Xjyuk

The equation of motion for a scalar field in curved spacetime $$fracpartialmathcalLpartialphi=frac1sqrt-gpartial_muleft[sqrt-gfracpartialmathcalLpartialleft(partial_muphiright)right]tag1$$ can be written in terms of the covariant derivative as $$fracpartialmathcalLpartialphi=nabla_muleft[fracpartialmathcalLpartialleft(partial_muphiright)right].tag2$$

Here $mathcalL$ is a Lagrangian scalar function.
How is Eq.$(2)$ obtained from Eq.$(1)$? The action of a covariant on a vector field $A_mu$ is given by $$nabla_mu A_nu=partial_mu A_nu-Gamma_munu^rhoA_rho.$$

I'll give hints instead of a complete solution, in case this ends up being categorized as homework.

I'll write $|g|$ instead of $-g$, because the former works for either sign convention (mostly-plus or mostly-minus). The quantity
$$
V^mu := fracpartial cal Lpartial(partial_muphi)
tag1
$$
is a vector (I mean, the components of a vector), so the question is how to derive
$$
frac1sqrtpartial_muleft(sqrt V^muright)
=nabla_mu V^mu
tag2
$$
for a vector $V$. Use the definition of $nabla_mu$ to see that (2) can also be written
$$
frac1sqrt V^mupartial_musqrt
= Gamma^alpha_alphamuV^mu.
tag3
$$
The left-hand side of (3) involves the quantity
$$
frac1sqrtpartial_musqrt
=
frac12partial_mulog|g|.
tag4
$$
We can think of $g$ as a square matrix, and $|g|$ is its determinant. The matrix is invertible, so $|g|$ is the product of its eigenvalues $lambda_n$. Rewrite the right-hand side of (4) in terms of $lambda_n$, and recall that if $A,B$ are diagonalizable square matrices, then the trace of their product is equal to the sum of the products of their eigenvalues. Use this write the right-hand side of (4) in terms of the matrices $g^-1$ and $partial_mu g$. Written in terms of indices, this becomes
$$
frac1sqrtpartial_musqrt
=frac12g^alphabetapartial_mu g_alphabeta.
tag5
$$
We can use this to rewrite the left-hand side of (3). Now write the $Gamma^alpha_alphamu$ on the right-hand side of (3) in terms of the metric and its derivatives to see that the identity (3) is indeed true.

Reference: Chapter 3 in Ward (1984), General Relativity, University of Chicago Press. Equation (3.4.10) in that book is equation (3) above.