/ 2/x\\
_______ | tan |-||
/ a - b |1 \2/|
2* / ----- *|- + -------|
\/ a + b \2 2 /
----------------------------------
/ 2/x\ \
| tan |-|*(a - b)| _________
| \2/ | / 2 2
|1 + ---------------|*\/ a - b
\ a + b /
$$\frac{2 \sqrt{\frac{a - b}{a + b}} \left(\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}\right)}{\sqrt{a^{2} - b^{2}} \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)}$$
/ / 2/x\\ \
_______ | |1 + tan |-||*(a - b) |
/ a - b / 2/x\\ | \ \2// | /x\
- / ----- *|1 + tan |-||*|-1 + -----------------------------|*tan|-|
\/ a + b \ \2// | / 2/x\ \ | \2/
| | tan |-|*(a - b)| |
| | \2/ | |
| |1 + ---------------|*(a + b)|
\ \ a + b / /
-----------------------------------------------------------------------
/ 2/x\ \
| tan |-|*(a - b)| _________
| \2/ | / 2 2
|1 + ---------------|*\/ a - b
\ a + b /
$$- \frac{\sqrt{\frac{a - b}{a + b}} \left(\frac{\left(a - b\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\left(a + b\right) \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\sqrt{a^{2} - b^{2}} \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)}$$
/ 2 2 \
| 2/x\ / 2/x\\ / 2/x\\ 2 2/x\ 2/x\ / 2/x\\ |
_______ | 3*tan |-| |1 + tan |-|| *(a - b) |1 + tan |-|| *(a - b) *tan |-| 3*tan |-|*|1 + tan |-||*(a - b)|
/ a - b / 2/x\\ |1 \2/ \ \2// \ \2// \2/ \2/ \ \2// |
2* / ----- *|1 + tan |-||*|- + --------- - ------------------------------- + ------------------------------- - -------------------------------|
\/ a + b \ \2// |4 4 / 2/x\ \ 2 / 2/x\ \ |
| | tan |-|*(a - b)| / 2/x\ \ | tan |-|*(a - b)| |
| | \2/ | | tan |-|*(a - b)| | \2/ | |
| 4*|1 + ---------------|*(a + b) | \2/ | 2 2*|1 + ---------------|*(a + b)|
| \ a + b / |1 + ---------------| *(a + b) \ a + b / |
\ \ a + b / /
-------------------------------------------------------------------------------------------------------------------------------------------------
/ 2/x\ \
| tan |-|*(a - b)| _________
| \2/ | / 2 2
|1 + ---------------|*\/ a - b
\ a + b /
$$\frac{2 \sqrt{\frac{a - b}{a + b}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\frac{\left(a - b\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(a + b\right)^{2} \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)^{2}} - \frac{\left(a - b\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \left(a + b\right) \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)} - \frac{3 \left(a - b\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}{2 \left(a + b\right) \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)} + \frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{4} + \frac{1}{4}\right)}{\sqrt{a^{2} - b^{2}} \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)}$$
/ 2 2 \
| 2/x\ / 2/x\\ / 2/x\\ 2 2/x\ 2/x\ / 2/x\\ |
_______ | 3*tan |-| |1 + tan |-|| *(a - b) |1 + tan |-|| *(a - b) *tan |-| 3*tan |-|*|1 + tan |-||*(a - b)|
/ a - b / 2/x\\ |1 \2/ \ \2// \ \2// \2/ \2/ \ \2// |
2* / ----- *|1 + tan |-||*|- + --------- - ------------------------------- + ------------------------------- - -------------------------------|
\/ a + b \ \2// |4 4 / 2/x\ \ 2 / 2/x\ \ |
| | tan |-|*(a - b)| / 2/x\ \ | tan |-|*(a - b)| |
| | \2/ | | tan |-|*(a - b)| | \2/ | |
| 4*|1 + ---------------|*(a + b) | \2/ | 2 2*|1 + ---------------|*(a + b)|
| \ a + b / |1 + ---------------| *(a + b) \ a + b / |
\ \ a + b / /
-------------------------------------------------------------------------------------------------------------------------------------------------
/ 2/x\ \
| tan |-|*(a - b)| _________
| \2/ | / 2 2
|1 + ---------------|*\/ a - b
\ a + b /
$$\frac{2 \sqrt{\frac{a - b}{a + b}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\frac{\left(a - b\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(a + b\right)^{2} \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)^{2}} - \frac{\left(a - b\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{4 \left(a + b\right) \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)} - \frac{3 \left(a - b\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{2} \right)}}{2 \left(a + b\right) \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)} + \frac{3 \tan^{2}{\left(\frac{x}{2} \right)}}{4} + \frac{1}{4}\right)}{\sqrt{a^{2} - b^{2}} \left(\frac{\left(a - b\right) \tan^{2}{\left(\frac{x}{2} \right)}}{a + b} + 1\right)}$$