right)^{2}}{2 \left(\cot^{2}{\left(2 x \right)} + 1\right)^{2} \cot{\left(4 x \right)}} - \frac{4 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1}$$
/ pi\
cos(4*x)*cos|4*x - --|
/ pi\ \ 2 /
- 2*cos|4*x - --| + 6*x + ----------------------
\ 2 / 2
$$6 x + \frac{\cos{\left(4 x \right)} \cos{\left(4 x - \frac{\pi}{2} \right)}}{2} - 2 \cos{\left(4 x - \frac{\pi}{2} \right)}$$
2
sin (4*x)*cot(4*x)
-2*sin(4*x) + 6*x + ------------------
2
$$6 x + \frac{\sin^{2}{\left(4 x \right)} \cot{\left(4 x \right)}}{2} - 2 \sin{\left(4 x \right)}$$
sin(8*x)
-2*sin(4*x) + 6*x + --------
4
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(8 x \right)}}{4}$$
1
-2*sin(4*x) + 6*x + ---------------
/ pi\
4*sec|8*x - --|
\ 2 /
$$6 x - 2 \sin{\left(4 x \right)} + \frac{1}{4 \sec{\left(8 x - \frac{\pi}{2} \right)}}$$
/ pi\
cos|8*x - --|
/ pi\ \ 2 /
- 2*cos|4*x - --| + 6*x + -------------
\ 2 / 4
$$6 x - 2 \cos{\left(4 x - \frac{\pi}{2} \right)} + \frac{\cos{\left(8 x - \frac{\pi}{2} \right)}}{4}$$
1
-2*sin(4*x) + 6*x + ---------------
/pi \
4*sec|-- - 8*x|
\2 /
$$6 x - 2 \sin{\left(4 x \right)} + \frac{1}{4 \sec{\left(- 8 x + \frac{\pi}{2} \right)}}$$
2
/ 2 \
4*tan(2*x) \1 - tan (2*x)/ *tan(4*x)
6*x - ------------- + -------------------------
2 2
1 + tan (2*x) / 2 \
2*\1 + tan (2*x)/
$$6 x + \frac{\left(1 - \tan^{2}{\left(2 x \right)}\right)^{2} \tan{\left(4 x \right)}}{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2}} - \frac{4 \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1}$$
tan(4*x) 4*tan(2*x)
6*x + ----------------- - -------------
/ 2 \ 2
2*\1 + tan (4*x)/ 1 + tan (2*x)
$$6 x + \frac{\tan{\left(4 x \right)}}{2 \left(\tan^{2}{\left(4 x \right)} + 1\right)} - \frac{4 \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1}$$
2 1
- ------------- + 6*x + ---------------
/ pi\ / pi\
sec|4*x - --| 4*sec|8*x - --|
\ 2 / \ 2 /
$$6 x + \frac{1}{4 \sec{\left(8 x - \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(4 x - \frac{\pi}{2} \right)}}$$
2 2/pi \
sin (4*x)*sin |-- + 4*x|
\2 /
-2*sin(4*x) + 6*x + ------------------------
sin(8*x)
$$6 x + \frac{\sin^{2}{\left(4 x \right)} \sin^{2}{\left(4 x + \frac{\pi}{2} \right)}}{\sin{\left(8 x \right)}} - 2 \sin{\left(4 x \right)}$$
2/ pi\
1 - cot |4*x + --|
4*tan(2*x) \ 4 /
6*x - ------------- + ----------------------
2 / 2/ pi\\
1 + tan (2*x) 4*|1 + cot |4*x + --||
\ \ 4 //
$$6 x + \frac{1 - \cot^{2}{\left(4 x + \frac{\pi}{4} \right)}}{4 \left(\cot^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1}$$
sin(4*x)
-2*sin(4*x) + 6*x + ----------
2*sec(4*x)
$$6 x - 2 \sin{\left(4 x \right)} + \frac{\sin{\left(4 x \right)}}{2 \sec{\left(4 x \right)}}$$
2
4*tan(2*x) 2*tan (2*x)
6*x - ------------- + -------------------------
2 2
1 + tan (2*x) / 2 \
\1 + tan (2*x)/ *tan(4*x)
$$6 x - \frac{4 \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1} + \frac{2 \tan^{2}{\left(2 x \right)}}{\left(\tan^{2}{\left(2 x \right)} + 1\right)^{2} \tan{\left(4 x \right)}}$$
2 1
- ------------- + 6*x + ------------------------
/ pi\ / pi\
sec|4*x - --| 2*sec(4*x)*sec|4*x - --|
\ 2 / \ 2 /
$$6 x - \frac{2}{\sec{\left(4 x - \frac{\pi}{2} \right)}} + \frac{1}{2 \sec{\left(4 x \right)} \sec{\left(4 x - \frac{\pi}{2} \right)}}$$
1
-2*sin(4*x) + 6*x + ---------------
4*csc(pi - 8*x)
$$6 x - 2 \sin{\left(4 x \right)} + \frac{1}{4 \csc{\left(\pi - 8 x \right)}}$$
2 1
- ------------- + 6*x + ----------
/ pi\ 4*csc(8*x)
sec|4*x - --|
\ 2 /
$$6 x - \frac{2}{\sec{\left(4 x - \frac{\pi}{2} \right)}} + \frac{1}{4 \csc{\left(8 x \right)}}$$
1
-2*sin(4*x) + 6*x + ----------
4*csc(8*x)
$$6 x - 2 \sin{\left(4 x \right)} + \frac{1}{4 \csc{\left(8 x \right)}}$$
/ pi\ sin(8*x)
- 2*cos|4*x - --| + 6*x + --------
\ 2 / 4
$$6 x + \frac{\sin{\left(8 x \right)}}{4} - 2 \cos{\left(4 x - \frac{\pi}{2} \right)}$$
2/ pi\
-1 + tan |4*x + --|
\ 4 /
-2*sin(4*x) + 6*x + ----------------------
/ 2/ pi\\
4*|1 + tan |4*x + --||
\ \ 4 //
$$6 x + \frac{\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} - 1}{4 \left(\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right)} - 2 \sin{\left(4 x \right)}$$
2/ pi\
-1 + tan |4*x + --|
4*cot(2*x) \ 4 /
6*x - ------------- + ----------------------
2 / 2/ pi\\
1 + cot (2*x) 4*|1 + tan |4*x + --||
\ \ 4 //
$$6 x + \frac{\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} - 1}{4 \left(\tan^{2}{\left(4 x + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1}$$
6*x - 4*cot(2*x)/(1 + cot(2*x)^2) + (-1 + tan(4*x + pi/4)^2)/(4*(1 + tan(4*x + pi/4)^2))