Integrate: Integrating ((arcsin((x^2 + a) / (b*x))) - c) / x

[nova1989]

Trying to integrate:

((arcsin((x^2 + a) / (b*x))) - c) / x dx

where a, b, c are constants.

No success so far. I've tried integration by parts, but the resulting integral is more complex than the starting integral above!

The free Wolfram online integrator doesn't even read the syntax correctly!

Any assistance appreciated. It's been many years since I last looked at a problem like this.

nova

 

[CRGreathouse]

I doubt that there is a closed form solution in 'usual' functions.

 

[CompuChip]

Mathematica gives something like
$$-\frac{\sin ^{-1}\left(x^2+a\right)}{b x}-\frac{2 i \sqrt{\frac{x^2}{a-1}+1} \sqrt{\frac{x^2}{a+1}+1} F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a+1}} x\right)|\frac{a+1}{a-1}\right)}{\sqrt{\frac{1}{a+1}} b \sqrt{-\left(x^2+a-1\right) \left(x^2+a+1\right)}}-c \log (x)$$
where $F(\phi, m)$ is some elliptic integral "of the first kind".

In short, I share CRGreathouse's doubts

 

[ssd]

I believe it does not have a solution in terms of simple functions.

 

[nova1989]

Many thanks to everyone for the replies. Seems I'm out of luck for a closed form solution.

However, if you do think of a possible solution at some point, I would be most grateful for your advice.

Best regards.

 

[CompuChip]

Yes, there is a solution. I posted it. I'm thinking of it now. If you insist, you can copy it. Or write n(x) and call it nova's function.