Sophie Germain prime counting heuristic

A Sophie Germain prime is a prime $p$ for which $2p+1$ is also prime, and let $C_2$ be the twin prime constant. Sophie Germain prime conjecture. The number of Sophie Germain primes $p\leq N$ is approximately

$$2C_2\int_2^N\frac{dx}{\log x\log(2x)}\sim\frac{2C_2N}{(\log N)^2}.$$

This is the prime-pair heuristic applied to the polynomials $x$ and $2x+1$; it remains unproved.