# Generalized Euler substitution

I've came across Euler substitutions today, then I saw the generalized substitution which can work on any case of the 3 cases using complex numbers but I didn't find any example on it, so can you please provide examples on this substitution, and extend it to the logarithmic rational functions like in the blockquote below:

Generalizations The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral $$\int \frac{d x}{\sqrt{-x^{2}+c}}$$, the substitution $$\sqrt{-x^{2}+c}=\pm i x+t$$ can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic. The substitutions of Euler can be generalized to a larger class of functions. Consider integrals of the form $$\int R_{1}\left(x, \sqrt{a x^{2}+b x+c}\right) \log \left(R_{2}\left(x, \sqrt{a x^{2}+b x+c}\right)\right) d x,$$ where $$R_{1}$$ and $$R_{2}$$ are rational functions of $$x$$ and $$\sqrt{a x^{2}+b x+c}$$. This integral can be transformed by the substitution $$\sqrt{a x^{2}+b x+c}=\sqrt{a}+x t$$ into another integral $$\int \tilde{R}_{1}(t) \log \left(\tilde{R}_{2}(t)\right) d t$$ where $$\tilde{R}_{1}(t)$$ and $$\tilde{R}_{2}(t)$$ are now simply rational functions of $$t$$. In principle, factorization and partial fraction decomposition can be employed to break the integral down into simple terms, which can be integrated analytically through use of the dilogarithm function. $${ }^{}$$