Smartphones

You can capture music streams with Audials. http://audials.com/en/how_to_record_stream_capture_music_videos_movies_from/apple_itunes.html

Best Way for copying, downloading & recording Music from Apple Music™ and Apple Radio with Audials
Step 1: Start Audials Tunebite, Tab “Streaming - Save Music”
Step 2: Click “Apple Music“ tile. Play back Apple Music iTunes Streaming in Browser
Step 3: Music will be automatically saved as file (e.g. MP3/AAC)
The Apple iTunes™ video streaming can be recorded and saved in the video file formats MP4, WMV, AVI, 3GP or others. The Audials Windows software tool will tag the file of the movie or the episode of a TV series afterwards.

Apple Music is all about streaming. You pay a flat fee to unlock access to Apple Music's entire catalogue, but you don't actually own the music you listen to.
To solve the above problems well, Macsome iTunes Music Converter is helpful and useful, which uses high recording and encoding to MP3 / AAC technology to convert these DRM Music files to MP3 formats, and then you can enjoy these files on any MP3 player devices as you like. (http://macsome.com/itunes-converter-mac/apple-music-converter.html)
Step 1. Free download the latest version of Apple Music Converter for Mac OS X, install and run it.
Step 2. Click the Add button to load your Apple Music files.
Choose the Music library in the drop-down list, and then choose the Apple Music files you want to convert.
Step 3. Click iTunes Converter Menu – Preferences – Advanced to set the output audio format, here we use the default one: Same as source.
Step 4. Click "Convert" button to start the converting.

If you have subscribed to Apple Music, you are able to download the Apple Music tracks for offline enjoyment. So I guess your question should be "how to save the Apple Music tracks on computer without losing".

To achieve it, either use the audio recording software or Apple Music converter. The former records the audio while playing Apple Music, the latter is to extract mp3 from Apple Music. Both ways work. Here is the review of the current workable Apple Music recorder and converter:

https://drm-wizard.com/best-apple-music-converter-to-mp3-review.html

for your reference. It introduces the way of how the application work to save Apple Music songs.

I recently met with a new tool named TunesKit Apple Music Converter in a forum. It was advertised to crack DRM protection from downloaded Apple music M4P files and convert Apple music to unprotected MP3, M4A, AAC, AC3. http://www.tuneskit.com/apple-music-converter-for-win.html

I'm trying to record Apple Music on PC and convert them to MP3 files. What's the easiest way to do that? I have installed the latest version of iTunes.
http://www.convertanymusic.com/streaming-audio-recorder/

Anything you have in Apple Music you can easily move to iTunes. Click the ... next to the track and select "Show in iTUnes Store." Then you'll own the tune even after your Apple Music trial is over.

Naturally, you'll have to shell out the buck or whatever it is, but it's the only fully legal way to keep the song on your device.

I love the simplicity of the software to create music that apple, with applemacsoft DRM Music Converter, you can view here : http://www.applemacsoft.com/drm-converter-windows.html
http://www.applemacsoft.com/drm-converter-windows.html

You can install iMusic on your Mac or windows computer, record any Apple music you're playing and iMusic would record automatically and saved as high quality. Follow this guide on how to record with iMusic. https://imusic.iskysoft.com/imusic/record-music-with-imusic.html