Generating random numbers from a distribution specified in R

I'm trying to generate random values from a given distribution given by the following function in R with the inverse cdf method found here described after the function

sigma=function(x){

  rc=0.5

  y=(x/rc) ; 

  C=(1/rc) ;

  Fun = log(1+C)-( C/(1+C) ) ; 

##### terms  

  a = 1/( y^2 - 1) ; b = 2/( sqrt(y^2 + 1 ) ) ;  c = atan( sqrt((y - 1)/y+1) ) ; 

  ff = a*(1 - b*c) ; 

##### My function   

  cst=2*pi

  sigma1 = (ff)/(cst*(rc^2)*Fun) ## function to generate random values

  # return(sigma1)

}

######## inverse CDF method from "found here" link

den<-sigma

#calculates the cdf by numerical integration
cdf<-function(x) integrate(den,-Inf,x)[[1]]

#inverts the cdf
inverse.cdf<-function(x,cdf,starting.value=0){
 lower.found<-FALSE
 lower<-starting.value
 while(!lower.found){
  if(cdf(lower)>=(x-.000001))
   lower<-lower-(lower-starting.value)^2-1
  else
   lower.found<-TRUE
 }
 upper.found<-FALSE
 upper<-starting.value
 while(!upper.found){
  if(cdf(upper)<=(x+.000001))
   upper<-upper+(upper-starting.value)^2+1
  else
   upper.found<-TRUE
 }
 uniroot(function(y) cdf(y)-x,c(lower,upper))$root
}

#generates 1000 random variables of distribution 'den'
vars<-apply(matrix(runif(1000)),1,function(x) inverse.cdf(x,cdf))
hist(vars)

And then the following error appears :

Error in integrate(den, -Inf, x) : the integral is probably divergent
Called from: integrate(den, -Inf, x)

Unfortunally I can't understand where is the problem in my function. However I tried another approach as follow: (which can be found here)

library(distr)
p <- sigma  # probability density function
dist <-AbscontDistribution(d=p)  # signature for a dist with pdf ~ p
rdist <- r(dist)                 # function to create random variates from p

set.seed(1)                      # for reproduceable example
X <- rdist(1000)                 # sample from X ~ p
x <- seq(-10,10, .01)
hist(X, freq=F, breaks=50, xlim=c(-5,5))
lines(x,p(x),lty=2, col="red")

And when I run the line dist <-AbscontDistribution(d=p), the following error is shown:

Error in seq.default(from = low1, to = upp1, length = ngrid) : 
  'from' cannot be NA, NaN or infinite
Além disso: Warning message:
In sqrt((y - 1)/y + 1) : NaNs produzidos

I'd like to know if someone can help me with this problem pointing me to some possible method that really works ?

I apologize for any error in my question.

If I was not very clear in my doubt, what I'm trying to do is something similar to what was done in this link, but with my function.

Again, thanks in advanced