Graphing y = (arctg(x-1)/(x^2-1))

Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 6811.41815542743$$
$$x_{2} = 2668.19670006129$$
$$x_{3} = -2847.16269001367$$
$$x_{4} = 10518.614624977$$
$$x_{5} = -5902.90626432681$$
$$x_{6} = 8119.83948181682$$
$$x_{7} = 4630.72814400825$$
$$x_{8} = 3540.40648032681$$
$$x_{9} = 5721.06977649485$$
$$x_{10} = 9864.40309784326$$
$$x_{11} = -4157.27399859556$$
$$x_{12} = -7648.04425945574$$
$$x_{13} = -9392.96605534623$$
$$x_{14} = -9829.1763422554$$
$$x_{15} = -2628.64211378136$$
$$x_{16} = 9428.26210891638$$
$$x_{17} = 7465.62855838261$$
$$x_{18} = -1972.45987214506$$
$$x_{19} = -3502.38001257541$$
$$x_{20} = 7901.76913183029$$
$$x_{21} = -8738.63741065811$$
$$x_{22} = 2232.13893691943$$
$$x_{23} = -10483.4807773401$$
$$x_{24} = -3720.70532998241$$
$$x_{25} = -6121.06675052115$$
$$x_{26} = 4848.79553643877$$
$$x_{27} = -4811.9787708472$$
$$x_{28} = -7211.78623703032$$
$$x_{29} = 10082.4736025948$$
$$x_{30} = -7429.9169956772$$
$$x_{31} = -10047.2791902854$$
$$x_{32} = 4194.59566306667$$
$$x_{33} = -9174.85839596281$$
$$x_{34} = 7029.48821321239$$
$$x_{35} = -2191.32254373413$$
$$x_{36} = -8956.74889420891$$
$$x_{37} = 5066.86349433144$$
$$x_{38} = -3939.00178115969$$
$$x_{39} = -3284.01993747129$$
$$x_{40} = -6339.22093528361$$
$$x_{41} = -10701.5796853305$$
$$x_{42} = -4375.52567475457$$
$$x_{43} = 7247.55835195549$$
$$x_{44} = -8520.52379145514$$
$$x_{45} = 6593.34819458712$$
$$x_{46} = 3976.53100048177$$
$$x_{47} = 5503.00069140277$$
$$x_{48} = -1753.39222987364$$
$$x_{49} = 3322.34766346881$$
$$x_{50} = -5466.56332152108$$
$$x_{51} = -6775.51286689925$$
$$x_{52} = 7683.69882142747$$
$$x_{53} = 10954.7556567109$$
$$x_{54} = -6557.36945599108$$
$$x_{55} = -7866.16832246646$$
$$x_{56} = -9611.07199900913$$
$$x_{57} = -6993.65165225979$$
$$x_{58} = -10919.6774375952$$
$$x_{59} = 4412.66145827765$$
$$x_{60} = 10736.6851401751$$
$$x_{61} = 8555.98027537042$$
$$x_{62} = 8774.05070872731$$
$$x_{63} = 1796.16694984546$$
$$x_{64} = 6375.27835002919$$
$$x_{65} = 3104.29218958228$$
$$x_{66} = 5284.93190799427$$
$$x_{67} = 2014.13690397986$$
$$x_{68} = -2410.03737475158$$
$$x_{69} = 9646.3325993535$$
$$x_{70} = 9210.19162862696$$
$$x_{71} = 3758.46779365714$$
$$x_{72} = -3065.61749619387$$
$$x_{73} = 8992.12116093852$$
$$x_{74} = -4593.75978881552$$
$$x_{75} = -5030.18462266333$$
$$x_{76} = 8337.90986483989$$
$$x_{77} = -8302.40786640037$$
$$x_{78} = 10300.5441120764$$
$$x_{79} = 6157.20864525689$$
$$x_{80} = -5684.73874002286$$
$$x_{81} = 0.679997123576428$$
$$x_{82} = 2450.16117528235$$
$$x_{83} = 5939.1391089851$$
$$x_{84} = -5248.37900812319$$
$$x_{85} = -8084.28944672083$$
$$x_{86} = 2886.24127305287$$
$$x_{87} = -10265.3806392786$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$

$$\lim_{x \to -1^-}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = -0.0833333333333333$$
$$\lim_{x \to 1^+}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = -0.0833333333333333$$
- the limits are not equal, so
$$x_{2} = 1$$
- is an inflection point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, 0.679997123576428\right]$$
Convex at the intervals
$$\left[0.679997123576428, \infty\right)$$