Someone has step by step to find the closed formula for the expected shortfall under the t-student distribution?

For Student’s t, ES is expressed as:

ES_(1-α) (R)=μ-σ (g_ν (t_ν^(-1) (α)))/α ((ν+(t_ν^(-1) (α))^2)/(ν-1))

where t_ν represents the Student’s t CDF and g_ν, its PDF.

Greetings,