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Is there any definite formula or definition by which logarithms and inverse trigonometric functions can be inter converted?

I come across this problem frequently in indefinite integration, for example: $\int\frac{1}{(x-a)(x-b)}$ can be evaluated using partial fractions to give an answer in terms of log, or it can be evaluated using the $\int\frac{1}{x^2+a^2}=\frac{1}{a}tan^{-1}\frac{x}{a}$ to give an answer in terms of arctan. Hence I wanted to know if these two answers can be converted to each other, as often the answer that I get and that is given are in terms of different functions.

I came across a similar question here but as far as I understood the interconversion involved the imaginary $i$ which I don't want. As I don't have a very good understanding of hyperbolic functions I couldn't understand the second part of the answer very well.

So far, I've found conversions between inverse trig functions and imaginary logs and inverse hyperbolic trig functions and logs, neither of which seem to help my cause.

(Sorry if I didn't write well, English isn't my first language)

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It is simple with the hyperboic functions. Consider for example $$y=\cosh(x)=\frac {e^x+e^{-x}}2$$ Let $e^x=t$ and solve $$2y=t+\frac 1t \implies t=y+\sqrt{y^2-1}\implies x=\log \left(y+\sqrt{y^2-1}\right)$$ $$\cosh^{-1}(x)=\log \left(x+\sqrt{x^2-1}\right)$$

Now, remember that $\cosh(ix)=\cos(x)$ and so on.

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