Definite Integration With Trig Functions and Euler's Number

Computing $$I=\int e^{-\tan ^{-1}(x)}\,dx$$ is very difficult (the result involves gaussian hypergeometric functions).

It would better to generate a table for $$J(t)=\int_0^t e^{-\tan ^{-1}(x)}\,dx$$ and interpolate between values $$\left( \begin{array}{cc} t & J(t) \\ 0 & 0.00000 \\ 1 & 0.66225 \\ 2 & 1.04267 \\ 3 & 1.34828 \\ 4 & 1.62339 \\ 5 & 1.88232 \\ 6 & 2.13129 \\ 7 & 2.37353 \\ 8 & 2.61092 \\ 9 & 2.84466 \\ 10 & 3.07555 \end{array} \right)$$

Interpolating in this table, we have $$\int_0^{6.0509} e^{-\tan ^{-1}(x)}\,dx=2.14377$$ So $$\text{CS}=22\times 2.14377-5.3872\times 6.0509=14.5655$$ while numerical integration would give $14.5654$