https://www.bra.se/bra/bra-in-english/home/crime-and-statistics/rape-and-sex-offences.html

Note the following charts on the above page:

Percentage exposed to sex offences

Reported sex offences

Relationship to the perpetrator ("Unknown" is what migrant/refugee rapes would fall under, and that hasn't changed as a proportion of overall rapes since 2005, the first year of the chart, meaning rapes by close family/acquaintances have gone up in tandem with other kinds)

Persons suspected of rape

Persons found guilty of sex offences

The increase in reported rapes after 2005 has been discussed before in numerous places.

From their own site:

“The number of reported rape offences has increased over the last ten years (2006-2015). The increase can be partially explained by the entry into force of new sex offence legislation on 1 April 2005. This legislation entails, among other things, that certain acts which were previously classified as sexual exploitation are now classified as rape. The effect of the statutory change appeared in the statistics such that the number of reported offences in respect of sexual coercion and exploitation declined in the years immediately following the statutory change while the number of reported rapes increased. As from 1 July 2013, the sex offence legislation was again made tougher; among other things rape was expanded to include cases where the victim reacts passively.”

Other crime stats: https://www.bra.se/bra/bra-in-english/home/crime-and-statistics/swedish-crime-survey.html (The rates stay relatively flat over the entire last decade with minor ups and downs)

Impact of Syrian refugee crisis in 2015-2016

Preliminary stats do show an uptick from the previous year:

http://www.bra.se/bra/nytt-fran-bra/arkiv/nyheter/2017-01-12-anmalda-brott-2016---preliminar-statistik.html

Reported Sexual Offences:

Rape:

2009: 5937

2010: 5960

2011: 6532

2012: 6324

2013: 6017

2014: 6697

2015: 5918

2016: 6560

Sexual coercion/exploitation:

Dropped 1%, graph hardly changes since 2005

Sexual molestation (I think this is what is translated as sexual assault):

2012: 7607

2013: 8797

2014: 9643

2015: 8840

2016: 10500

Tens of thousands of immigrants from Muslim countries arrived every year going back to 2012 at least. The upticks/downticks are through these new arrivals.

It also says:

Reported assault increased by 5 per cent to 54 200 crimes, while reported molestation fell 2 percent to 55,400 crimes.

I have no idea which chart those numbers are coming from, either way not big swings.

TL;DR - There isn't a rape epidemic in Sweden by any definition of the word "epidemic". In spite of an exponential increase in the number of foreign immigrants arriving throughout the past decade. It's laughable to compare Sweden's crime rates today to the United States in any manner.