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Works on Linux, OS X and Windows.
In addition to having great flexibility over demuxer, decoder, processer, encoder, muxer choice and settings, ffmpeg can crop, stream, merge audio and video from different sources and perform many other tasks.
Pro Free and open source
Licensed under LGPL.
Pro Screen capture
You can use ffmpeg to record your desktop along with audio.
Pro Extensive list of formats supported
Among the more common formats ffmpeg can handle are h.264, HEVC(h.265), mp3, AAC, mpeg-4, wmv3, ProRes, QuickTime, SWF, Speex, FLAC, VP9 and many more. To get a full list in terminal type: ffmpeg-formats
Pro Excellent streaming support
Because ffmpeg allows transcoding on the fly and supports multiple streaming protocols such as rtmp, rtsp, http, ftp, hls, you can use it to stream to services such as twitch.tv or set up your own streaming solution.
You can use both local realtime recordings or another stream as a source, transcode it if necessary, and output it to a different stream.
ffmpeg -i rtmp://server/live/originalStream -c:a copy -c:v libx264 -vpre slow -f flv rtmp://server/live/h264Stream
Con Steep learning curve
Unless you use a front-end (that has reduced functionality), ffmpeg might be intimidating at first.