<s>[INST] Integrating reciprocals of functions with known antiderivatives If
$$\int_{}^{} f(x)\,dx$$
is known, is there a way to directly find
$$\int_{}^{} \frac{1}{f(x)}\,dx$$
 [/INST] Consider the function $f(x)$ in a power series. Let 
\begin{align}
f(x) = \sum_{n=0}^{\infty} a_{n} \, x^{n}
\end{align}
then it can be seen that
\begin{align}
\frac{1}{f(x)} = \frac{1}{a_{0}} - \frac{a_{1}}{a_{0}^{2}} \, x + \frac{a_{1}^{2} - a_{2}}{a_{0}^{3}} \, x^{2} + \cdots
\end{align}
Integration of each shows
\begin{align}
\int f(x) \, dx &= a_{0} x + \frac{a_{1}}{2} \, x^{2} + \frac{a_{2}}{3} \, x^{3} + \cdots \\
\int \frac{dx}{f(x)} &= \frac{x}{a_{0}} - \frac{a_{1}}{2 \, a_{0}^{2}} \, x^{2} + \frac{a_{1}^{2} - a_{2}}{3 \, a_{0}^{3}} \, x^{3} + \cdots
\end{align}
</s>