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How do you 2tg(6x)/(1+tg^2(6x)) in partial fractions?

An expression to simplify:

The solution

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  2*tan(6*x) 
-------------
       2     
1 + tan (6*x)
$$\frac{2 \tan{\left(6 x \right)}}{\tan^{2}{\left(6 x \right)} + 1}$$
(2*tan(6*x))/(1 + tan(6*x)^2)
General simplification [src]
sin(12*x)
$$\sin{\left(12 x \right)}$$
sin(12*x)
Fraction decomposition [src]
2*tan(6*x)/(1 + tan(6*x)^2)
$$\frac{2 \tan{\left(6 x \right)}}{\tan^{2}{\left(6 x \right)} + 1}$$
  2*tan(6*x) 
-------------
       2     
1 + tan (6*x)
Numerical answer [src]
2.0*tan(6*x)/(1.0 + tan(6*x)^2)
2.0*tan(6*x)/(1.0 + tan(6*x)^2)
Powers [src]
               /   6*I*x    -6*I*x\           
           2*I*\- e      + e      /           
----------------------------------------------
/                        2\                   
|    /   6*I*x    -6*I*x\ |                   
|    \- e      + e      / | / -6*I*x    6*I*x\
|1 - ---------------------|*\e       + e     /
|                       2 |                   
|     / -6*I*x    6*I*x\  |                   
\     \e       + e     /  /                   
$$\frac{2 i \left(- e^{6 i x} + e^{- 6 i x}\right)}{\left(- \frac{\left(- e^{6 i x} + e^{- 6 i x}\right)^{2}}{\left(e^{6 i x} + e^{- 6 i x}\right)^{2}} + 1\right) \left(e^{6 i x} + e^{- 6 i x}\right)}$$
2*i*(-exp(6*i*x) + exp(-6*i*x))/((1 - (-exp(6*i*x) + exp(-6*i*x))^2/(exp(-6*i*x) + exp(6*i*x))^2)*(exp(-6*i*x) + exp(6*i*x)))
Expand expression [src]
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                                                                                                                                                                                                                                                                                                                                                                40*tan (x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              12*tan (x)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              12*tan(x)                                                                                                                                                                                                                                                                                                                                                              
- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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         6            2            4                                        10956*tan (x)                                                                       4612*tan  (x)                                                                        780*tan (x)                                                                         36*tan  (x)                                                                          36*tan (x)                                                                         780*tan  (x)                                                                        4612*tan (x)                                                                       10956*tan  (x)                                           6            2            4                                        10956*tan (x)                                                                       4612*tan  (x)                                                                        780*tan (x)                                                                         36*tan  (x)                                                                          36*tan (x)                                                                         780*tan  (x)                                                                        4612*tan (x)                                                                       10956*tan  (x)                                           6            2            4                                        10956*tan (x)                                                                       4612*tan  (x)                                                                        780*tan (x)                                                                         36*tan  (x)                                                                          36*tan (x)                                                                         780*tan  (x)                                                                        4612*tan (x)                                                                       10956*tan  (x)                                 
  1 - tan (x) - 15*tan (x) + 15*tan (x) - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------   1 - tan (x) - 15*tan (x) + 15*tan (x) - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------   1 - tan (x) - 15*tan (x) + 15*tan (x) - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- - --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------
                                                 12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8                                                     12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8                                                     12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8             12             6            2            10             4             8   
                                          1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)                                           1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)                                           1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)   1 + tan  (x) - 452*tan (x) - 30*tan (x) - 30*tan  (x) + 255*tan (x) + 255*tan (x)
$$\frac{12 \tan^{5}{\left(x \right)}}{- \tan^{6}{\left(x \right)} + 15 \tan^{4}{\left(x \right)} - 15 \tan^{2}{\left(x \right)} + 1 - \frac{36 \tan^{16}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{780 \tan^{14}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{4612 \tan^{12}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{10956 \tan^{10}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{10956 \tan^{8}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{4612 \tan^{6}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{780 \tan^{4}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{36 \tan^{2}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1}} - \frac{40 \tan^{3}{\left(x \right)}}{- \tan^{6}{\left(x \right)} + 15 \tan^{4}{\left(x \right)} - 15 \tan^{2}{\left(x \right)} + 1 - \frac{36 \tan^{16}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{780 \tan^{14}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{4612 \tan^{12}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{10956 \tan^{10}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{10956 \tan^{8}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{4612 \tan^{6}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{780 \tan^{4}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{36 \tan^{2}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1}} + \frac{12 \tan{\left(x \right)}}{- \tan^{6}{\left(x \right)} + 15 \tan^{4}{\left(x \right)} - 15 \tan^{2}{\left(x \right)} + 1 - \frac{36 \tan^{16}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{780 \tan^{14}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{4612 \tan^{12}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{10956 \tan^{10}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{10956 \tan^{8}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{4612 \tan^{6}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} - \frac{780 \tan^{4}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1} + \frac{36 \tan^{2}{\left(x \right)}}{\tan^{12}{\left(x \right)} - 30 \tan^{10}{\left(x \right)} + 255 \tan^{8}{\left(x \right)} - 452 \tan^{6}{\left(x \right)} + 255 \tan^{4}{\left(x \right)} - 30 \tan^{2}{\left(x \right)} + 1}}$$
-40*tan(x)^3/(1 - tan(x)^6 - 15*tan(x)^2 + 15*tan(x)^4 - 10956*tan(x)^8/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 4612*tan(x)^12/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 780*tan(x)^4/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 36*tan(x)^16/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 36*tan(x)^2/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 780*tan(x)^14/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 4612*tan(x)^6/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 10956*tan(x)^10/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8)) + 12*tan(x)^5/(1 - tan(x)^6 - 15*tan(x)^2 + 15*tan(x)^4 - 10956*tan(x)^8/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 4612*tan(x)^12/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 780*tan(x)^4/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 36*tan(x)^16/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 36*tan(x)^2/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 780*tan(x)^14/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 4612*tan(x)^6/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 10956*tan(x)^10/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8)) + 12*tan(x)/(1 - tan(x)^6 - 15*tan(x)^2 + 15*tan(x)^4 - 10956*tan(x)^8/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 4612*tan(x)^12/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 780*tan(x)^4/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) - 36*tan(x)^16/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 36*tan(x)^2/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 780*tan(x)^14/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 4612*tan(x)^6/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8) + 10956*tan(x)^10/(1 + tan(x)^12 - 452*tan(x)^6 - 30*tan(x)^2 - 30*tan(x)^10 + 255*tan(x)^4 + 255*tan(x)^8))
Trigonometric part [src]
  2*cot(6*x) 
-------------
       2     
1 + cot (6*x)
$$\frac{2 \cot{\left(6 x \right)}}{\cot^{2}{\left(6 x \right)} + 1}$$
           2            
------------------------
/        1    \         
|1 + ---------|*cot(6*x)
|       2     |         
\    cot (6*x)/         
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(6 x \right)}}\right) \cot{\left(6 x \right)}}$$
    1    
---------
csc(12*x)
$$\frac{1}{\csc{\left(12 x \right)}}$$
sin(12*x)
$$\sin{\left(12 x \right)}$$
             2             
        4*sin (6*x)        
---------------------------
/         4     \          
|    4*sin (6*x)|          
|1 + -----------|*sin(12*x)
|        2      |          
\     sin (12*x)/          
$$\frac{4 \sin^{2}{\left(6 x \right)}}{\left(\frac{4 \sin^{4}{\left(6 x \right)}}{\sin^{2}{\left(12 x \right)}} + 1\right) \sin{\left(12 x \right)}}$$
            /pi      \       
       2*csc|-- - 6*x|       
            \2       /       
-----------------------------
/       2/pi      \\         
|    csc |-- - 6*x||         
|        \2       /|         
|1 + --------------|*csc(6*x)
|         2        |         
\      csc (6*x)   /         
$$\frac{2 \csc{\left(- 6 x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 6 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(6 x \right)}}\right) \csc{\left(6 x \right)}}$$
      1       
--------------
   /       pi\
sec|12*x - --|
   \       2 /
$$\frac{1}{\sec{\left(12 x - \frac{\pi}{2} \right)}}$$
            2*sec(6*x)            
----------------------------------
/         2        \              
|      sec (6*x)   |    /      pi\
|1 + --------------|*sec|6*x - --|
|       2/      pi\|    \      2 /
|    sec |6*x - --||              
\        \      2 //              
$$\frac{2 \sec{\left(6 x \right)}}{\left(\frac{\sec^{2}{\left(6 x \right)}}{\sec^{2}{\left(6 x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(6 x - \frac{\pi}{2} \right)}}$$
       2*sec(6*x)       
------------------------
/       2     \         
|    sec (6*x)|         
|1 + ---------|*csc(6*x)
|       2     |         
\    csc (6*x)/         
$$\frac{2 \sec{\left(6 x \right)}}{\left(1 + \frac{\sec^{2}{\left(6 x \right)}}{\csc^{2}{\left(6 x \right)}}\right) \csc{\left(6 x \right)}}$$
            /      pi\       
       2*cos|6*x - --|       
            \      2 /       
-----------------------------
/       2/      pi\\         
|    cos |6*x - --||         
|        \      2 /|         
|1 + --------------|*cos(6*x)
|         2        |         
\      cos (6*x)   /         
$$\frac{2 \cos{\left(6 x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(6 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(6 x \right)}}\right) \cos{\left(6 x \right)}}$$
   /       pi\
cos|12*x - --|
   \       2 /
$$\cos{\left(12 x - \frac{\pi}{2} \right)}$$
       2*sin(6*x)       
------------------------
/       2     \         
|    sin (6*x)|         
|1 + ---------|*cos(6*x)
|       2     |         
\    cos (6*x)/         
$$\frac{2 \sin{\left(6 x \right)}}{\left(\frac{\sin^{2}{\left(6 x \right)}}{\cos^{2}{\left(6 x \right)}} + 1\right) \cos{\left(6 x \right)}}$$
2*sin(6*x)/((1 + sin(6*x)^2/cos(6*x)^2)*cos(6*x))