_______
/ x / 1 x \
/ ----- *(x + 1)*(x - 1)*|--------- - ----------|
\/ x + 1 |2*(x + 1) 2| / _______\
\ 2*(x + 1) / | / x |
1 + ---------------------------------------------------- + asin| / ----- |
___________ \\/ x + 1 /
/ x
x* / 1 - -----
\/ x + 1
$$\operatorname{asin}{\left(\sqrt{\frac{x}{x + 1}} \right)} + 1 + \frac{\sqrt{\frac{x}{x + 1}} \left(x - 1\right) \left(x + 1\right) \left(- \frac{x}{2 \left(x + 1\right)^{2}} + \frac{1}{2 \left(x + 1\right)}\right)}{x \sqrt{- \frac{x}{x + 1} + 1}}$$
/ / x \ / x \\
_______ | (-1 + x)*|-1 + -----| (-1 + x)*|-1 + -----||
/ x / x \ | -1 + x -1 + x \ 1 + x/ \ 1 + x/|
/ ----- *|-1 + -----|*|-1 + ------ + --------- + --------------------- + ---------------------|
\/ 1 + x \ 1 + x/ | 2*x 2*(1 + x) 4*x / x \|
| 4*(1 + x)*|1 - -----||
\ \ 1 + x//
--------------------------------------------------------------------------------------------------
___________
/ x
x* / 1 - -----
\/ 1 + x
$$\frac{\sqrt{\frac{x}{x + 1}} \left(\frac{x}{x + 1} - 1\right) \left(\frac{x - 1}{2 \left(x + 1\right)} + \frac{\left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)}{4 \left(x + 1\right) \left(- \frac{x}{x + 1} + 1\right)} - 1 + \frac{\left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)}{4 x} + \frac{x - 1}{2 x}\right)}{x \sqrt{- \frac{x}{x + 1} + 1}}$$
/ 2 2 2 \
| / x \ / x \ / x \ / x \ / x \ / x \ / x \ / x \ / x \ |
_______ | 3*|-1 + -----| 3*(-1 + x)*|-1 + -----| |-1 + -----| *(-1 + x) 3*|-1 + -----| (-1 + x)*|-1 + -----| 3*(-1 + x)*|-1 + -----| 3*|-1 + -----| *(-1 + x) (-1 + x)*|-1 + -----| |-1 + -----| *(-1 + x)|
/ x / x \ | 3 3 -1 + x -1 + x \ 1 + x/ -1 + x \ 1 + x/ \ 1 + x/ \ 1 + x/ \ 1 + x/ \ 1 + x/ \ 1 + x/ \ 1 + x/ \ 1 + x/ |
/ ----- *|-1 + -----|*|--- + --------- - ------ - -------- + -------------- - --------- - ----------------------- - ---------------------- + --------------------- - --------------------- - ----------------------- - ------------------------ - ----------------------- - -----------------------|
\/ 1 + x \ 1 + x/ |2*x 2*(1 + x) 2 2 4*x x*(1 + x) 2 2 / x \ 2 / x \ 4*x*(1 + x) 2 / x \ / x \|
| x (1 + x) 4*x 8*x 4*(1 + x)*|1 - -----| (1 + x) *|1 - -----| 2 / x \ 2*x*(1 + x)*|1 - -----| 4*x*(1 + x)*|1 - -----||
| \ 1 + x/ \ 1 + x/ 8*(1 + x) *|1 - -----| \ 1 + x/ \ 1 + x/|
\ \ 1 + x/ /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
___________
/ x
x* / 1 - -----
\/ 1 + x
$$\frac{\sqrt{\frac{x}{x + 1}} \left(\frac{x}{x + 1} - 1\right) \left(- \frac{x - 1}{\left(x + 1\right)^{2}} - \frac{\left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)}{\left(x + 1\right)^{2} \left(- \frac{x}{x + 1} + 1\right)} - \frac{3 \left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)^{2}}{8 \left(x + 1\right)^{2} \left(- \frac{x}{x + 1} + 1\right)^{2}} + \frac{3}{2 \left(x + 1\right)} + \frac{3 \left(\frac{x}{x + 1} - 1\right)}{4 \left(x + 1\right) \left(- \frac{x}{x + 1} + 1\right)} - \frac{3 \left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)}{4 x \left(x + 1\right)} - \frac{x - 1}{x \left(x + 1\right)} - \frac{\left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)^{2}}{4 x \left(x + 1\right) \left(- \frac{x}{x + 1} + 1\right)} - \frac{\left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)}{2 x \left(x + 1\right) \left(- \frac{x}{x + 1} + 1\right)} + \frac{3 \left(\frac{x}{x + 1} - 1\right)}{4 x} + \frac{3}{2 x} - \frac{\left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)^{2}}{8 x^{2}} - \frac{3 \left(x - 1\right) \left(\frac{x}{x + 1} - 1\right)}{4 x^{2}} - \frac{x - 1}{x^{2}}\right)}{x \sqrt{- \frac{x}{x + 1} + 1}}$$