intros x; unfold arcsinh.
assert (0 < x + sqrt (x ^ 2 + 1)). {
destruct (Rle_dec x 0);
[ | assert (0 < x) by (apply Rnot_le_gt; assumption);
apply Rplus_lt_le_0_compat; auto; apply sqrt_pos].
replace (x ^ 2) with ((-x) ^ 2) by ring.
assert (sqrt ((- x) ^ 2) < sqrt ((-x)^2+1)). {
apply sqrt_lt_1_alt.
split;[apply pow_le|]; lra.
}
pattern x at 1; replace x with (- (sqrt ((- x) ^ 2))).
- assert (t:= sqrt_pos ((-x)^2)); lra.
- simpl; rewrite Rmult_1_r, sqrt_square, Ropp_involutive; auto; lra.
}
assert (0 < x ^ 2 + 1). {
apply Rplus_le_lt_0_compat;[simpl; rewrite Rmult_1_r; apply Rle_0_sqr|lra].
}
replace (/sqrt (x ^ 2 + 1)) with
(/(x + sqrt (x ^ 2 + 1)) *
(1 + (/(2 * sqrt (x ^ 2 + 1)) * (INR 2 * x ^ 1 + 0)))).
2:{ replace (INR 2 * x ^ 1 + 0) with (2 * x) by (simpl; ring).
replace (1 + / (2 * sqrt (x ^ 2 + 1)) * (2 * x)) with
(((sqrt (x ^ 2 + 1) + x))/sqrt (x ^ 2 + 1));
[ | field; apply Rgt_not_eq, sqrt_lt_R0; assumption].
apply Rmult_eq_reg_l with (x + sqrt (x ^ 2 + 1));
[ | apply Rgt_not_eq; assumption].
field.
split;apply Rgt_not_eq; auto; apply sqrt_lt_R0; assumption. }
apply (derivable_pt_lim_comp (fun x => x + sqrt (x ^ 2 + 1)) ln).
+ apply (derivable_pt_lim_plus).
* apply derivable_pt_lim_id.
* apply (derivable_pt_lim_comp (fun x => x ^ 2 + 1) sqrt x).
-- apply derivable_pt_lim_plus.
++ apply derivable_pt_lim_pow.
++ apply derivable_pt_lim_const.
-- apply derivable_pt_lim_sqrt; assumption.
+ apply derivable_pt_lim_ln; assumption.
Qed.

Lemma arcsinh_lt : forall x y, x < y -> arcsinh x < arcsinh y.
intros x y xy.