integrate tan(x) for real argument

[Carsten Knoll]

I want to calculate the antiderivative for tan(x)

In [4]: x = sp.Symbol('x', real=True)

In [5]: sp.integrate(sp.tan(x), x)
Out[5]: -log(sin(x)**2 - 1)/2


This result is technically correct, but I would (for didactic purposes)
prefer something like

-log(cos(x)).

How can I achieve this?

Thanks.
Carsten.

[Aaron Meurer]

trigsimp() will convert the sin to cos. To remove the negative and
square, you'll have to use expand_log(force=True), which gives a
complex constant (which can be ignored, since this is an integral).

It would definitely be better if integrate at least gave log(1 -
sin(x)**2) so that the result is real.

Aaron Meurer

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[Ondřej Čertík]

On Fri, Feb 19, 2016 at 8:45 AM, Aaron Meurer <asme...@gmail.com> wrote:
> trigsimp() will convert the sin to cos. To remove the negative and
> square, you'll have to use expand_log(force=True), which gives a
> complex constant (which can be ignored, since this is an integral).
>
> It would definitely be better if integrate at least gave log(1 -
> sin(x)**2) so that the result is real.

Here is a code that does this:

In [1]: expand_log(integrate(tan(x), x).trigsimp(), force=True)
Out[1]: -log(cos(x)) - I*pi/2

Ondrej

>
> Aaron Meurer
>
> On Fri, Feb 19, 2016 at 8:07 AM, Carsten Knoll <Carste...@gmx.de> wrote:
>> I want to calculate the antiderivative for tan(x)
>>
>> In [4]: x = sp.Symbol('x', real=True)
>>
>> In [5]: sp.integrate(sp.tan(x), x)
>> Out[5]: -log(sin(x)**2 - 1)/2
>>
>>
>> This result is technically correct, but I would (for didactic purposes)
>> prefer something like
>>
>> -log(cos(x)).
>>
>> How can I achieve this?
>>
>> Thanks.
>> Carsten.
>>
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>> To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/56C71381.4090204%40gmx.de.
>> For more options, visit https://groups.google.com/d/optout.
>
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[Aaron Meurer]

It looks like the meijerg algorithm gives the better form:

In [33]: integrate(tan(x), x, meijerg=True)
Out[33]: -log(cos(x))

Aaron Meurer

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[Carsten Knoll]

Thanks. Both variants are helpful!

Carsten