<s>[INST] How can I resolve rational indefinite integral? $$\int\left( {20.56\over x^2-1.27}+x^{55}\right) dx$$
I came to something like this
$$20.56 \int {1\over (x-1)^2 - \frac{27}{100} }~dx  + {x^{56}\over56} + \text{Constant}$$
Please can you help me to resolve this?
 [/INST] Your integral is of the form
$$I = \int \left(\dfrac{a}{x^2 - b^2} + x^n \right)dx$$
Hence,
\begin{align}
I & = \int \left(\dfrac{a}{2b}\dfrac1{x - b} - \dfrac{a}{2b}\dfrac1{x + b} + x^n \right)dx \\
& = \dfrac{a}{2b} \log(\vert x-b \vert) - \dfrac{a}{2b} \log(\vert x+b \vert) + \dfrac{x^{n+1}}{n+1} + \text{constant}\\
& = \dfrac{a}{2b} \log\left(\left \vert \dfrac{x-b}{x+b} \right \vert \right) + \dfrac{x^{n+1}}{n+1} + \text{constant}
\end{align}
In your case, $a = 20.56$, $b = \sqrt{1.27}$ and $n = 55$.
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