Тригонометрическая часть
[src]cos(6*a)
-------- - sin(4*a)*sin(6*a)
sec(4*a)
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \frac{\cos{\left(6 a \right)}}{\sec{\left(4 a \right)}}$$
// / /pi \ \\
|| 0 for And|im(a) = 0, |-- + 4*a| mod pi = 0||
|| \ \2 / /| // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|| | || | || | || |
|| / pi\ | || 2 | || 2*cot(2*a) | || 2*cot(3*a) |
|< 2*cot|2*a + --| |*|<-1 + cot (3*a) | - |<------------- otherwise |*|<------------- otherwise |
|| \ 4 / | ||-------------- otherwise | || 2 | || 2 |
||------------------ otherwise | || 2 | ||1 + cot (2*a) | ||1 + cot (3*a) |
|| 2/ pi\ | \\1 + cot (3*a) / \\ / \\ /
||1 + cot |2*a + --| |
\\ \ 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
1 1
---------------------- - -----------------
/pi \ csc(4*a)*csc(6*a)
csc|-- - 6*a|*sec(4*a)
\2 /
$$\frac{1}{\csc{\left(- 6 a + \frac{\pi}{2} \right)} \sec{\left(4 a \right)}} - \frac{1}{\csc{\left(4 a \right)} \csc{\left(6 a \right)}}$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| 1 |
||-1 + --------- | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|< 2 |*|< | - |< |*|< |
|| tan (2*a) | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise /
||-------------- otherwise |
|| 2 |
\\ csc (2*a) /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| / 2 \ | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|< 2 | sin (4*a) | |*|< | - |< |*|< |
||sin (2*a)*|-1 + -----------| otherwise | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise /
|| | 4 | |
\\ \ 4*sin (2*a)/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| 2 / 1 \ | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
||4*tan (a)*|-1 + ---------| | || | || | || |
|| | 2 | | || 2 | || 2*tan(2*a) | || 2*tan(3*a) |
|< \ tan (2*a)/ |*|<1 - tan (3*a) | - |<------------- otherwise |*|<------------- otherwise |
||-------------------------- otherwise | ||------------- otherwise | || 2 | || 2 |
|| 2 | || 2 | ||1 + tan (2*a) | ||1 + tan (3*a) |
|| / 2 \ | \\1 + tan (3*a) / \\ / \\ /
|| \1 + tan (a)/ |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \tan{\left(3 a \right)}}{\tan^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan^{2}{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(3 a \right)}}{\tan^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2 \ / pi\
2*\1 - tan (3*a)/*tan|2*a + --|
4*tan(2*a)*tan(3*a) \ 4 /
- ------------------------------- + ------------------------------------
/ 2 \ / 2 \ / 2 \ / 2/ pi\\
\1 + tan (2*a)/*\1 + tan (3*a)/ \1 + tan (3*a)/*|1 + tan |2*a + --||
\ \ 4 //
$$\frac{2 \cdot \left(1 - \tan^{2}{\left(3 a \right)}\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(3 a \right)} + 1\right) \left(\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(2 a \right)} \tan{\left(3 a \right)}}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$
/ 1 for And(im(a) = 0, 5*a mod pi = 0)
|
| 2
<-1 + cot (5*a)
|-------------- otherwise
| 2
\1 + cot (5*a)
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(5 a \right)} - 1}{\cot^{2}{\left(5 a \right)} + 1} & \text{otherwise} \end{cases}$$
/ pi\ / pi\
cos(4*a)*cos(6*a) - cos|4*a - --|*cos|6*a - --|
\ 2 / \ 2 /
$$\cos{\left(4 a \right)} \cos{\left(6 a \right)} - \cos{\left(4 a - \frac{\pi}{2} \right)} \cos{\left(6 a - \frac{\pi}{2} \right)}$$
cos(6*a)
------------- - sin(4*a)*sin(6*a)
/pi \
csc|-- - 4*a|
\2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \frac{\cos{\left(6 a \right)}}{\csc{\left(- 4 a + \frac{\pi}{2} \right)}}$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|< 2 / 2 \ |*|< | - |< |*|< |
||sin (2*a)*\-1 + cot (2*a)/ otherwise | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise /
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| 2 | // 1 for And(im(a) = 0, 3*a mod pi = 0)\
|| csc (2*a) | || | // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
||-1 + -------------- | || 1 | || | || |
|< 2/pi \ |*|<------------- otherwise | - |< 1 |*|< 1 |
|| csc |-- - 2*a| | || /pi \ | ||-------- otherwise | ||-------- otherwise |
|| \2 / | ||csc|-- - 6*a| | \\csc(4*a) / \\csc(6*a) /
||------------------- otherwise | \\ \2 / /
|| 2 |
\\ csc (2*a) /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{1}{\csc{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{1}{\csc{\left(6 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} - 1}{\csc^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 6 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
/ 2 \ / 2 \
\1 - tan (2*a)/*\1 - tan (3*a)/ 4*tan(2*a)*tan(3*a)
------------------------------- - -------------------------------
/ 2 \ / 2 \ / 2 \ / 2 \
\1 + tan (2*a)/*\1 + tan (3*a)/ \1 + tan (2*a)/*\1 + tan (3*a)/
$$\frac{\left(1 - \tan^{2}{\left(2 a \right)}\right) \left(1 - \tan^{2}{\left(3 a \right)}\right)}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)} - \frac{4 \tan{\left(2 a \right)} \tan{\left(3 a \right)}}{\left(\tan^{2}{\left(2 a \right)} + 1\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| 2/ pi\ |
|| sec |2*a - --| | // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|| \ 2 / | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ || | || |
||-1 + -------------- | || | || 1 | || 1 |
|< 2 |*|< 1 | - |<------------- otherwise |*|<------------- otherwise |
|| sec (2*a) | ||-------- otherwise | || / pi\ | || / pi\ |
||------------------- otherwise | \\sec(6*a) / ||sec|4*a - --| | ||sec|6*a - --| |
|| 2/ pi\ | \\ \ 2 / / \\ \ 2 / /
|| sec |2*a - --| |
|| \ 2 / |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{1}{\sec{\left(6 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{1}{\sec{\left(6 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
$$\frac{1}{\sec{\left(10 a \right)}}$$
/pi \ /pi \
sin|-- + 4*a|*sin|-- + 6*a| - sin(4*a)*sin(6*a)
\2 / \2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \sin{\left(4 a + \frac{\pi}{2} \right)} \sin{\left(6 a + \frac{\pi}{2} \right)}$$
1
--------------
/pi \
csc|-- - 10*a|
\2 /
$$\frac{1}{\csc{\left(- 10 a + \frac{\pi}{2} \right)}}$$
/pi \
cos(4*a)*sin|-- + 6*a| - sin(4*a)*sin(6*a)
\2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \sin{\left(6 a + \frac{\pi}{2} \right)} \cos{\left(4 a \right)}$$
2 2 -cos(10*a) + cos(2*a) 2 2
1 - 2*cos (2*a) - 2*cos (3*a) - --------------------- + 4*cos (2*a)*cos (3*a)
2
$$- \frac{\cos{\left(2 a \right)} - \cos{\left(10 a \right)}}{2} + 4 \cos^{2}{\left(2 a \right)} \cos^{2}{\left(3 a \right)} - 2 \cos^{2}{\left(2 a \right)} - 2 \cos^{2}{\left(3 a \right)} + 1$$
1 1
--------------------------- - -----------------
/pi \ /pi \ csc(4*a)*csc(6*a)
csc|-- - 6*a|*csc|-- - 4*a|
\2 / \2 /
$$\frac{1}{\csc{\left(- 6 a + \frac{\pi}{2} \right)} \csc{\left(- 4 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(4 a \right)} \csc{\left(6 a \right)}}$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|| // 0 for And(im(a) = 0, 2*a mod pi = 0)\ | || | || | || |
|| || | | ||/ 1 for And(im(a) = 0, 3*a mod pi = 0) | ||/ 0 for And(im(a) = 0, 4*a mod pi = 0) | ||/ 0 for And(im(a) = 0, 6*a mod pi = 0) |
|| || 2 | | ||| | ||| | ||| |
| 2 \ || 4*cot (a) | |*|<| 2 | - |<| 2*cot(2*a) |*|<| 2*cot(3*a) |
||\-1 + cot (2*a)/*|<-------------- otherwise | otherwise | ||<-1 + cot (3*a) otherwise | ||<------------- otherwise otherwise | ||<------------- otherwise otherwise |
|| || 2 | | |||-------------- otherwise | ||| 2 | ||| 2 |
|| ||/ 2 \ | | ||| 2 | |||1 + cot (2*a) | |||1 + cot (3*a) |
|| ||\1 + cot (a)/ | | \\\1 + cot (3*a) / \\\ / \\\ /
\\ \\ / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(a \right)}}{\left(\cot^{2}{\left(a \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\ // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|| | || | || | || |
|| 2 | || 2 | || 2*cot(2*a) | || 2*cot(3*a) |
|<-1 + cot (2*a) |*|<-1 + cot (3*a) | - |<------------- otherwise |*|<------------- otherwise |
||-------------- otherwise | ||-------------- otherwise | || 2 | || 2 |
|| 2 | || 2 | ||1 + cot (2*a) | ||1 + cot (3*a) |
\\1 + cot (2*a) / \\1 + cot (3*a) / \\ / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 a \right)} - 1}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
1 1
----------------- - ---------------------------
sec(4*a)*sec(6*a) / pi\ / pi\
sec|4*a - --|*sec|6*a - --|
\ 2 / \ 2 /
$$- \frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)} \sec{\left(6 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 a \right)} \sec{\left(6 a \right)}}$$
/pi \
sin|-- + 10*a|
\2 /
$$\sin{\left(10 a + \frac{\pi}{2} \right)}$$
1 1
----------------- - -----------------
sec(4*a)*sec(6*a) csc(4*a)*csc(6*a)
$$\frac{1}{\sec{\left(4 a \right)} \sec{\left(6 a \right)}} - \frac{1}{\csc{\left(4 a \right)} \csc{\left(6 a \right)}}$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| | // 1 for And(im(a) = 0, 3*a mod pi = 0)\
|| / 2 \ | || | // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|< 2 | sin (4*a) | |*|< /pi \ | - |< |*|< |
||sin (2*a)*|-1 + -----------| otherwise | ||sin|-- + 6*a| otherwise | \\sin(4*a) otherwise / \\sin(6*a) otherwise /
|| | 4 | | \\ \2 / /
\\ \ 4*sin (2*a)/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\sin{\left(6 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| 2 / 1 \ | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 3*a mod pi = 0)\
|| |
/ 2 \ || 2 |
\-1 + cot (2*a)/*|<-1 + cot (3*a) |
// 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\ ||-------------- otherwise |
|| | || | || 2 |
|| 2*cot(2*a) | || 2*cot(3*a) | \\1 + cot (3*a) /
- |<------------- otherwise |*|<------------- otherwise | + ----------------------------------------------------------------------
|| 2 | || 2 | 2
||1 + cot (2*a) | ||1 + cot (3*a) | 1 + cot (2*a)
\\ / \\ /
$$\left(\frac{\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)}{\cot^{2}{\left(2 a \right)} + 1}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\ 2 / 2 \ // 1 for And(im(a) = 0, 3*a mod pi = 0)\
- |< |*|< | + sin (2*a)*\-1 + cot (2*a)/*|< |
\\sin(4*a) otherwise / \\sin(6*a) otherwise / \\cos(6*a) otherwise /
$$\left(\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(2 a \right)}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| 4 2 / 1 \ | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|<4*cos (a)*tan (a)*|-1 + ---------| otherwise |*|< | - |< |*|< |
|| | 2 | | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise /
|| \ tan (2*a)/ |
\\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\4 \left(-1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \cos^{4}{\left(a \right)} \tan^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 1 for And(im(a) = 0, 5*a mod pi = 0)
<
\cos(10*a) otherwise
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod \pi = 0 \\\cos{\left(10 a \right)} & \text{otherwise} \end{cases}$$
// 1 for And(im(a) = 0, 3*a mod pi = 0)\
|| |
|| 2 | / pi\
2*|<-1 + cot (3*a) |*tan|2*a + --|
// 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\ ||-------------- otherwise | \ 4 /
|| | || | || 2 |
|| 2*cot(2*a) | || 2*cot(3*a) | \\1 + cot (3*a) /
- |<------------- otherwise |*|<------------- otherwise | + ---------------------------------------------------------------------
|| 2 | || 2 | 2/ pi\
||1 + cot (2*a) | ||1 + cot (3*a) | 1 + tan |2*a + --|
\\ / \\ / \ 4 /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\frac{\cot^{2}{\left(3 a \right)} - 1}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}\right)$$
2
1 - tan (5*a)
-------------
2
1 + tan (5*a)
$$\frac{1 - \tan^{2}{\left(5 a \right)}}{\tan^{2}{\left(5 a \right)} + 1}$$
// 1 for And(im(a) = 0, 3*a mod pi = 0)\ / pi\
2*|< |*tan|2*a + --|
// 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\ \\cos(6*a) otherwise / \ 4 /
- |< |*|< | + ---------------------------------------------------------------
\\sin(4*a) otherwise / \\sin(6*a) otherwise / 2/ pi\
1 + tan |2*a + --|
\ 4 /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right) \tan{\left(2 a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}\right)$$
$$\cos{\left(10 a \right)}$$
/pi \
cos(6*a)*sin|-- + 4*a| - sin(4*a)*sin(6*a)
\2 /
$$- \sin{\left(4 a \right)} \sin{\left(6 a \right)} + \sin{\left(4 a + \frac{\pi}{2} \right)} \cos{\left(6 a \right)}$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|| // 0 for And(im(a) = 0, 2*a mod pi = 0)\ | || | || | || |
| 2 \ || | |*| 1 for And(im(a) = 0, 3*a mod pi = 0) | - | 0 for And(im(a) = 0, 4*a mod pi = 0) |*| 0 for And(im(a) = 0, 6*a mod pi = 0) |
||\-1 + cot (2*a)/*|<1 - cos(4*a) | otherwise | ||< otherwise | ||< otherwise | ||< otherwise |
|| ||------------ otherwise | | \\\cos(6*a) otherwise / \\\sin(4*a) otherwise / \\\sin(6*a) otherwise /
\\ \\ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(\cot^{2}{\left(2 a \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1 - \cos{\left(4 a \right)}}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for And(im(a) = 0, 2*a mod pi = 0)\
|| |
|| / 2 \ | // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|| 2/ pi\ | cos (2*a) | | // 1 for And(im(a) = 0, 3*a mod pi = 0)\ || | || |
|
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\cos{\left(4 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\cos{\left(6 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// / /pi \ \\
|| 0 for And|im(a) = 0, |-- + 4*a| mod pi = 0|| // 1 for And(im(a) = 0, 3*a mod pi = 0)\ // 0 for And(im(a) = 0, 4*a mod pi = 0)\ // 0 for And(im(a) = 0, 6*a mod pi = 0)\
|< \ \2 / /|*|< | - |< |*|< |
|| | \\cos(6*a) otherwise / \\sin(4*a) otherwise / \\sin(6*a) otherwise /
\\cos(4*a) otherwise /
$$\left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(4 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 3 a \bmod \pi = 0 \\\cos{\left(6 a \right)} & \text{otherwise} \end{cases}\right)\right)$$