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Replacing mathematical functions of expressions with different mathematical functions of the same expression

Hello!

I am trying to replace hyperbolic trig with it's expanded form. As a concrete example I would like to replace

arcsinh(z) = ln( z + sqrt(z^2 + 1) )

Now I can do this if I knew that it was actually arcsinh(z) using subs, specifically using the command;

test = arcsinh(z)
test2 = test.subs_expr(arcsinh(z) == (log (z + sqrt((z^2 + 1))) )

The problem is that this seems to work only if it's an exact string match. Meaning if I tried;

test = arcsinh(1/3*x + 1/3)
test2 = test.subs_expr(arcsinh(z) == (log (z + sqrt((z^2 + 1))) )

I get

test2 = arcsinh(1/3*x + 1/3)

Is there a way to replace anything of the form

arcsinh(stuff) -> log (stuff + sqrt((stuff)^2 + 1))

Or does sage have this built in somewhere? I want to ideally make a sage function that I can pass a (randomly generated) mathematical expression to and have it expand the hyperbolic pieces for me. Thus I won't know the argument of the hyperbolic beforehand most of the time.

To give a concrete example, I would like to have something along the following:

a = random(1,1000)
b = random(1,1000)
f = a*arcsinh(b*x + a^2) - b
f2 = f.magicsimplifyfunction()

and get out

f2 = a*log (b*x + a^2 + sqrt((b*x + a^2)^2 + 1)) - b

Where the magicsimplifyfunction is the function that will work on any such f, not tailored to that specific f.

Thanks!

Replacing mathematical functions of expressions with different mathematical functions of the same expression

Hello!

I am trying to replace hyperbolic trig with it's expanded form. As a concrete example I would like to replace

arcsinh(z) = ln( z + sqrt(z^2 + 1) )

Now I can do this if I knew that it was actually arcsinh(z) using subs, specifically using the command;

test = arcsinh(z)
test2 = test.subs_expr(arcsinh(z) == (log (z + sqrt((z^2 + 1))) )

The problem is that this seems to work only if it's an exact string match. Meaning if I tried;

test = arcsinh(1/3*x + 1/3)
test2 = test.subs_expr(arcsinh(z) == (log (z + sqrt((z^2 + 1))) )

I get

test2 = arcsinh(1/3*x + 1/3)

Is there a way to replace anything of the form

arcsinh(stuff) -> log (stuff + sqrt((stuff)^2 + 1))

To give a concrete example, I would like to have something along the following:

a = random(1,1000)
b = random(1,1000)
f = a*arcsinh(b*x + a^2) - b
f2 = f.magicsimplifyfunction()

and get out

f2 = a*log (b*x + a^2 + sqrt((b*x + a^2)^2 + 1)) - b

Where the magicsimplifyfunction is the function that will work on any such f, not tailored to that specific f.

Thanks!

Edit for clarity:

I need a solution that doesn't require me to know the argument of arcsinh beforehand. So the magicsimplyfunction would work something like this:

f = 5*arcsinh(3*x + 1) - 2*e^x + 6*x*arcsinh(x^2)

Applying the simplify function would then "capture" the arguments 3*x+1 and x^2 as dummy variable z1 and z2 respectively and replace them so it would look like the following:

f = 5*arcsinh(z1) - 2*e^x + 6*x*arcsinh(z2)
z1 = 3*x + 1
z2 = x^2

Then I would apply subs_expr to get the following:

f = 5*(log(z1 + sqrt(z1^2 + 1))) - 2*e^x +6*x*(log(z2 + sqrt(z2^2 + 1)))

Then compose back in (or more accurately use another subs_expr) z1 and z2 to finally get

f = 5*(log(3*x+1 + sqrt((3*x+1)^2 + 1))) - 2*e^x +6*x*(log(x^2 + sqrt(x^4 + 1)))

The important thing here is nowhere in the process of executing the "magicsimplifyfunction" command did I specify 3x+1 or x^2. Because in most cases I won't know that's the argument before I am trying to expand it.