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## Sage and Mathematica integration

Last week I was trying to integrate a power of the Hyperbolic Tangent  (tanh) in sage, so I first try,

```sage: n,x = var('n,x')
sage: integrate(tanh(x)^n, x)
```

but Sage didn’t integrate it. So I impose $n$ to be an integer,

```
sage: n,x = var('n,x')
sage: assume(n, 'integer')
sage: integrate(tanh(x)^n, x)

and still nothing. However, for specific values of $n$ it worked,

sage: for n in range(1,6):
...  integrate(tanh(x)^n, x)
...

the results were shown.
In the other hand, Mathematica could solve the integration in general,

Integrate[Tanh[x]^n, x]

in terms of Hyperbolic Functions (which Maxima does not manages). Even in the case of specific values of $n$, the given results were much nicer because the answer were given in term of hyperbolic functions instead of exponential.
Then, I decide to try the Sage-Mathematica synapses.
What do we need?

Mathematica installed in the computer.
The License information of Mathematica (even if you have already registered it)
A working Sage installation.

Proceed…
Open a terminal and call a sage subshell,

\$ sage -sh

and call the mathematica kernel,

\$ math

Here you will be asked to provide the license information. NOTE: It is possible that if runs without the information of the license, in that case you are ready to use Mathematica within Sage.
After provide the information you are ready.
Using it!
The way of using is a bit weird, at the beginning,  for integrate the sine function, use this,

sage: mathematica.Integrate(sin(x), x)

Explaination:
Since we are welling to use Mathematica kernel, the first word would be mathematica, followed by a Mathematica command separated by a point. Then, using Sage notation the argument.
This would work! However, the answer is presented in Mathematica notation. If you’d like to have the answer in Sage notation, use something like

sage: eq = mathematica.Integrate(sin(x), x); eq._sage_()

That’s it.
I’d like to thank to schilly for his help.

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