2
/ 2 \ 8*x *cos(4*x)*sin(4*x)
2*x*atan\cos (4*x)/ - ----------------------
4
1 + cos (4*x)
$$- \frac{8 x^{2} \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{\cos^{4}{\left(4 x \right)} + 1} + 2 x \operatorname{atan}{\left(\cos^{2}{\left(4 x \right)} \right)}$$
/ / 4 2 \ \
| 2 | 2 2 4*cos (4*x)*sin (4*x)| |
| 16*x *|cos (4*x) - sin (4*x) + ---------------------| |
| | 4 | |
| \ 1 + cos (4*x) / 16*x*cos(4*x)*sin(4*x) / 2 \|
2*|- ----------------------------------------------------- - ---------------------- + atan\cos (4*x)/|
| 4 4 |
\ 1 + cos (4*x) 1 + cos (4*x) /
$$2 \left(- \frac{16 x^{2} \left(- \sin^{2}{\left(4 x \right)} + \cos^{2}{\left(4 x \right)} + \frac{4 \sin^{2}{\left(4 x \right)} \cos^{4}{\left(4 x \right)}}{\cos^{4}{\left(4 x \right)} + 1}\right)}{\cos^{4}{\left(4 x \right)} + 1} - \frac{16 x \sin{\left(4 x \right)} \cos{\left(4 x \right)}}{\cos^{4}{\left(4 x \right)} + 1} + \operatorname{atan}{\left(\cos^{2}{\left(4 x \right)} \right)}\right)$$
/ / 4 2 \ / 4 6 2 2 2 \ \
| | 2 2 4*cos (4*x)*sin (4*x)| 2 | 3*cos (4*x) 8*cos (4*x)*sin (4*x) 5*cos (4*x)*sin (4*x)| |
16*|- 12*x*|cos (4*x) - sin (4*x) + ---------------------| - 3*cos(4*x)*sin(4*x) + 32*x *|1 - ------------- - --------------------- + ---------------------|*cos(4*x)*sin(4*x)|
| | 4 | | 4 2 4 | |
| \ 1 + cos (4*x) / | 1 + cos (4*x) / 4 \ 1 + cos (4*x) | |
\ \ \1 + cos (4*x)/ / /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
4
1 + cos (4*x)
$$\frac{16 \left(32 x^{2} \left(1 + \frac{5 \sin^{2}{\left(4 x \right)} \cos^{2}{\left(4 x \right)}}{\cos^{4}{\left(4 x \right)} + 1} - \frac{3 \cos^{4}{\left(4 x \right)}}{\cos^{4}{\left(4 x \right)} + 1} - \frac{8 \sin^{2}{\left(4 x \right)} \cos^{6}{\left(4 x \right)}}{\left(\cos^{4}{\left(4 x \right)} + 1\right)^{2}}\right) \sin{\left(4 x \right)} \cos{\left(4 x \right)} - 12 x \left(- \sin^{2}{\left(4 x \right)} + \cos^{2}{\left(4 x \right)} + \frac{4 \sin^{2}{\left(4 x \right)} \cos^{4}{\left(4 x \right)}}{\cos^{4}{\left(4 x \right)} + 1}\right) - 3 \sin{\left(4 x \right)} \cos{\left(4 x \right)}\right)}{\cos^{4}{\left(4 x \right)} + 1}$$