I would like to bring clarity to the issue on whether or not Nietzsche was a misogynist. The short and sweet answer is: no, the long answer: fuck no...

My poor attempt at ice breaking aside, let's delve into the long answer, for real. Let's start by diving into the details of what the heck Nietzsche's position on "Woman" even is... let us venture to his very first aphorism written in Birth of Tragedy, as it's a seed sprouted for us, and we can trace it through the rest of his work as his thoughts grow upon this topic, we can even will discuss possible criticism of his own concept in Birth of Tragedy:

We shall have gained much for the science of æsthetics, when once we have perceived not only by logical inference, but by the immediate certainty of intuition, that the continuous development of art is bound up with the duplexity of the Apollonian and the Dionysian: in like manner as procreation is dependent on the duality of the sexes*, involving perpetual conflicts with only periodically intervening reconciliations...both these so heterogeneous tendencies run parallel to each other, for the most part openly at variance, and* continually inciting each other to new and more powerful births*, to perpetuate in them the strife of* this antithesis*, which is but* seemingly bridged over by their mutual term... — Nietzsche, BoT § 1

Let us first approach this by leveling a criticism at this from Nietzsche's later stance...which he will criticize for being too dialectical but also too democratic with...

Duality of Sexes: In nature, yes there is a bimodal representation that highlights the extremes in sex, but we can see here that Nietzsche details these extremes as an antithesis of parallel heterogeneous tendencies ... in other words he's viewing the TWO from the antithesis of values... (the dialectical approach). He overcomes this position by Human All Too Human (HATH), and we can see by the time we get to Thus Spoke Zarathustra (TSZ), and Beyond Good and Evil (BGE), that instead of through an antithesis of values Nietzsche understands it takes both male and female reproductive material to produce a sex, such that sex is that which intertwines both male and female (BGE 2) as essentially coming from the same material, but the overwhelming majority are partitioned, and noncommunicating, this is what Deleuze details in Anti-Oedipus when he says:

everyone is bisexual, everyone has two sexes, but partitioned, noncommunicating; the man is merely the one in whom the male part, and the woman the one in whom the female part, dominates statistically. — Deleuze, Anti Oedipus

So there's still an antagonism through the oppositions, but rather than an antithesis of two heterogeneous values, sex is more so that which grows out of the interplay of male and female sexual reproductive forces, an active becoming rather than a rigid dichotomy through statistical dominance of genetic expression via the male/female sexual reproductive material. Just as the human grows out of that which intertwines the animal and the superman, hence the human is the rope intertwining two opposites, enmeshing them as one (Prologue TSZ). One can think of this rope as the flower whose roots bind the two concepts together. For the male, however, the plant's roots are statistically dominated in the male spectrum of the soil, and yes, males even have roots that stem in the feminine but are drowned out by the noise of the genetic statistical dominance of and vice versa.

So why the hell does any of this matter? Well, we can see through out Nietzsche's work a running theme of overcoming oneself in one's opposite. From the very get go all the way through Ecce Homo. More or less, analogous of how Nietzsche see's Humans as the "hybrid" (a coming together of two) of "phantom and plant" (Prologue TSZ) that grows (hence plant; like the Tree on the Hill in TSZ) out of a multiplicity of wills (phantom), each fighting for its own dominion, its own triumphant affirmation... sex is a coming together of the two into a single expressed through statistical dominance. So again why does this matter?

Because for Nietzsche: The highest presentment of man was through the doctrine of Athena (in man's opposite, a woman) as seen in 177 of Miscellaneous Maxims and Opinions (Part 1 of HATH Book 2) :

The presentment of the highest man, the most simple and at the same time the most complete, has hitherto been beyond the scope of all artists. Perhaps, however, the Greeks, in the doctrine of Athena, saw farther than any men did before or after their time. — Nietzsche, HATH Book 2 § 177

What exactly is the doctrine of Athena to Nietzsche? Well, as someone pointed out before, Athena is known as Athena the Wise, for her harsh wisdom, Athena the Serpent, Nietzsche's love of harsh wisdom, Nietzsche's serpent, Nietzsche's harsh wisdom the harsh wisdom that the Semites turned upside down into the "original sin" a crime against nature ... Athena the Serpent, aka Satan ... the serpent persuades Eve (woman) to eat from the Tree of Knowledge of Good and Evil, which God had forbidden, arguing that doing so would not lead to death but rather to enlightenment.

"For God doth know that in the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil" (Genesis 3:5)

This moment marks a rupture in innocence, a transition from an undivided, pre-moral existence into a realm of self-awareness, knowledge, and internalized suffering. However as Nietzsche details (and as Deleuze so adequately details in a summary of Nietzsche) the following on Birth of Tragedy 9:

Even the Titans do not yet know the incredible Semitic and Christian inventions, bad conscience, fault and responsibility. At the time of the Birth of Tragedy Nietzsche contrasts the Titan and Promethean crime to original sin. But he does it in dark and symbolic terms, because this opposition is his negative secret like the mystery of Ariadne is his positive one. He writes that, "in original sin, curiosity, mendacious deception, susceptibility to seduction, lust — in short a series of pre-eminently feminine affects was considered the origin of evil . . . Thus the Aryans understand sacrilege as something masculine; while the Semites understand sin as feminine. This is not Nietzschean misogyny; Ariadne is Nietzsche's first secret, the first feminine power, the anima, the inseparable fiancée of Dionysian affirmation." — Deleuze

Nietzsche declares this in Ecce Homo as he discusses his invention of the Dionysian dithyrambs:

What language will such a spirit speak, when he speaks unto his soul? The language of the dithyramb. I am the inventor of the dithyramb. Hearken unto the manner in which Zarathustra speaks to his soul Before Sunrise (iii. 48). Before my time such emerald joys and divine tenderness had found no tongue. Even the profoundest melancholy of such a Dionysus takes shape as a dithyramb. As an example of this I take "The Night-Song,"—the immortal plaint of one who, thanks to his superabundance of light and power, thanks to the sun within him, is condemned never to love.... Such things have never been written, never been felt, never been suffered: only a God, only Dionysus suffers in this way. The reply to such a dithyramb on the sun's solitude in light would be Ariadne. ... Who knows, but I, who Ariadne is! — Nietzsche, Ecce Homo

If you're unaware of who Ariadne is ... Ariadne represents the feminine instinct, the anima, the nurturing, Dionysian acceptance of life’s suffering. She is a Grecian symbol of amor fati, the willing embrace of fate. For Nietzsche, she is the secret key to his philosophy, (Yes-saying) to life, the inseparable companion (Fettered-Heart Free-Spirit BGE 87) to Dionysian affirmation that protects man from being torn to pieces by his own minotaur of bad conscience as Nietzsche details in Beyond Good and Evil 29:

It’s the business of very few people to be independent: - that is a right of the strong. And whoever attempts it - even with the best right to it, but without being compelled to - shows by that action that he is probably not only strong but exuberantly daring. He is entering a labyrinth; he is increasing a thousand-fold the dangers which life already brings with it, not the least of which is the fact that no one’s eyes see how and where he goes astray, gets isolated, and is torn to pieces by some cavern-dwelling Minotaur of conscience. Suppose such a person comes to a bad end, that happens so far away from men’s understanding that they feel nothing and have no sympathy: - and he cannot go back anymore! — Nietzsche, BGE § 29

We can see that to Nietzsche the metaphorical representation of Ariadne as a part of the human instinct is a vital element he uses in his dithyrambs, that nurturing divine tenderness which sooths the soul of the suffering, to overcome the horrors of existence... a necessary counterpart to the wild, overwhelming nature of Dionysus—she embodies a form of surrender that is not weakness, but rather an affirmation of the necessity of suffering and passion. A form of Yea-Saying to even the shit in life. A style of thinking that is abolished by the Semitic and Christian way of life by posing the feminine instincts as a sin, a divorce of man from God.

Nietzsche does not see ressentiment (it's your fault) and bad conscience (it's my fault) and their common fruit (responsibility) as simple psychological events but rather as the fundamental categories of Semitic and Christian thought, of our way of thinking and interpreting existence in general. Nietzsche takes on the tasks of providing a new interpretation and another way of thinking. — Deleuze

Nietzsche’s critique of "woman" is often misunderstood as misogyny when, in reality, it is largely a critique of the Semitic and Christian moral construction of womanhood. There are several aphorisms that point to this very fact, we can see them in The Gay Science (GS), BGE, and Twilight of Idols (TI):

GS 68: Will and Willingness.—Some one brought a youth to a wise man and said, "See, this is one who is being corrupted by women!" The wise man shook his head and smiled. "It is men," he called out, "who corrupt women; and everything that women lack should be atoned for and improved in men,—for man creates for himself the ideal of woman, and woman moulds herself according to this ideal."
BGE 86: In the background of all their personal vanity, women themselves have still their impersonal scorn—for "woman".
TI 13: Man created woman—out of what? Out of a rib of his god,—of his “ideal.” — Nietzsche

Nietzsche argues that in previous, stronger societies, women played a powerful and indirect role through their instincts, cunning, and nature. That the feminism in his time was mostly a masculinization of women though seeking equal footing in a rational, legalistic, "masculine" way, such that women lose their unique strength, becoming "boring" (tedious) instead of seductive, dangerous, or fascinating. Due to the feminine instinct being exiled as Sin in the pious interpretation of existence. Deleuze details this succinctly in a passage from his book on Nietzsche pg 21:

The imputation of wrongs and responsibilities, the bitter recrimination, the perpetual accusation, the ressentiment - this is the pious interpretation of existence. "It's your fault, it's your fault", until the accused, in turn, says, "it's my fault" and the desolated world resounds with all these moans and their echoes. "Everywhere where responsibilities have been sought it is the instinct of revenge that has sought them. This instinct of revenge has gained such a hold on humanity through the centuries that all of metaphysics, psychology, history and above all morality bear its imprint. — Deleuze

Nietzsche does not advocate for the oppression of women, but he argues that their traditional role—shaped by centuries under Semitic and Christian moral constructs—once carried a distinct power and vitality that modernity is now eroding without providing a worthy alternative. Instead, modernity further undermines feminine instincts by further imposing masculinized roles, via the democratic enlightenment, another step in stripping away the unique strengths that once defined the feminine. This masculinizing of the feminine is what Nietzsche rails on about in BGE 238:

To be mistaken in the fundamental problem of "man and woman," to deny here the profoundest antagonism and the necessity for an eternally hostile tension, to dream here perhaps of equal rights, equal training, equal claims and obligations: that is a TYPICAL sign of shallow-mindedness; and a thinker who has proved himself shallow at this dangerous spot—shallow in instinct! — Nietzsche, BGE § 238

Reducing woman to mere equality with man in the modern sense is a form of democratic leveling—erasing the essential differences that give her unique power and vitality. But what happens when the profoundest antagonism, the necessary and eternally hostile tension that drives growth and elevation, is lost? Without this tension—without man and woman as opposing yet complementary forces—neither can truly ascend. As Nietzsche shows his thoughts in BGE 236, that noble men believe the eternally feminine nurtures them aloft and vice versa:

I have no doubt that every noble woman will oppose what Dante and Goethe believed about woman—the former when he sang, "ELLA GUARDAVA SUSO, ED IO IN LEI," and the latter when he interpreted it, "the eternally feminine draws us ALOFT"; for THIS is just what she believes of the eternally masculine. — Nietzsche

Because of the nature of Dionysian affirmation, the overcoming of oneself in their opposite, in this case, as we already detailed, for Nietzsche, the highest presentment of man was through the doctrine of Athena: "Careless, mocking, forceful—so does wisdom wish us: she is a woman, and never loves any one but a warrior." That harsh wisdom that the Semites equate as coming from Satan. That harsh wisdom that Nietzsche himself is an advocate of.

How can Nietzsche be a misogynist when the feminine instinct is woven into the very fabric of his highest equation in Dionysian affirmation—amor fati—manifested through the Grecian symbolism of Ariadne, the embodiment of eternal embrace and becoming? And if he were a misogynist, why would he declare the highest presentment of man was shaped under the doctrine of the woman Athena?

You would think old Bertie would at least have a clue having written about the history of western philosophy... but apparently he missed the fundamental thrust of Nietzsche's philosophy entirely. Russell falls into the trap of projecting his own dialectially moral framework onto Nietzsche, distorting the very essence of his thought. Russell's critique not only misrepresents Nietzsche but also misses the profound, liberating aspects of Nietzsche's works, especially when it comes to Nietzsche's metaphors surrounding "woman," which are far from being misogynistic, but rather rooted in his equation for the affirmation of life: Amor Fati