Can the Inverse Hyperbolic Substitution Solve this Tricky Integral?

[AKJ1]

Homework Statement

(x+ln(x+sqrt(x^2-1)))^3 / (sqrt(x^2-1))

The Attempt at a Solution

I have tried so many different things with this integral but cannot seem to get anywhere with it. Its almost so nicely an inverse coshx but not quite.

Any help?

 

[Dr. Courtney]

Have you tried MMa or Wolfram Alpha?

 

[Ray Vickson]

Homework Statement

(x+ln(x+sqrt(x^2-1)))^3 / (sqrt(x^2-1))

The Attempt at a Solution

I have tried so many different things with this integral but cannot seem to get anywhere with it. Its almost so nicely an inverse coshx but not quite.

Any help?

Have you tried Maple?

 

[AKJ1]

I have tried putting it in various forms into wolfram but returned nothing for step by step results. I tried taking the derivative of the said solution, but couldn't make any sense of it.

 

[ehild]

Try the substitution x=cosh(y).

 

[AKJ1]

Try the substitution x=cosh(y).

Thank you! This worked out quite nicely!

 

[SammyS]

It makes sense that ehild's suggested substitution worked.

After all: $\ \text{arcosh} (x) = \ln \left(x + \sqrt{x^{2} - 1} \right); x \ge 1 \$

 

[ehild]

It makes sense that ehild's suggested substitution worked.

After all: $\ \text{arcosh} (x) = \ln \left(x + \sqrt{x^{2} - 1} \right); x \ge 1 \$

and AKJ1 has noticed it...