Evaluate the following integral: $\int_\limits{0}^{\pi /4}\dfrac{\tan(x)^2 dx}{1+x^2}$ I tried to find and antiderivative but I think it's imposible so I would like to know if there is another way to find its value.
 A: Just as you, I suppose that the antiderivative is impossible to find and, more than likely, $$I=\int\dfrac{\tan^2(x) }{1+x^2}dx$$ will not show any explicit expression.
One thing you could to is to consider the Taylor expansions of $\tan(x)$ and $\frac 1{1+x^2}$ and get $$\dfrac{\tan^2(x) }{1+x^2}=x^2-\frac{x^4}{3}+\frac{32 x^6}{45}-\frac{18 x^8}{35}+\frac{8672
   x^{10}}{14175}-\frac{264332 x^{12}}{467775}+O\left(x^{14}\right)$$ and integrate one term at the time.
Suppose that we make the expansion to $O\left(x^{2n}\right)$ and let us call $J_{n}$ the result of the integral for the given bounds. Computing the values, we should find
$$\left(
\begin{array}{cc}
 n & J_n \\
 2 & 0.161491 \\
 3 & 0.141568 \\
 4 & 0.160295 \\
 5 & 0.153797 \\
 6 & 0.157698 \\
 7 & 0.155817 \\
 8 & 0.156862 \\
 9 & 0.156303 \\
 10 & 0.156614 \\
 11 & 0.156441 \\
 12 & 0.156538 \\
 13 & 0.156483 \\
 14 & 0.156515 \\
 15 & 0.156497
\end{array}
\right)$$ while numerical integration would give $0.156503$.
Inverse symbolic calculators are unable to identify the result.
